Multidisk problems in H Optimization: A method for analyzing numerical algorithmshelton/BILLSPAPERSscanned/DHM02.pdf · 2010-09-21 · Multidisk Problems in H1Optimization 1115 to

Multidisk Problems in H∞ Optimization:A Method for Analysing Numerical AlgorithmsHARRY DYM, J. WILLIAM HELTON & ORLANDO MERINO

ABSTRACT. The theory of block Toeplitz operators and blockHankel operators is exploited to analyze numerical algorithmsfor solving optimization problems of the form

γ∗ = inff∈AN

supeiθ‖Γ(eiθ, f (eiθ))‖m×m,

where Γ(eiθ, z) is a smooth positive semidefinite matrix valuedfunction of eiθ, z = (z1, . . . , zn) and z̄ = (z1, . . . , zN), and ANis a prescribed set of N-tuples f = (f1, . . . , fN) of functions thatare analytic in the open unit disk D of the complex plane C. Thealgorithms under consideration are based on writing the equa-tions that an optimum must satisfy (in terms of “primal” and“dual” variables f , γ, Ψ) as T(Ψ , f , γ) = 0 and then invokinga Newton algorithm (or something similar) to solve these equa-tions. The convergence of Newton’s method depends criticallyupon whether or not the differential T ′ is invertible. For the classof problems under consideration, this is a very difficult issue toresolve. However, it is relatively easy to determine when T ′ is aFredholm operator with Fredholm index equal to zero, as will beshown in this paper. Fortunately, it turns out that this weakercondition seems to characterize effective numerical algorithmsand is reasonably easy to check. Explicit tests for the differen-tial T ′ to be a Fredholm operator of index zero are presented andcompared with numerical experiments on a few randomly chosentwo and three disk problems. The experimental results lend cre-dence to our contention that whether or not the differential T ′is a Fredholm operator of index zero determines the numericalbehavior for “almost all” multidisk problems.

1111Indiana University Mathematics Journal c©, Vol. 51, No. 5 (2002)

1112 HARRY DYM, J.WILLIAM HELTON & ORLANDO MERINO

1. INTRODUCTION

1.1. The optimization problem MOPT. A number of optimization prob-lems can be formulated as follows:(MOPT) Given a nonnegative scalar valued function Γ on ∂D×CN (that measures

performance and will be termed a performance function), find γ∗ ≥ 0and f∗ in AN which solve

γ∗ = inff∈AN

supeiθ

Γ(eiθ, f (eiθ)).Here AN is a prescribed set of N-tuples (f1, . . . , fN) of functions that are analyticin the open unit disk D of the complex plane C.

In this paper we consider the case where the performance function is of theform Γ (eiθ, z) = ‖Γ(eiθ, z)‖m×m,where Γ is a smooth positive semidefinite (and hence automatically selfadjoint)m ×m matrix valued function and ‖M‖m×m is the largest singular value of thematrix M. This is a mathematically appealing type of performance function thatis continuous but typically is not differentiable, since the matrix norm is not dif-ferentiable for most matrix valued functions. (Think of |x|.) Such performancefunctions Γ arise in engineering, cf. [10]. For example, the well known Nehariproblem and the H∞ multidisk problem can be incorporated into this framework,as can many other problems.

This paper develops the mathematics needed to understand and develop nu-merical algorithms. These algorithms are based on writing the equations that anoptimum must satisfy (in terms of “primal” and “dual” variables f , γ, Ψ) as

T(Ψ , f , γ) = 0

and then invoking a Newton algorithm (or something similar) to solve these equa-tions. Such algorithms are called primal-dual algorithms and play a major role inthe optimization of matrix valued functions. Newton’s method involves the dif-ferential T ′ and uses its inverse critically. Indeed the “second order convergence”of Newton’s method is completely governed by whether or not this differential isinvertible.

In this paper we obtain the differential T ′ of T and find that it is difficult todetermine when it is invertible. However, it is relatively easy to determine whenT ′ is a Fredholm operator with Fredholm index equal to zero. Fortunately, it turnsout that this weaker condition seems to characterize effective numerical algorithms(and is reasonably easy to check for multidisk problems). This is perhaps notso surprising if one recalls that Fredholm operators of index zero are compactperturbations of invertible operators.

Multidisk Problems in H∞ Optimization 1113

Recall that the H∞ multidisk problem is one where we have a special class ofΓ that are block diagonal, with block diagonal entries equal to the matrix valuedperformance functions

(1.1) Γp(eiθ, f (eiθ)) = (Kp(eiθ)− f(eiθ))T (Kp(eiθ)− f(eiθ)),where Kp(eiθ), p = 1, . . . , v, and f(eiθ) arem×m matrix valued functions. Inother coordinates, MOPT with such Γ are often called Integral Quadratic Con-straint (IQC) problems. Here we determine explicit tests for the differential T ′ tobe a Fredholm operator of index zero for the multidisk problem and compare theimplications of these tests with numerical experiments on a few randomly chosentwo and three disk problems.

The experimental results lend credence to our contention that whether or notthe differential T ′ is a Fredholm operator of index zero determines the numericalbehavior for “almost all” problems in the class under consideration. Moreover,both our theory and our experiments lead us to assert that the v disk problem inm ×m matrix function space is well behaved for a broad range of Newton typemethods when v =m, in contrast with other values of v.

Earlier work, [10, Parts III and IV], gave a reasonably complete theory ofMOPT problems for performance functions Γ which are smooth. Also, [12], and[10, Part V] gave optimality conditions for MOPT and some numerical algo-rithms based on it. However, the cited sources did not analyze these algorithms.That is the task of the present paper.

1.2. Assumptions. We shall assume throughout this paper that Γ(eiθ, z) is apositive semidefinite matrix valued function that is twice continuously differen-tiable in z = (z1, . . . , zN) and z = (z1, . . . , zN), and that

Γ(eiθ, z), ∂Γ∂z`

(eiθ, z),∂Γ∂z`

(eiθ, z),

∂2Γ∂z`∂zj

(eiθ, z),∂2Γ∂z2`(eiθ, z),

∂2Γ∂z`2 (e

iθ, z)

are at least continuously differentiable in θ (for all points eiθ on the unit circleT = ∂D).

Some sample problems are sketched below.

Everything is independent of θ. An important special case of the MOPT prob-lem that has received considerable attention, cf. the survey [18], is the case whereΓ and f do not depend on θ. Then f is an N- tuple of complex numbers ora 2N-tuple of real numbers and Γ is a matrix valued function of f alone. Theoptimization problem MOPT becomes

γ∗ = inff∈CN

‖Γ(f )‖m×m.


The matrix Nehari problem. To illustrate MOPT we recall the classical Nehariproblem: Given K, a boundedm×m matrix valued function on the unit circle, findits distance to the Hardy space of bounded matrix valued analytic functions H∞m×m.That is, with some poetic license1, find

(1.2) dist(K,H∞m×m) := inff∈H∞m×m

supθ‖K(eiθ)− f(eiθ)‖m×m,

or, equivalently,

(1.3) dist(K,H∞m×m)2 = inff∈H∞m×m

supθ

∥∥K(eiθ)− f(eiθ)∥∥2m×m.

Since ‖BTB‖m×m = ‖B‖2m×m for anym×mmatrix B and its conjugate transpose

BT , we may rewrite (1.3) as

(1.4) dist(K,H∞m×m)2

= inff∈H∞m×m

supθ‖(K(eiθ)− f(eiθ))T (K(eiθ)− f(eiθ))‖m×m.

To put the Nehari Problem in MOPT notation, take

(1.5) Γ(eiθ, Z) = (K(eiθ)− Z)T (K(eiθ)− Z),where Z = (zij)mi,j=1 denotes a matrix with N = m2 independent entries. It isclear that Γ(eiθ, Z) is analytic in zij and zij , i, j = 1, . . . , m, and continuous inθ if K is continuous in θ, and that in this case MOPT gives an optimal value γsuch that

(1.6) γ = dist(K,H∞m×m)2.

Also, in view of the convexity of the L∞m×m norm, a local solution is a globalsolution too. Hence solutions to MOPT correspond to solutions to the Nehariproblem: the minimizers f ∈ H∞m×m are the same, while the optimal values arerelated by equation (1.6).

We now use the Nehari problem to illustrate one important aspect of MOPTproblems, namely, that solutions are often not unique when m > 1. To illustratethis we consider the function

(1.7) K(eiθ) =(e−iθ 0

0 0

).

One can easily check with the help of Theorem 2 of [12] (which also appears asTheorem 17.1.1 of [10]) that the function zero gives the optimal distance from K

1We are using supremum instead of essential supremum.


to H∞m×m, which turns out to be 1. If f ∈ H∞ satisfies |f(eiθ)| ≤ 1 for all θ,then

(1.8) supθ

∥∥∥∥∥K(eiθ)−(

0 00 f(eiθ)

)∥∥∥∥∥2×2

= 1,

so any such f gives rise to a solution to MOPT.

A two disk problem. The two disk problem is a natural generalization of theNehari problem (which we look at as a one disk problem) to “two disks”. Given apair K1 and K2 of continuousm×mmatrix valued functions on the unit circle (whichwe think of as the centers of matrix function disks) and two performance functions

(1.9) Γp(eiθ, Z) = (Kp(eiθ)− Z)T (Kp(eiθ)− Z), p = 1,2,

the two disk problem is to find the smallest number γ and a function f in the space ofbounded analytic functions H∞m×m, so that

(1.10) Γ 1(eiθ, f (eiθ)) ≤ γIm and Γ 2(eiθ, f (eiθ)) ≤ γIm.In formula (1.9), Z denotes a matrix Z = (zij)mi,j=1 and N = m2. Note that(1.10) holds if and only if f simultaneously lies inside both of the matrix functiondisks. This is the MOPT problem for the performance function

Γ :=Γ 1 0

0 Γ 2

.The multidisk problem. The analysis in the preceding subsection is easily ex-

tended to v disks,

(1.11) Γp(eiθ, f (eiθ)) = (Kp(eiθ)− f(eiθ))T (Kp(eiθ)− f(eiθ)),p = 1, . . . , v.

Here Kp and f are m ×m matrix valued functions and we seek the smallest γsatisfying Γp(eiθ, f (eiθ)) ≤ γImfor p = 1, . . . , v and all θ.

The multidisk problem is the MOPT problem for the performance function

(1.12) Γ := diag(Γ 1, . . . , Γv) =

Γ 1 0 · · · 0

0 Γ 2 · · · 0

......

. . ....

0 0 · · · Γv

,

where Γp, p = 1, . . . , v, is given by (1.11).


Notation and usage.

• The acronym mvf denotes matrix valued function. If B is a matrix (or a mvf),then BT stands for the conjugate transpose of B, Bτ stands for the plain trans-pose of B and, if B is square, then trB = traceB.

• Γ(eiθ, z) = Γ(eiθ, z1, . . . , zN) is a positive semidefinite (and hence automati-cally selfadjoint) smooth m ×m mvf on T × CN . Thus m and N are positiveintegers that are determined by Γ .

• ∂Γ∂z( · , f ) =

[∂Γ∂z1

( · , f ), . . . , ∂Γ∂zN

( · , f )]

is anm×Nmmvf with N blocks of sizem×m in a row. Multiplication on theright by an m ×m matrix (or mvf) B, and the application of the trace, shouldbe understood as performed on each block entry. Thus, for example,

tr{∂Γ∂z( · , f )B

}=[

tr{∂Γ∂z1

( · , f )B}, . . . , tr

{∂Γ∂zN

( · , f )B}]

is a 1×N mvf.• (∂Γ/∂z)T is a block column mvf with N blocks of size m ×m: (∂Γ/∂zj)T ,j = 1, . . . , N. Since Γ is selfadjoint,

(∂Γ∂zj

)T= ∂Γ∂zj

, j = 1, . . . , N.

• The abbreviations

a := ∂Γ∂z( · , f ) = [a1, a2, . . . , aN] and aϕ =

N∑j=1

ajϕj

will be used when it is clear from the context what f is and when ϕ ∈ H∞N .• Γ(e(iθ), ·) is said to be plurisubharmonic (PLUSH) if the Nm×Nm matrix

Γz̄z(eiθ, ·) =[∂2Γ∂zi∂zj

(eiθ, ·)]Ni,j=1

is positive semidefinite on CNm. Γ is said to be strictly PLUSH if Γzz̄ is strictlypositive definite on CNm. If m = 1, then it is often the case that Γ is strictlyPLUSH, while if m > 1 it is more reasonable to expect that only the weakercondition PLUSH holds.


• H2r×n stands for the Hardy space of r × n mvf ’s with entries in the (scalar)

Hardy space H2 for the open unit disc with inner product

〈F,G〉 = 12π

∫ 2π

0tr{G(eiθ)TF(eiθ)

}dθ.

• (H2k×m)+ = {F ∈ H2

k×m | the first k × k block of the constant term f(0) isupper triangular (not strictly)} when k ≤m.

• H2+ = (H2k×m)+

1.3. Optimality equations for MOPT. The optimality conditions that are thesource of our computer algorithms are obtained by associating a mixture of theprimal problem MOPT and a dual problem which we now state:

Find γ := minf

maxΨ1

2π

∫ 2π

0tr{Γ(eiθ, f (eiθ))Ψ(eiθ)} dθ(PDMOPT)

subject to: Ψ ≥ 0, Ψ ∈ L1m×m, ,

∫ 2π

0trΨ(eiθ)dθ = 2π, f ∈ H∞N .

In this paper we shall always assume that the indicated minimum and maximumexist.

Solutions f , Ψ to the primal-dual problem produce f which are solutions toMOPT, providing that f is continuous. Solving the first order optimality con-ditions for the primal-dual problem gives excellent candidates for local solutionsto MOPT. Earlier results on optimality conditions assume that for each θ thecolumns of the N ×m2 mvf

(1.13) Uf (eiθ) =

∂Γ1,1∂z1

( · , f ) · · · ∂Γ`,k∂z1

( · , f ) · · · ∂Γm,m∂z1

( · , f )...

...

∂Γ1,1∂zN

( · , f ) · · · ∂Γ`,k∂zN

( · , f ) · · · ∂Γm,m∂zN

( · , f )

are linearly independent; see e.g. [12, Theorem 2] and [10, Theorem 17.1.1]. Butthis is only possible if

(1.14) N ≥ the number of entries in the mvf Γ ,which is not the case for the multidisk problem when v > 1, since then N =m2

and Γ is a vm × vm mvf. Fortunately, it turns out that we can get by with less


stringent assumptions that are formulated in terms of the mvf

(1.15) Vf (eiθ, B) =

tr{∂Γ∂z1

(eiθ, f (eiθ))B}

...

tr{∂Γ∂zN

(eiθ, f (eiθ))B}

tr{[γIm − Γ(eiθ, f (eiθ))]B}

.

Here Γ is an m ×m mvf and B is an m ×m matrix. For each θ, Vf (eiθ, B)defines a linear map from Cm×m into CN+1. The basic condition that we wantis that the null space of this map be zero, at least when restricted to appropriateclasses of matrices B (orm×m matrix valued measures dµ). In order to comparethis condition with the condition imposed on Uf that was discussed earlier, it isuseful to note that the first N entries of Vf (eiθ, B) can be reexpressed as

(1.16)

tr{a1B}

...tr{aNB}

= Uf (eiθ) vec(B),

where vec(B) is the m2 × 1 vector that is obtained by stacking the successivecolumns of B on top of each other. Thus,

(1.17) null space {Uf (eiθ)} = 0 ⇒ null space {Vf (eiθ, B)} = 0,

but not vice versa.If f is a continuous function on the circle and γIm−Γ(eiθ, f (eiθ)) ≥ 0, then

we shall say that the triple (γ, f (eiθ), Γ(eiθ, f (eiθ)) is measure nondegenerate iffor everym×m matrix valued measure dµ on the circle T

(1.18) Vf (eiθ, dµ) = 0 and dµ ≥ 0 -⇒ dµ = 0.

Here too, the condition null space {Uf (eiθ)} = 0 automatically implies the va-lidity of (1.18), but not vice versa.

Theorem 1.1. Let Γ(eiθ, z) be a positive semidefinite mvf that satisfies the smooth-ness conditions specified in Section 1.2, and assume that the PDMOPT problem hasa solution (Ψ , f , γ) ∈ L1

m×m ×H∞N ×R such that:(1) Ψ ∈ L2

m×m and is positive semidefinite for a.e. point eiθ ∈ T;(2) f is continuous and γIm − Γ(eiθ, f (eiθ)) ≥ 0 a.e.;(3) The triple (γ, f , Γ) is measure multidisk nondegenerate (i.e., (1.18) holds).


Then Ψ , f and γ must satisfy the following conditions:

Ψ(γIm − Γ( · , f )) = 0 a.e.,(1.19a)

PH2N

[tr

[∂Γ∂z

T( · , f )Ψ]] = 0,(1.19b)

12π

∫tr{Ψ}dθ − 1 = 0.(1.19c)

The proof of this theorem is given in Section 6.

1.4. Complementarity and strict complementarity. Condition (1.19a) in Theo-rem 1.1 holds if and only if the function Ψ satisfies

(1.20) rangeΨ ⊂ null {γI − Γ( · , f )},for a.e. θ, or equivalently, if and only if

(1.21) rangeΨ ⊥ range{γI − Γ( · , f )}for a.e. θ. Because of this, condition (1.19a) is often called a complementaritycondition. A stronger condition on a solution (Ψ , f , γ) to the primal-dual problem(PDMOPT) is the strict complementarity condition. It states that, in addition to(1.21), the two range spaces are orthogonal complements. In other words theirsum spans Cm. This forces equality in (1.20).

Strict complementarity is an extremely important condition here as well as inmany areas of optimization. It is the main ingredient in the assumption SCOMwhich is formulated below in Section 2.2 and is invoked in the main theorems ofthis paper.

Remark 1.2. Assumption (3) of Theorem 1.1 requiring measure non-degeneracy may not be necessary. We do not know an example which insuresthat it is needed.

Factoring the dual variable. Throughout the rest of the paper we assume thatthe dual variable Ψ = GTG has an outer spectral factor G ∈ (H2

k×m)+ withrankG = k a.e.. We remark that if Ψ = GTG, where G ∈ H2

k×m, then thereis no loss of generality in assuming that G ∈ (H2

k×m)+. Indeed, one may choose a(constant) unitary matrix U so that G1 = UG is in H2+; this uses the QR factoriza-tion of matrices. Note that GT1G1 = GTG.

We now reexpress the optimality condition (1.19a) of Theorem 1.1 in termsof G.

Lemma 1.3. If G ∈ H2+ and if γI − Γ( · , f ) ≥ 0, then the following statementsare equivalent:(1) (γI − Γ( · , f ))GTG = 0 a.e.,


(2) PH2+[G(γI − Γ( · , f ))] = 0,(3) G(γI − Γ( · , f )) = 0 a.e..

The proof appears in Section 6.

1.5. Numerical Algorithms. Many numerical algorithms can be based onsolving the optimality equations (1.19a)-(1.19c) in Theorem 1.1 or a variation ofthem. We shall list two algorithms in this paper and analyze the second of them.We hope that the method we use for this analysis can be readily adapted to mostnatural algorithms, although this remains to be seen.

The main idea is that solving equations (1.19a)-(1.19c) in Theorem 1.1 forG, f , and γ, is equivalent to solving an operator equation of the form

(1.22) T

Gfγ

= 0

for the same unknowns. It is common to approach such problems using Newton’smethod or some variation thereof.

Newton’s method. Newton’s method is an iterative scheme for solving operatorequations of the form

(1.23) T[x] = 0.

In terms of the differential T ′ of T , the Newton step for updating a given x is:

(1.24) x̃ = x − (T ′x)−1T[x],

whenever the differential is invertible. We refer to (1.24) as the Newton iteration,and to the repeated application of (1.24) as the Newton algorithm.

A standard fact [13] about Newton iteration is the following result.

Theorem 1.4. Let B1 and B2 be normed linear spaces, and suppose the operatorT : B1 → B2 is two times continuously Frechet differentiable in a neighborhood V ofx∗ ∈ B1. Assume also that T[x∗] = 0, and that T ′x∗ has a bounded inverse (T ′x∗)−1.Then there exist a neighborhood W of x∗ in B1 and a constant c > 0 such that forevery x ∈ W the linear operator T ′x has a bounded inverse, and x̃ := x−(T ′x)−1T[x]satisfies ∥∥x̃ − x∗∥∥B1

≤ c∥∥x − x∗∥∥2

B1.

In the present setting, the differential

T ′(G,f ,γ) :

H2+

H2N

R

→H2+

H2N

R


is the key term in the linearization

(1.25) 0 = T(G, f , γ)+ T ′(G,f ,γ)[(∆,ϕ,η)]of (1.22) that is used to solve for (∆,ϕ,η) at each iteration. Variations are tomodify the operator T in some way which has better numerical properties, butwhich still allows one to construct a primal-dual solution from the answer. IfT ′(G,f ,γ) is invertible at the optimum (G, f , γ), then Newton’s method has localsecond order convergence (which is very good).

Interior point methods often mix following “the central path” with Newtonsteps. The theory of interior point methods [19], [1] tells us that one can expectlocal second order convergence to a solution (G, f , γ) providing that T ′(G,f ,γ) isinvertible at this solution. Indeed invertibility of T ′(G,f ,γ) is sufficient but notnecessary for good local behavior (even without following the central path). Ourconcern in this article is the theoretical analysis of such invertibility, since it isoften extremely informative.

Algorithms for solving MOPT.

THE ALGORITHM GTG. The first algorithm we present uses the operator

(1.26) T :

Gfγ

→

PH2+[G(γI − Γ( · , f ))]PH2

N

[tr[− ∂Γ∂z( · , f )TGTG

]]1

2π

∫tr{GTG}dθ − 1

.

ALGORITHM GTG. Given x(r) = (G(r), f (r), γ(r)), to update to x(r+1)

carry out steps I-III below:

I. Subproblem. Solve T(x)+ T ′x(δx) = 0 for δx = (δG,δf , δγ).II. Line search. Perform a linear search to determine a step parameter t ≥ 0 that

minimizes ‖T(x + tδx)‖.III. Update. Set x(r+1) = (G(r+1), f (r+1), γ(r+1)), where f (k+1) = f (k) + tδf ,

γ(k+1) = γ(k) + tδγ, and G(k+1) = G(k) + tδG. If γI − Γ( · , f ) ≥ 0 is notsatisfied, γ(k+1) is chosen as supθ Γ(eiθ, f (k+1)(eiθ)).

THE ALGORITHM G+GT . The second algorithm we present uses the addi-tive decomposition Ψ = G+GT . In this setting G is always square, that is, k =m,


and the operator

T :

Gfγ

=

PH2+[(G +GT)(γI − Γ( · , f ))+ (γI − Γ( · , f ))(G +GT)]PH2

N

[tr[− ∂Γ∂z( · , f )T (G +GT)

]]1

2π

∫tr{G +GT}dθ − 1

.

ALGORITHM G+GT . Given x(r) = (G(r), f (r), γ(r)), to update to x(r+1)

carry out steps I-III below:I. Subproblem. Solve T(x)+ T ′x(δx) = 0 for δx = (δG,δf , δγ).

II. Line search. Perform a linear search to determine a step parameter t ≥ 0 thatminimizes supθ Γ(eiθ, f (k)(eiθ))+ tδf(eiθ)).

III. Update. Set x(r+1) = (G(r+1), f (r+1), γ(r+1)), where f (k+1) = f (k) + tδf ,γ(k+1) = supθ Γ(eiθ, f (k+1)(eiθ)), and, providing that G + GT ≥ 0 is satis-fied, G(k+1) = G(k) + tδG. If G+GT ≥ 0 does not hold for this choice of t,then a smaller step is selected for G to ensure positivity.

This gives a function space analog of Haeberly and Overton’s XZ +ZX algorithm[6], [1] given by those authors for finite dimensional matrix optimization.

In this paper we describe a theory for analyzing algorithms of the type givenabove. We apply the theory to Algorithm GTG since it is the first one we looked at.A theoretical study of AlgorithmG+GT is in progress. Numerical experiments (seeSection 1.7) strongly suggest that Algorithm G + GT has a considerably broaderrange of effectiveness than Algorithm GTG.

The main problem in analyzing algorithms. A wide range of mathematicalquestions arise in the analysis of numerical algorithms. However, our philoso-phy in this paper is different from the traditional one which is commonplace inmathematics, wherein one wants as many theorems as possible about the situa-tion. By contrast, when dealing with numerical algorithms, if (as is usually thecase) the goal is to have a theory of a class of algorithms which helps to developnew algorithms, then (although having many theorems is satisfying) what is re-ally important is knowing the smallest set of theorems which correlate stronglywith the performance of algorithms. The point is that when presented with a newcandidate for an algorithm we want a few simple calculations that can be carriedout quickly and serve to predict whether the algorithm will be successful or not.A major objective is to avoid many fruitless computer experiments. This puts apremium on identifying simple properties which are critical in practice and iden-tifying other properties which can be ignored. As indicated above, a question thatis central to analyzing the performance of optimization algorithms is:

When is T ′(G,f ,γ) invertible? This is a very difficult question to answer in thesettings under consideration. Instead, we shall focus on the following question(the pertinent definitions are recalled below).


When is T ′(G,f ,γ) a Fredholm operator with Fredholm index equal to zero? An af-firmative answer seems to be a good indicator of an effective algorithm. Fredholmindex zero is a weaker notion than invertibility, but we found that in our setting itis easy to check, and moreover, we believe that it gives the information we need topredict and evaluate the performance of a substantial class of algorithms. (See theconjecture in Section 1.6.)

In Section 1.7 we compare the implications of our tests with numerical exper-iments on a few randomly chosen two and three disk problems. The experimentalresults lend credence to our contention that whether or not T ′ has Fredholm indexzero determines numerical behavior for “almost all” multidisk problems.

Definition 1.5. Let X, Y be Banach spaces, letA : X → Y be a bounded linearoperator from X into Y and let B(X, Y) denote the space of all such operators.Then A is a Fredholm operator if:

(i) n(A) := dim nullA <∞;(ii) rangeA is closed in Y ;

(iii) r(A) := dim(Y \ rangeA) < ∞.If A is a Fredholm operator, the number

ι(A) := n(A)− r(A)

is called the Fredholm index of A. Let F0 denote the set of Fredholm operators ofindex 0.

If A and B are Hilbert spaces, then condition (iii) may be written asdim(rangeA)⊥ < ∞. It is well known that compact perturbations of Fredholmoperators are Fredholm. It is clear from the definition that when X = Y , thebounded invertible operators are Fredholm operators with index equal to zero.Also, a self adjoint bounded linear operator on a Hilbert space that is Fredholmclearly has index zero.

Another issue that is completely suppressed is a detailed analysis of the variouschoices of function spaces on which T and consequently T ′ can act. The reasonfor ignoring this can be seen in the light of well known theorems to the effect thatthe non zero spectrum of a Toeplitz operator generated by a fixed smooth symbolis reasonably independent of the choice of the space on which the operator acts; seee.g., [4]. Also empirically we find that the behavior in the L2 norm that is analysedhere corresponds to the behavior observed experimentally. Further justification fornot doing analysis on complicated spaces stems from [11], which studied the OPTproblem rather then the more complicated MOPT problem. That paper analyseda special case of the algorithm studied here, but took the underlying space to beC∞. The analysis showed that the key Toeplitz operator there is invertible on L2 ifand only if it is invertible on the space C∞ equipped with a Frechet norm. Thus,our advice to anyone interested in the development of algorithms is to focus theireffort on how T ′ behaves on L2.


1.6. T and T ′ for the multidisk problem. The main theorem of this paper,Theorem 2.3, is formulated for general PDMOPT problems. Instead of statingthe fully general result in the introduction, we consider the special case of theAlgorithm GTG, applied to the H∞ multidisk problem; see Theorem 1.9. Thefirst step is to specialize Theorem 1.1 to the multidisk case. Fortunately, the dualvariable Ψ in the PDMOPT optimality equations for the multidisk problem canbe taken block diagonal: Ψ = diag(Ψ1, . . . ,Ψv). We shall assume that each ofthese diagonal blocks is factorable. The main result is formulated in the nextsubsection.

Optimality conditions for the multidisk problem. We first observe that the mea-sure non-degenerate condition (1.18) when specialized to the multidisk problembecomes the multidisk measure non-degenerate condition that is defined for γ, f ,Kp, p = 1, . . . , v, as follows. If dµp, p = 1, . . . , v, is a set of positive semidefinitem×m matrix valued measures on the unit circle such that

(1.27)

v∑p=1

{Kp(eiθ)− f(eiθ)}dµp = 0,

{γIm − Γp(eiθ, f (eiθ))}dµp = 0 for p = 1, . . . , v

⇒ dµp = 0 for p = 1, . . . , v.

Theorem 1.6. Let the functions Kp(eiθ), p = 1, . . . , v, that appear in (1.11)be continuously differentiable, and assume that the PDMOPT problem for the H∞

multidisk problem has a solution (Ψ , f , γ) ∈ L1vm×vm ×H∞m2 ×R such that:

(1) Ψ = diag((G1)TG1, . . . , (Gv)TGv), where Gp ∈ (H2kp×m)+ is outer with

rankGp = kp a.e. for p = 1, . . . , v;(2) Ψ ∈ L2

vm×vm;(3) f is continuous and γIm − Γ(eiθ, f (eiθ)) ≥ 0;(4) γ, f , Kp, p = 1, . . . , v, is multidisk measure non-degenerate (i.e., (1.27) holds).Then Gp, f and γ must satisfy the following conditions:

PH2+[Gp(γI − Γp( · , f ))] = 0 for p = 1, . . . , v,(1.28a)

β :=v∑p=1

GpTGp(Kp − f)T ∈ eiθH2

m×m,(1.28b)

v∑p=1

12π

∫tr{(Gp)TGp}dθ − 1 = 0.(1.28c)

This theorem is an easy consequence of Theorem 1.1 and Lemma 1.3. Theproof is given in Section 5. Similar conclusions are available in [10] and [12]


under the more restrictive condition that the columns of Uf (eiθ) are linearlyindependent. See also [15] for a related (though more abstract) result for the twodisk problem.

The operator T for the Algorithm GTG in the multidisk setting is

(1.28) T :

G

f

γ

→

diag{PH2+[G1(γI − Γ 1( · , f ))], . . . , PH2+[G

v(γI − Γv( · , f ))]}PH2m×m

[−

v∑p=1(Kp − f)GpTGp

]1

2π

∫tr{GTG}dθ − 1

.

The Fredholmness of T ′. Our goal is to determine when T ′ for the H∞ multi-disk problem is Fredholm of index 0. As we shall see, the most valuable indicatorof this is the number µpnull(θ) that is given by Definition 1.7 below. We shallimpose the following extra condition in addition to those that are imposed in theformulation of Theorem 1.6.

The outer factors Gp of Ψp are continuous with constant rank kp forp = 1, . . . , v.

Definition 1.7. For each θ, let µpnull(θ) denote the dimension of the follow-ing vector subspace of Cm×m:

{B ∈ Cm×m | Ψp(eiθ)BT = 0 and Ψp(eiθ)(Kp(eiθ)− f(eiθ))TB = 0,

for p = 1, . . . , v}.

Definition 1.8. For each θ, let µdnull(θ) denote the dimension of the follow-ing space of matrix tuples:

{{B1, . . . , Bv} ∈ Ck1×k1 × · · · ×Ckv×kv |

v∑p=1(Kp(eiθ)− f(eiθ))Gp(eiθ)TBpGp(eiθ) = 0

}.

Note that if the functions Kp, f and Gp are given numerically, then the num-bers µpnull(θ) and µdnull(θ) can be computed numerically at each θ. The notationµpnull(θ) and µdnull(θ) stems from the connection with the null space of σprimal

and σdual in the multidisk case; see Section 5.


Conclusions for multidisk problems.

Theorem 1.9. Let the functions Kp(eiθ), p = 1, . . . , v, that appear in (1.11) becontinuously differentiable, and assume that the PDMOPT problem for the multidiskproblem has a solution (Ψ , f , γ) ∈ L1

vm×vm ×H∞m×m ×R such that:(1) Ψ = diag(Ψ1, . . . ,Ψv) is block diagonal and continuously differentiable;(2) Ψp = (Gp)TGp has a continuous outer spectral factor Gp ∈ (H2

kp×m)+ withrankGp = kp for p = 1, . . . , v and all θ;

(3) f is continuously differentiable and γIm − Γ(eiθ, f (eiθ)) ≥ 0;(4) γ, f , Kp, p = 1, . . . , v, is multidisk measure non-degenerate (i.e., (1.27) holds);(5) Strict complementarity holds at (Ψ , f , γ);(6) µpnull(θ) = 0 for all θ;(7) µdnull(θ) = 0 for all θ.Then the operator T ′ is Fredholm with index zero. If µpnull(θ) 6= 0 or µdnull(θ) 6= 0for some θ, i.e., if (6) or (7) fails, then T ′ is not Fredholm.

Remark 1.10. Measure non-degeneracy looks similar to µdnull(θ) = 0, enoughso that Assumption (4) of Theorem 1.9 may eventually be subsumed by Assump-tion (7). This has not been fully confirmed yet; see Section 2.2 for some prelimi-nary discussion. At this point, however, we do know, from the substantial numberof computer experiments we have run, that in practice Assumption (4) plays norole. On the other hand, whether or not µpnull(θ) = 0 has a big effect.

The proof of Theorem 1.9 is given in Section 5. The extra smoothness as-sumptions that are imposed on Ψ(eiθ) and f(eiθ) in the formulation of this the-orem are added to insure the existence of continuous factors H(eiθ) in item (3)of the definition of SCOM in Section 2.2. Much of the theory can be developedwithout these restrictions, but the added generality seems to have little practicalimportance in the analysis of numerical algorithms, at least at this time. For atheorem of wider scope, which is formulated for the general MOPT problem andnot just the H∞ multidisk problem, see Theorem 2.3.

At this point we state a conjecture that is based on both numerical experi-ments (that are summarized in Section 1.7) and theoretical considerations. If theconjecture is true, then Theorem 1.9 becomes a very powerful tool for predictingthe effectiveness of Newton’s method and its variants.

Conjecture 1.11. For generic multidisk optimization problems, if T ′ at solu-tions to PDMOPT is a Fredholm operator of index zero, then T ′ is invertible.

One rationale for this conjecture is the general fact that the class F0 of Fred-holm operators of index 0 is an open dense subset of B(X,X), see [2, Section1.11]. Naturally, the set of invertible operators is an open dense subset of B(X,X),and so is an open subset of F0 which is dense in F0. The presumption behind theconjecture is that in the class of Kp’s that produce T ′(G,f ,γ) in F0, almost all Kp’sproduce T ′ in the open dense set of invertible operators. Further, in this paper we


see that T ′ has the formT ′ = L+ C,

where L is a block Toeplitz matrix and C is a compact operator. The conditionsin Theorem 1.9 (primarily µpnull(θ) = 0 and µdnull(θ) = 0 for all θ), imply thatL is Fredholm of index 0. The conjecture thus asserts that for most selections ofKp, the resulting C has no special relationship to L and so yields an operator L+Cwhich is invertible.

Also in the paper we do a little more work than is immediately necessary inanalysing the Toeplitz operator L. Possibly this will be useful in further investiga-tion. For example, in Section 2.2 we give the dimension of the null space of L.The proof is long and is not included.

Rules of thumb. We observe in experiments that in multidisk situations, whenthe computed values of Ψp(eiθ) have rank that is independent of θ, then theAlgorithm GTG produces T ′ with Fredholm index zero if v =m. This section givesa theoretical justification for this observation, namely, we prove this to be true insituations where the Ψp, p = 1, . . . , v all have rank one (that is, kp = 1) andthe Ψp and f are in “general position,” a notion we now define. In computerexperiments we see that when all disk constraints are active (that is, kp > 0) thereis a dichotomy: Either

(1) Ψp(eiθ) has rank independent of θ, in which case the Ψp , p = 1, . . . , v, allhave rank one and general position holds, or

(2) for some p, Ψp(eiθ) has rank equal to 0 on some intervals in [0,2π].

Now we introduce some definitions. A set of matrices Lp ∈ Cm×kp , p = 1,. . . , v, is said to have linearly independent ranges provided that any selection ofnonzero vectors xp ∈ rangeLp, p = 1, . . . , v is linearly independent. Note thatthe set of matrix v-tuples {L1, . . . , Lv} ∈ Cm×k1 × · · · × Cm×kv that satisfy theconstraint

(1.29)v∑p=1

rankLp ≤m,

and have linearly independent ranges is an open dense subset of the set Cm×k1 ×· · · ×Cm×kv , i.e., this is a generic property.

Definition 1.12. The functions Ψ = diag(Ψ1, . . . ,Ψv), f are in general posi-tion if for each point eiθ:

(1) The matrices Ψp for p = 1, . . . , v have linearly independent ranges;(2) The matrices (K1−f)Ψ1, . . . , (Kv −f)Ψv have linearly independent ranges;(3) The matrices (K1 − f), . . . , (Kv − f) are all invertible.


Definition 1.13. For each θ, let

µD(θ) :=m− rank{Ψ1(eiθ)+ · · · + Ψv(eiθ)} ,µR(θ) :=m− rank{[(K1 − f)Ψ1(K1 − f)T

+ · · · + (Kv − f)Ψv(Kv − f)T ](eiθ)}.Proposition 1.14. Under the hypotheses of Theorem 1.9,

µpnull(θ) = µD(θ)µR(θ).

If Ψ and f are in general position, then∑vp=1 rankΨp ≤m, and µpnull(θ), which is

independent of θ, equals

µpnull =(m−

v∑p=1

rankΨp)2

and then µdnull(θ) = 0. If in addition v = m and all constraints are active, thenµpnull = 0.

The proof of this proposition is given in Section 6.When

∑vp kp < m, Proposition 1.14 says µpnull ≠ 0, which says that the dif-

ferential T ′ will not be invertible (see Theorem 1.9). Since∑vp=1 kp < m is what

typically occurs when v < m, Proposition 1.14 suggests that for almost all v-diskproblems with v < m, the differential T ′ will not be invertible. Consequently,if v < m, algorithms based solely on Newton’s method will behave badly. Inparticular, this is so for the Nehari problem whenm> 1.

When v > m, the spectral factors G of Ψ do not exist, so the hypotheses ofour theorems break down.

When v = m, we conclude from the lemma and our studies that Newtonbased methods are well behaved for almost all situations.

Some functional analysts might be both a bit disappointed and a bit surprisedthat the conditions for the invertibility of T ′(f ,Ψ , γ) for any given problem cannot be checked a priori. At best they hold for “generic” problems, which doesnot say anything about a particular problem. This, however, is a precise analog ofthe situation in linear programming [20], where the main result says that if “strictcomplementarity” holds at the solution to a given LP problem, then numerous in-terior point methods are better than first order convergent to that solution. Strictcomplementarity does not always hold, it is merely generic, and one never knowsuntil after an algorithm is run whether or not the optimum satisfies strict comple-mentarity. Nonetheless, the theorem alluded to is extremely valuable in analyzingand assessing interior point methods in LP.


It is important to bear in mind that even if conditions on (G, f , γ) are foundto insure that T ′(G,f ,γ) is invertible on an open dense set of (G, f , γ) in some suit-able topology, there is no guarantee that for an open dense set of Γ the optimizing(G, f , γ) correspond to invertible differentials T ′G,f ,γ . However, it does suggestthat. Proving such results could easily take many people many years, and wouldlikely have little effect on computational practice. Even when we allow many as-sumptions, as we shall see, quite a bit of mathematics must be developed. Thus,we assume things like smoothness freely.

1.7. Numerical experiments. We implemented Algorithms GTG and G +GT . We always took Gp to be an m ×m matrix valued function rather than themore aggressive choice of k×m. We found Algorithm G+GT to be the most broadlyeffective and would recommend it to the interested reader.

One caveat about the experiments is that they were run on a course grid (atmost 128 points.) This is all that was practical since for convenience we representfunctions on the disk inefficiently as values on grid points, or as power series.Possibly the coarse grid compromises some of our findings.

For the multidisk problem, there are of course various cases. In one case, forexample, not all the performance measures Γp matter; in another case they all domatter. This leads us to call a performance measure Γp active at an optimum ifΨp is not identically 0. For example, Γ 2 matters if and only if at an optimum it isactive for at least one θ. In what follows we only report on what happens when allthe constraints turn out to be active.

Our numerical experiments for v ≤ m, in generic situations, led to the fol-lowing observations:• For both algorithms we always found that rank(Ψp) ≤ 1, for all θ and indeed

for v ≤m, that rank(Ψp) = 1. While higher rank Ψp are usually possible thesealgorithms have a strong disposition not to produce them.

• For both algorithms we found in a few randomly selected problems that µpnull(θ)is independent of θ in many cases (though not all), and that µpnull = 0 predictsthat T ′ is invertible, thus substantiating the conjecture of Section 1.6.

• Also if µpnull(θ) is independent of θ, we found it to be given by the formula(m−

∑p

rankΨp)2= (m− v)2.

• In practice µpnull determines a finer structure of the null space of T ′. WhenT ′ is not Fredholm one would expect that its null space is within finite dimen-sions of being invariant under multiplication by eiθ, and thus it makes senseto count its dimension as a module over H∞. This number is easy to computeexperimentally via the formula

moddim :=dim null space of T ′(G∗,f∗,γ∗)

grid size.


We found in all experiments that if µpnull(θ) is constant, then µpnull = moddim.• The reason Algorithm G + GT works better than expected appears to be that

even when T ′ at optimum is not invertible, e.g., if v < m, at each iteration thecurrent T ′ has good conditioning until the very end of the computer run (thenit explodes). The Algorithm GTG certainly does not have this property and itsbehavior is indeed dominated by the invertibility of T ′ at optimum, just as ourtheory predicts.

The table below gives our experimental findings for both algorithms.

m v (m− v)2 moddim dim nullT ′ Experim.rank(Gp)

2 1 1 1 ∞ 1 for ∀p2 2 0 0 0 1 for ∀p3 1 4 4 ∞ 1 for ∀p3 2 1 1 ∞ 1 for ∀p

In this paper, we shall develop the theory underlying the GTG algorithm butdo not analyze the G + GT algorithm. Nevertheless the experiments reported onhere were run for both algorithms and suggest that µpnull has a strong connectionwith the Algorithm G + GT as well as for the Algorithm GTG. The AlgorithmG +GT is currently under study.

Comparisons with the matrix case and [1] are interesting. [1] have primal anddual non degeneracy conditions that are similar to the constraint m = v in ourcase. They prefer their XZ +ZX algorithm, which is the analog of our AlgorithmG + GT . Experimentally we found that in our setting m = v is stronger thanneeded to get good results (but not necessarily second order convergence).

2. GENERAL THEOREMS

This section begins by computing the differential of T and then analyzing its in-vertibility. We ultimately state a result about Newton type methods for the generalMOPT problem. The subsequent sections show how this result specializes to themultidisk problem, and gives the proof of the results stated in the introduction.

2.1. Calculation of the differential of T . In this section we shall calculatethe differential of the basic operator T that is defined by formula (1.26). We shallassume the smoothness conditions specified in Section 1.1, and we shall make useof (the first two terms of ) the formula

F(z0 +w) = F(z0)+ 2<{∂F∂z(z0)w

}+ 1

2

{2wT

∂2F∂z̄∂z

(z0)w + 2<w̄T ∂2F∂z2w

}+ o(‖w‖2) =


= F(z0)+N∑j=1

∂F∂zj

(z0)wj +N∑j=1

∂F∂zj

(z0)wj

+ 12

{2

N∑i,j=1

wi∂2F∂zi∂zj

(z0)wj +N∑i,j=1

wi∂2F∂zi∂zj

(z0)wj

+N∑i,j=1

wi∂2F∂zi∂zj

(z0)wj}+ o(‖w‖2),

which is valid for smooth functions F : CN → R. Here, ∂2F/(∂z̄∂z) and ∂2F/∂z2

are N × N matrices with ij entries equal to ∂2F/(∂zi∂zj) and ∂2F/(∂zi∂zj),respectively.

(It is perhaps useful to recall that, in terms of the notation zj = xj + iyj andzj = xj − iyj ,

∂F∂zj

= 12

{∂F∂xj

− i ∂F∂yj

}and

∂F∂zj

= 12

{∂F∂xj

+ i ∂F∂yj

}

(for real or complex valued functions F) and that if F is analytic in the variablesz1, . . . , zN , then ∂F/∂zj = 0.) Therefore,

T ′(G,f ,γ) :

H2+

H2N

R

→H2+

H2N

R

is given by

(2.1) T ′(G,f ,γ)

∆ϕη

=

PH2+[∆(γI−Γ( · , f ))]−PH2+

[G∂Γ∂z( · , f )ϕ+G∂Γ

∂z( · , f )ϕ

]+ηG

−PH2N

tr

[{GT∆+∆TG} ∂Γ

∂z( · , f )T+GTG

{∂2Γ∂z∂z

( · , f )ϕ+ ∂2Γ∂z2 ( · , f )ϕ

}]

2< 12π

∫tr{GT∆}dθ

.

Here, the second row of T ′(G,f ,γ) is the N × 1 mvf with entries


− PH2 tr{[∆TG +GT∆] ∂Γ

∂zi

}

−N∑j=1

PH2 tr

[G∂2Γ∂zi∂zj

GTϕj +G ∂2Γ∂zi∂zj

GTϕj

]

for i = 1, . . . , N.

2.2. The assumptions PSCON and SCOM. From now on we shall be work-ing with Γ and tuples (Ψ , f , γ) which satisfy certain assumptions. The first as-sumption includes the smoothness of Γ that is presumed throughout.

(PSCON) Γ is a positive semidefinite mvf that meets the smoothness assump-tions that are specified in Section 1.2, and is plurisubharmonic.

The next assumption guarantees that (Ψ , f , γ) satisfies a function theoretic formof the strict complementarity condition that was briefly mentioned in Section 1.4.

(SCOM) 1. The triple (Ψ , f , γ) is continuous, with Ψ positive semidefinite, f ∈H∞N and γIm − Γ(eiθ, f (eiθ)) ≥ 0. Moreover, it meets the strictcomplementarity condition

(2.2) Cm = rangeΨ(eiθ)⊕ range(γIm − Γ(eiθ, f (eiθ)))at every point eiθ.

2. There exists a G ∈ H∞k×m that is outer and continuous with rank kat every point eiθ such that Ψ = GTG.

3. There exists an H ∈ H∞`×m, ` =m−k, that is outer and continuouswith rank ` at every point eiθ, such that

H(γIm − Γ( · , f ))HT = I`×`.The function G is called the outer spectral factor of Ψ ; it is unique up to a constantk× k unitary left multiplier.

2.3. T ′ = L+ C with L selfadjoint and C compact.

Proposition 2.1. If assumptions SCOM and PSCON are in force, then thedifferential of T may be written in the form T ′(G,f ,γ) = L + C, where L and C mapH2+ ⊕ H2

N ⊕ C into itself, C is compact and L is the selfadjoint operator given by theformula


L

∆ϕη

=

PH2+[∆(γI − Γ( · , f ))]− PH2+

[G∂Γ∂z( · , f )ϕ

]+ ηG

−PH2N

[tr{(GT∆) ∂Γ

∂z( · , f )T

}]− PH2

N

[tr

{GTG

∂2Γ∂z ∂z

( · , f )ϕ}]

+ 0

12π

∫tr{GT∆}dθ + 0+ 0

.

Proof. The issue reduces to analyzing L because of the fact that a Hankeloperator

Hϑ : g ∈ H2 → PH2⊥ϑg

with symbol ϑ ∈ H∞ + C is compact, as is its adjoint. Here C stands for the classof continuous functions on T. In the present setting the operator C := T ′ − L canbe expressed in the form

C

∆ϕη

=

C12(ϕ)

C21(∆)+ C22(ϕ)

C31(∆)

,where

C12(ϕ) = −PH2+

[G∂Γ∂zϕ]= −

N∑j=1

PH2+

[G∂Γ∂zj

ϕj

],

C21(∆) = −PH2N

tr

{G(∂Γ∂z

)T ∆T}

is an N × 1 mvf with entries

−PH2 tr{G∂Γ∂zi

∆T} , i = 1, . . . , N,

C22(ϕ) = −PH2N

tr

{G∂2Γ∂z 2G

Tϕ}

is an N × 1 mvf with entries

−N∑j=1

PH2

[tr

{G

∂2Γ∂zi ∂zj

GTϕj

}], i = 1, . . . , N,


and

C31(∆) = 12π

∫ 2π

0tr{G∆T}dθ.

Thus, in view of the preceding remarks, C is compact. It is also readilychecked that L is selfadjoint with the help of the formula

(2.3) 〈∆, L12(ϕ)〉H2+ = 〈L21(∆),ϕ〉H2N.

We now verify (2.3).

〈L21(∆),ϕ〉H2N= −

∫ N∑j=1

tr[GT∆( ∂Γ

∂zj( · , f )

)T]ϕj(2.4)

= −∫ N∑j=1

tr

[∂Γ∂zj

( · , f )TϕjGT∆]

= −∫ N∑j=1

tr[∆(G ∂Γ

∂zj( · , f )ϕj

)T]

= 〈∆, L12(ϕ)〉H2+ .

❐

2.3. Fredholmness of T ′. Proposition 2.1 implies that T ′ is Fredholm of in-dex 0 if and only if L is Fredholm of index 0. We shall give conditions guaranteeingthat L is Fredholm, but first we need a few definitions.

Key definitions.

Definition 2.2. Let a = (∂Γ/∂z)( · , f ) = (a1, . . . , aN) : T → (Cm×m)N (sothat aϕ = ∑N

j=1 ajϕj) and G : T → Ck×m be continuous functions and let Adenote them×m matrix with ij entry

Aij = tr

{G∂2Γ∂zi∂zj

( · , f )GT}.

(i) Let σprimal and σdual denote the multiplication operators

σprimal : ϕ ∈ H2N -→

(GaϕAϕ

)∈ L2

k×m × L2N,

σdual : ∆1 ∈ (H2k×k)+ -→ tr{GaTGT∆1} ∈ L2

N.

(ii) The multiplication operator σprimal is said to be regular if it is generated bya function which has a trivial null space at every point eiθ ∈ T.

(iii) The multiplication operator σdual is said to be regular if the span of the k×kmatrices (GaTj G

T )(eiθ) is equal to Ck×k at every point eiθ ∈ T.


A key result.

Theorem 2.3. Assume that Γ , G, f , and γ meet the following constraints:(1) The assumptions PSCON and SCOM are satisfied;(2) The operators σprimal and σdual are regular.Then T ′(G,f ,γ) is Fredholm of index 0.

Remark 2.4. The reader may wonder why the measure nondegeneracy condi-tion does not appear explicitly in the preceding theorem (as well as the next one).The reason is that, although it intervenes in the theorems of Section 1 to helpinsure that T(G, F, γ) = 0 at solutions of the PDMOPT, the differential T ′ ofT is then calculated and studied at an arbitrary “point” (G, F, γ). The propertiesof T ′ and L etc as operators do not involve the measure non degeneracy per se.Therefore it does not appear in the statements of the current theorems. It is “only”when you try to connect with the original problem that these assumptions comeinto play.

The last theorem is a consequence of Theorem 3.9. The proof is given inSection 3. Note that the most damning assumption we made is that Ψ has a nicespectral factor G. However, the numerical experiments that are discussed in [12]indicate that this is often true.

Measure non-degeneracy vs. σdual regular. If ν is a selfadjoint matrix valuedmeasure, then

(2.5) dµ = GT dνG

is also, and, in addition, dµ satisfies

(2.6) (γIm − Γ)dµ = 0.

Strict complementarity of γIm−Γ and Ψ suggests that (2.6) implies (2.5), althoughwe have not checked this.

A matrix valued measure dµ of the form (2.5) satisfies the measure non-degeneracy condition Vf (eiθ, dµ) = 0 if and only if

tr(ajGT dνG) = 0 for j = 1, . . . , N.

This implies that dν = 0 if and only if the span of the matrices (GaTj GT )(eiθ),

j = 1, . . . , N is equal to Ck×k, the set of all complex k × k matrices. This is ex-actly the σdual regularity condition. Thus we see that measure non-degeneracyand σdual regularity are closely related. Measure non degeneracy is, however, aless restrictive condition because it only need hold for matrix valued measures dµwhich are nonnegative.


The null space of L. We have considerably stronger theorems that tell us thesize of the null space of L. We begin with a key definition.

Definition 2.5. Let a = (a1, . . . , aN) : T → (Cm×m)N (so that aϕ =∑Nj=1 ajϕj) and let G : T→ Ck×m be continuous functions.(i) Let Lprimal and Ldual denote the operators given by

Lprimalϕ :=PH2+Gaϕ

PH2NAϕ

and Ldual∆1 :=

PH2N

tr{GaTGT∆1}1

2π

∫tr{GGT∆1}dθ

,where ∆1 ∈ H2

k×k and ϕ ∈ H∞N . We call σprimal (resp. σdual) the symbol ofthe operator Lprimal (resp. Ldual).

(ii) The operator Lprimal (resp. Ldual) is said to be regular if σprimal (resp. σdual)is regular.

Additional analysis leads to a refinement of Theorem 2.3, which is based onthe observation that the null space of L effectively splits into two parts. One comesfrom the primal problem and one from the dual problem. The next theorem,which is stated without proof in order to save space, serves as a sample.

Theorem 2.6. Assume that Γ , G, f , and γ meet the following constraints:(1) The assumptions PSCON and SCOM are satisfied;(2) The operators σprimal and σdual are regular.Then the dimension of the space of all triples (∆,ϕ,η) ∈ H2+×H2

N×R in the null spaceof L is equal to the dimension of the space of all triples (∆1,ϕ,η) ∈ (H2

k×k)+×H2N×R

such that

Lprimalϕ =Gη

0

and Ldual∆1 = 0.

We remark that Lprimalϕ = 0 may not force ϕ to be 0.

3. PROOFS

3.1. The null space of σdual. Before launching into proofs, we shall showthat Uf (eiθ)TUf (eiθ) invertible implies that σdual is regular. Although we shallnot need this result in the rest of this paper, it does serve to illustrate the connec-tions between a number of the conditions that are often imposed in optimizationproblems of this sort.

Proposition 3.1. Let Γ(eiθ, z) satisfy the smoothness conditions specified in Sec-tion 1.2, let f ∈ H∞N be continuous and recall the notation

a`(·) := ∂Γ∂z`

( · , f (·)), ` = 1, . . . , N, and a := (a1, . . . , aN).


The condition Uf (eiθ)TUf (eiθ) is invertible for all θ, implies N ≥m2 and isequivalent to each of the following two conditions:

(i) The span of the set {aT1 , aT2 , . . . , aTN} is equal to Cm×m for all θ;(ii) For each θ, the map of B ∈ Cm×m , aB = (a1B, . . . , aNB) ∈ (Cm×m)N has

a trivial null space.

Proof. This is done by careful bookkeeping applied to the definition of Uf .Formula (1.16) is helpful. ❐

3.2. Changing the variables of L. We shall analyze the invertibility of L withthe help of a change of variables. If Q ∈ H∞m×m, then

∆ -→ ∆̃ := ∆Qis a map of (H2

k×m)+ into H2k×m. Moreover, if Q(0) is upper triangular, then this

map sends (H2k×m)+ into itself. If bothQ and its pointwise inverseQ−1 belong to

H∞m×m, then these mappings are onto. The adjoint of this mapping with respectto the matrix inner product 〈A,B〉 = tr{ABT} is

(3.1) ∆→ ∆QT .We consider the map M0 from (H2

k×m)+ ⊕H2N ⊕ R into itself that is defined

by the rule M0(∆,ϕ,η) := (∆Q,ϕ,η).The adjoint of this map in the ( 1

2π)∫ 2π0 tr{ABT}dθ inner product is the

Toeplitz like operator MT0 defined by

MT0 (∆,ϕ,η) := (PH2+{∆QT},ϕ,η).Our goal is to choose Q so that invertibility of the map

(3.2) L̃ :=MT0 LM0

is easy to analyze. We shall assumeQ is in (H∞m×m)+, and hence that the constantterm Q(0) is upper triangular.

The abbreviations

a := ∂Γ∂z( · , f ) = [a1, a2, . . . , aN] and aϕ =

N∑j=1

ajϕj

that were introduced earlier will be useful.It is clear from formula (3.2) that L̃ is selfadjoint. Moreover,

L̃

∆ϕ

η

=

PH2+[∆Q(γI − Γ( · , f ))QT ]− PH2+[GaϕQ

T]+ ηPH2+[GQT]

−PH2N[tr{GT∆QaT}]− PH2

N[Aϕ]+ 0

12π

∫tr{GT∆Q}dθ + 0+ 0

.


The exhibited formula for L̃ follows easily from the formula for L with the help ofthe following result.

Lemma 3.2. If Q is in (H∞m×m)+ and F ∈ L2k×m, then

PH2+{(PH2+F)QT} = PH2+{FQT}.

Proof. If F ∈ L2k×m, then F = F+ + F−, where F+ ∈ H2+ and F− ∈ (H2+)⊥.

Therefore,FQT = F+QT + F−QT .

But now as 〈C, F−QT 〉 = 〈CQ,F−〉 = 0 for every C ∈ H2+, it follows that F−QT ∈(H2+)⊥. Thus

PH2+{FQT} = PH2+{F+QT},as claimed. ❐

The next conclusion is immediate from formula (3.2).

Proposition 3.3. If Q and Q−1 belong to H∞m×m, then L is invertible (resp.Fredholm of index k) if and only if L̃ is invertible (resp. Fredholm of index k).

3.3. A nice form for L̃.

The spectral factors G, H and Q. Now we select Q ∈ (H∞m×m)+ to meet ourends. There are two closely related ways to define Q.(1) Define ρ by ρ := Ψ [−1] + (γI − Γ), where Ψ [−1] denotes the Moore-Penrose

inverse of Ψ . Then, in view of the strict complementarity assumption SCOM,the inequality ρ(eiθ) ≥ δIm×m holds a.e. for some δ > 0. Take Q to be theouter function in H∞m×m satisfying ρ−1 = QTQ.

Then Q is invertible in H∞m×m, and

ρ = Q−1Q−T and QρQT = Im×m.

(2) Note that Q may be expressed in terms of the outer mvf ’s G ∈ (H∞k×m)+ andH ∈ H∞`×m that appear in the formulas H(γI − Γ)HT = I`×` and Ψ = GTGthat were introduced in the assumption SCOM as

Q =GH

.Here rankG = k and rankH = `. To see this, recall that at the optimumchoice of (G, f , γ), G{γI − Γ( · , f )} = 0.

Therefore, upon introducing the singular value decomposition

G = U[D 0

]V,


where U is k×k unitary, D is k×k positive diagonal, and V ism×m unitaryat each point θ, it follows that

Ψ [−1] = VTD−2 0

0 0

V, GΨ [−1]GT = Ik×k,

and

rangeΨ [−1]GT = rangeVTIk×k

0

= rangeGT

= rangeΨ [−1] = rangeΨ ,whereas,

range{γIm − Γ( · , f )} = rangeVT 0

I`×`

.Thus,

γIm − Γ( · , f ) = VT0 0

0 E2

Vfor some positive definite ` × ` matrix E and hence, upon setting

H =[0 E−1

]V,

we see that

HΨ [−1] = 0 and HρHT = H{γIm − Γ( · , f )}HT = I`×`.Therefore,

Q =GH

has the property

Q{γIm − Γ( · , f )}QT =0 0

0 I`×`

.L̃ in nice coordinates. The next step is to partition ∆ as

∆ = (∆1 ∆2

)


with ∆1 ∈ (H2k×k)+ and ∆2 ∈ H2

k×` (so that ∆1(0) is upper triangular, while∆2(0) is arbitrary). It then follows that L̃ acting on

∆ϕη

can be re-expressed in the form

(3.3)

PH2+[(0 ∆2)]− PH2+[(GaG

T GaHT)ϕ]+ ηPH2+[(GGT 0)]

−PH2N[tr(∆1GaTGT +∆2HaTGT)]− PH2

N[Aϕ]+ 0

12π

∫tr{GGT∆1}dθ + 0+ 0

.

Here we used the fact that the first optimality condition implies that GHT =0, that is, the range of GT and the range of HT are orthogonal complements.Moreover, the following conventions are in force:

PH2+

[(GaGT GaHT

)ϕ]= PH2+

[ N∑j=1

(GajGT GajHT

)ϕj],

and

PH2N[tr(∆1GaTGT +∆2HaTGT)] = PH2

N

tr(∆1GaT1GT +∆2HaT1GT)

...tr(∆1GaTNGT +∆2HaTNGT)

.

3.4. L̃ and Fredholmness. We shall show in this section that the operator L̃given by formula (3.3) in Section 3 is a Toeplitz operator. Also, we establish con-ditions under which L̃ is Fredholm of index zero, and we shall discuss invertibilityissues.

Assumptions and notation. Throughout this section we suppose that Γ(eiθ, z)satisfies PSCON and that (Ψ , f , γ) satisfies SCOM. In addition, we assume thatG is a continuous outer function in (H2

k×m)+ and that Q is a continuous invert-ible outer function in (H2

m×m)+, which is obtained by stacking the given matrixfunction G with the (k −m) ×m matrix valued analytic function H that wasconsidered in the preceding subsection. Thus, a = (a1, . . . , aN) is a continuousmvf whose entries aj take m ×m matrix values and A is a continuous positivesemidefinite mvf from T into CN×N .


L̃ is Toeplitz. We now introduce a multiplication operator that plays a key rolein our discussion of L̃ and T ′.

Definition 3.4. For ` =m− k, let P2 denote the projection operator

(3.4)P2 : (H2

k×m)+ = (H2k×k)+ ×H2

k×` -→ (H2k×m)+∆ = (∆1,∆2) 7 -→ ∆Π2 = (0,∆2),

where Π2 designates the matrix 0k×k 0k×`

0`×k I`×`

,and let M denote the operator

(3.5)

M : (H2k×m)+ ×H2

N -→ L2k×m × L2

N∆ϕ

7 -→ ∆Π2 −GaQTϕ− tr[∆QaTGT]−Aϕ

.Note that we are representing the values of M in block column format. The

map M is defined pointwise, so for fixed θ we may think of M as a map fromCk×m×CN to itself. Also, a similar comment applies to P2, and one may view thepointwise action of P2 as multiplication (on the right) of k ×m matrices by theblock matrix Π2.

We now confirm that the map M is related to L̃ in an obvious way.

Definition 3.5. Let P denote the orthogonal projection operator

(3.6) P : L2k×m × L2

N -→ (H2k×m)+ ×H2

N

and let L : (H2k×m)+ ×H2

N → (H2k×m)+ ×H2

N be the operator given by

(3.7) L(·) = P(M(·)).

Formula (3.7) suggests that L is a Toeplitz operator. This is indeed the case.

Proposition 3.6. The operatorL is a Toeplitz operator T̃ with continuous symboldefined on (H2

k×m)+ ×H2N . Moreover,

(3.8) L̃

∆ϕ

η

=L

∆ϕ

0

+

PH2+[GQT]η

0

12π

∫tr{GT∆Q}dθ

.


Proof. To see this, represent H2k×m ×H2

N as H2km+N and use a shift on some

of the entries of H2k×m ×H2

N to obtain (H2k×m)+ ×H2

N . Clearly, the action of Mon (H2

k×m)+ ×H2N corresponds to the action of a suitable multiplication operator

with continuous matrix symbol M̃ on elements in H2km+N . Hence the operator

P(M(·)) = L is a Toeplitz operator with continuous symbol. ❐

L is selfadjoint.

Proposition 3.7. The operators M and L are self adjoint with respect to theindicated inner products.

Proof. For each fixed θ, one can view M as a mapping from Ck×m × CN toitself. We choose the inner product on this space to be an Euclidean one,

〈(∆,ϕ), (δ,ψ)〉Ck×m×CN = tr(δT∆)+ (ψTϕ).We have that for all (∆,ϕ), (δ,ψ) ∈ Ck×m ×CN ,

〈(∆,ϕ),M[(δ,ψ)]〉Ck×m×CN= tr{(δΠ2 −GaQTψ)T∆} + (− tr[δQaTGT ]−Aψ)Tϕ= tr{(Π2δT −ψTQaTGT)∆} + (− tr(GaQTδT )−ψTAT)ϕ= tr(Π2δT∆−GaQTδTϕ)+ tr(−ψTQaTGT∆)−ψTATϕ= tr(δT∆Π2 − δTGaQTϕ)+ψT(− tr(QaTGT∆)−Aϕ)= tr{δT(∆Π2 −GaQTϕ)} +ψT(− tr(∆QaTGT)−Aϕ)= 〈M[∆,ϕ], (δ,ψ)〉Ck×m×CN .

If now (∆,ϕ), (δ,ψ) represent functions in (H2k×m)+ ×H2

N , then

〈(∆,ϕ),L(δ,ψ)〉L2k×m×L2

N

= 〈(∆,ϕ),P(M[δ,ψ])〉L2k×m×L2

N

= 〈(∆,ϕ),M[δ,ψ]〉L2k×m×L2

N

= 12π

∫ 2π

0〈(∆,ϕ),M[δ,ψ]⟩

Ck×m×CN dθ

= 12π

∫ 2π

0〈M[∆,ϕ], (δ,ϕ)〉Ck×m×CN dθ

= 〈M[∆,ϕ], (δ,ψ)〉L2k×m×L2

N

= 〈P[M[∆,ϕ] ], (δ,ψ)〉L2k×m×L2

N

= 〈L(∆,ϕ), (δ,ψ)〉L2k×m×L2

N.

The rest of the proof is straightforward. ❐


The null space ofM. In this subsection we derive a characterization of the nullspace of M that is useful for our study of the operator L̃.

Proposition 3.8. Let (∆,ϕ) ∈ Ck×m×CN . Partition ∆ as (∆1,∆2) ∈ Ck×k×Ck×(m−k). Then M[∆,ϕ] = 0 if and only if the following conditions hold:

∆2 = 0;(i)

σprimal(ϕ) = 0;(ii)

σdual(∆1) = 0.(iii)

Proof. If M[∆,ϕ] = 0, then

(3.9)[0 ∆2

]−G

N∑`=1

a`ϕ`[GT HT

]= 0

and

(3.10) − tr{∆1GaTGT} − tr{∆2HaTGT} −Aϕ = 0.

From (3.9) it follows that

(3.11) 0 = GN∑`=1

a`ϕ`GT = GaϕGT

and

(3.12) ∆2 = GN∑`=1

a`ϕ`HT = GaϕHT .

Combine (3.12) and (3.10) to obtain

(3.13) tr{∆1GaTGT} + tr{G

N∑`=1

a`ϕ`HTHaTGT}+Aϕ = 0.

Next, multiply by ϕT on the left to obtain

(3.14) tr{∆1G(aϕ)TGT} + tr{GaϕHTH(aϕ)TGT} +ϕTAϕ = 0.

The first term on the left hand side of (3.14) is zero by (3.11) and each of theremaining two terms is nonnegative. Thus, we see that

(3.15) GaϕHT = 0


and

(3.16) Aϕ = 0.

From (3.11), (3.15), the invertibility of QT =(GT HT

), and (3.16), we conclude

that σprimal(ϕ) = 0.The left hand side of (3.15) is precisely ∆2 by formula (3.12), that is, we have

(3.17) ∆2 = 0.

Now substitute (3.16) and (3.17) into (3.10) to obtain σdual(∆1) = 0.We now prove the converse. Suppose that equations (i)-(iii) hold. Then, since∆Π2 = ∆2 = 0, we see that

M

∆ϕ

= ∆Π2 −GaQTϕ− tr[∆QaTGT]−Aϕ

= −GaϕQT

− tr[∆1GaTGT]−Aϕ

= 0.

❐

L̃ as a Fredholm operator of index zero.

Theorem 3.9. Suppose that Γ(eiθ, z) satisfies PSCON and let (Ψ , f , γ) be suchthat SCOM is satisfied. Let G be a continuous outer function of rank k in (H2

k×m)+such that GTG = Ψ (the existence of such an outer function is guaranteed by SCOM)and assume that the null space of Vf (eiθ) restricted to matrices of the form

{GT(eiθ)BG(eiθ) | B ∈ Ck×k}

is equal to zero for every point eiθ. Then σdual is regular.If, in addition, the operator σprimal is regular, then L̃ and T ′ are Fredholm oper-

ators of index zero.

Proof. Suppose first that B ∈ Ck×k is orthogonal to the span of

{(GaTj GT )(eiθ) | j = 1, . . . , N},

for some choice of eiθ. Then

0 = 〈B, (GaTj GT )(eiθ)〉 = tr{(GajGT )(eiθ)B}= tr{aj(eiθ)GT (eiθ)BG(eiθ)}

for j = 1, . . . , N. Moreover, by the SCOM assumption,

{γIm − Γ(eiθ, f (eiθ))}G(eiθ)T = 0.


Consequently,G(eiθ)TBG(eiθ) belongs to the null space ofVf (eiθ, ·). There-fore, by assumption, G(eiθ)TBG(eiθ) = 0, and hence, as G(eiθ) is right invertibleby assumption, B = Ok×k. This completes the proof that σdual is regular.

Now assume in addition that σprimal is also regular. We claim that for each θ,the operator M acting on Cm×m×CN has a trivial null space. To see this, supposeM[∆,ϕ] = 0 for an element (∆,ϕ) ∈ Cm×m × CN . Then, by (i) and (ii) ofProposition 3.8, we must have ∆2 = 0 and

(3.18) σprimal(ϕ) =GaϕAϕ

= 0.

Since σprimal is regular, it follows that ϕ = 0.Next, from item (iii) of Proposition 3.8 we see that

(3.19) 〈∆1, Ga`GT 〉 = 0 for ` = 1, . . . , N.

Therefore, since σdual is regular, ∆1 = 0. Thus, ∆ = 0 and M has a trivial kernel.SinceM is selfadjoint by Proposition 3.7, we conclude thatM is an invertible mul-tiplication operator for each point θ. Hence, L is a self adjoint Toeplitz operatorwith continuous, pointwise invertible symbol. Theorem 2.94 on page 96 of [2]guarantees that such operators are Fredholm. Thus, we conclude that L is a Fred-holm operator. Moreover, since L is selfadjoint by Proposition 3.7, the Fredholmindex of L is zero.

By Proposition 3.6, L̃ (and therefore T ′) is Fredholm with index 0 if and onlyif L is is Fredholm with index 0, since the operator that corresponds to the secondcolumn on the right hand side of formula (3.8) is compact. ❐

Proof of Theorem 2.3. The desired conclusions are immediate from the lastthree paragraphs of the proof of Theorem 3.9. ❐

4. THE H∞ ONE DISK (NEHARI) PROBLEM

Our main interest in this paper is the multidisk problem. Nevertheless, the Nehariproblem is a good place to start, since it is a one disk problem and hence thecalculations for this setting serve as a model for the multidisk problem.

Derivatives. Recall that for the Nehari problem

(4.1) Γ(eiθ, Z) = (K(eiθ)− Z)T (K(eiθ)− Z),where Γ , K (and f ) are m ×m mvf ’s and Z = (zij)mi,j=1 with N =m2 indepen-dent entries. We shall need formulas for the first and second derivatives of Γ . Inparticular, it is readily seen that

(4.2) ak` := ∂∂zk`

Γ(eiθ, Z) = −(K(eiθ)− Z)TEk`


and

(4.3)∂2

∂zrs∂zk`Γ(eiθ, Z) = ETrsEk` = EsrEk`,

where Ek` is the m ×m constant matrix with a one in the k` position and zeroselsewhere.

Lemma 4.1. The performance function Γ(eiθ, f ) for the Nehari problem isplurisubharmonic. It is strictly plurisubharmonic if and only if m = 1.

Proof. By formula (4.3),

m∑r ,s,k,`=1

uTrs∂2Γ

∂zrs∂zkùk` =

m∑r ,s,k,`=1

uTrsEsrEkùk` =∥∥∥ m∑k,`=1

Ekùk`∥∥∥2

for every choice of the N = m2 column vectors u11, . . . , umm of size m × 1.Therefore, Γ is plurisubharmonic. Moreover, if m > 1, then there exist nonzerovectors uk` such that Ekùk` = 0. Thus, Γ is strictly plurisubharmonic if and onlyif m = 1. ❐

We remark that, since

(4.4) EsrEk` ={

0 when r 6= k,Es` when r = k,

the last sum is also equal to

m∑s,k,`=1

uTksEskEkùk` =m∑k=1

∥∥∥ m∑`=1

Ekùk`∥∥∥2

Moreover, much the same sort of analysis leads to the auxiliary conclusionthat them2 ×m2 matrix with entries

tr

{∂2Γ

∂zrs∂zk`

}= tr{EsrEk`}

is positive definite, since

m∑s,r ,k,`=1

ϕrs tr{EsrEk`}ϕk` =m∑

k,`=1

|ϕk`|2.


The optimality condition. In the Nehari case, the MOPT optimality condi-tions given in Theorem 1.1 can be stated as follows:

[γI − (K − f)T (K − f)]GT = 0,(4.5a)

PH2m×m{[K − f]GTG} = 0,(4.5b)

12π

∫tr{GTG}dθ = 1.(4.5c)

Proof. (4.5a) is immediate from Lemma 1.3 and (4.5c) is obvious. Therefore,we turn to (4.5b). For each k, ` we have

0 = PH2 tr[aTk`GTG] = −PH2 tr[ETk`(K − f)GTG]

= −PH2[(K − f)GTG]k`. ❐

The null space of σprimal. We turn now to the analysis of the null space ofσprimal in the Nehari case. For this analysis it is convenient to think of ϕ as anm×m matrix with entries ϕk`, k, ` = 1, . . . ,m, rather than anm2 × 1 columnvector. Then

(4.6) ϕ =m∑

k,`=1

Ek`ϕk`.

Lemma 4.2. In the setting of the Nehari problem, a matrix B ∈ Cm×m is in thenull space of σprimal at the point eiθ ∈ T if and only if

(4.7) G(eiθ)(K(eiθ)− f(eiθ))TB = 0 and G(eiθ)BT = 0.

Proof. If σprimal(B) = 0, then the first statement in (4.7) follows directlyfrom the first block row in the definition of σprimal and formula (4.2). Next, thesecond block row of the formula for σprimal implies that

0 = 〈AB,B〉 =∑r ,s

∑k,`

tr{G(eiθ)BrsEsrEk`Bk`G(eiθ)T}

= tr{G(eiθ)BTBG(eiθ)T}.

But this clearly implies that B G(eiθ)T = 0 also, as claimed. ❐

We remark that if K(eiθ)−f(eiθ) is invertible when f is a solution to MOPT,then an alternate way of writing the first condition in (4.7) at an optimum isG(K − f)−1B = 0. This is because the optimality condition

G(eiθ)(γIm − (K(eiθ)− f(eiθ))T (K(eiθ)− f(eiθ))) = 0


can be rewritten as

γG(eiθ) = G(eiθ)(K(eiθ)− f(eiθ))T (K(eiθ)− f(eiθ)).

But this implies that

G(eiθ)(K(eiθ)− f(eiθ))T = γG(eiθ)(K(eiθ)− f(eiθ))−1,

which gives the result.

The null space of σdual. In the setting of the Nehari problem, σdual mapsthe k× k matrix B into them×m matrix with entries

tr

{G∂Γ∂zij

( · , f )TGTB}= − tr{G[(K − f)TEij]TGTB}

= − tr{GEji(K − f)GTB}= − tr{Eji(K − f)GTBG}= −((K − f)GTBG)ij,

for i, j = 1, . . . ,m. Thus, we have established the following result:

Lemma 4.3. In the setting of the Nehari problem, a matrix B ∈ Ck×k is in thenull space of σdual at the point eiθ ∈ T if and only if

(K(eiθ)− f(eiθ))G(eiθ)TBG(eiθ) = 0.

The U condition. The condition for Uf given by (1.13) to be invertiblespecializes as follows.

Lemma 4.4. In the Nehari case, UTf (eiθ)Uf (eiθ) is invertible if and only ifK(eiθ)− f(eiθ) is invertible.

Proof. Direct calculation gives

Uf (eiθ) = −ΠT diag{K(eiθ)− f(eiθ), . . . , K(eiθ)− f(eiθ)}Π,where the mvf inside the curly brackets on the right is block diagonal with midenticalm×m blocks and Π is anm2 ×m2 permutation matrix. ❐

Conclusions for the Nehari case. The conclusions for the Nehari case aresubsumed in the conclusions for the multidisk problem: just take v = 1. Inparticular, it should be noted that if m > 1, then the differential T ′ is not aFredholm operator in this setting.


5. MULTIDISK MOPT

The Nehari problem prescribes a disk in matrix function space and seeks an ana-lytic mvf which lies inside it. Now we specify v disks

(5.1) Γp(eiθ, f (eiθ)) = (Kp(eiθ)− f(eiθ))T (Kp(eiθ)− f(eiθ)),p = 1, . . . , v.

Here Γp and f arem×mmatrix valued functions and we seek an analytic functionf and the smallest γ satisfying

Γp(eiθ, f (eiθ)) ≤ γIm for p = 1, . . . , v and all θ.

Multiperformance MOPT. To any problem with multiple performance func-tions Γp, p = 1, . . . , v, we can associate a single block matrix (now of sizemv ×mv)

Γ :=

Γ 1 0 · · · 0

0 Γ 2 · · · 0

......

. . ....

0 0 · · · Γv

and consider the corresponding MOPT problem. Fortunately, the dual variable Ψin the corresponding PDMOPT problem (which is formulated in Subsection 1.3)can be taken block diagonal.

Theorem 5.1. Let Γ(eiθ, z) be a positive semidefinite mvf that meets the smooth-ness conditions specified in Section 1.2 and assume that the primal dual problemPDMOPT has a solution (Ψ , f , γ) ∈ L1

vm×vm ×H∞N ×R such that:(1) Ψ = diag(Ψ1, . . . ,Ψv) is block diagonal;(2) Ψp = (Gp)TGp has an outer spectral factor Gp ∈ (H2

kp×m)+ with rankGp =kp a.e. for p = 1, . . . , v;

(3) Ψp ∈ L2m×m for p = 1, . . . , v;

(4) f is continuous and γI − Γ(eiθ, f (eiθ)) ≥ 0;(5) The conditions

v∑p=1

tr

{∂Γp∂zj

dµp}= 0,

trv∑p=1

{γIm − Γp(eiθ, f (eiθ))}dµp = 0, and

dµp ≥ 0

for p = 1, . . . , v imply that dµp = 0 for p = 1, . . . , v.


Then Gp, f and γ must satisfy the following conditions:(a) G1(γI − Γ 1) = · · · = Gv(γI − Γv) = 0,

(b) PH2N

tr{(G1)TG1 ∂Γ 1

∂z

T

+ · · · + (Gv)TGv ∂Γv∂z

T}= 0,

(c)v∑p=1

∫tr(Gp

TGp)dθ = 2π .

Proof. This theorem is an easy consequence of Theorem 1.1 and the specialstructure of the multidisk problem. ❐

It is also readily checked by straightforward calculation that the mvf Uf (eiθ)for the full performance function Γ is simply related to the mvf ’s Upf (eiθ) for theperformance functions Γp, p = 1, . . . , v. The set of columns ofUf is equal to theunion of the set of columns Upf , p = 1, . . . , v, supplemented by zero columns.Thus,

(5.2) UfUTf =v∑p=1

Upf (Upf )T .

However, this does not seem to be useful at the moment, since the columns ofUf (eiθ) are linearly independent if and only if Uf (eiθ)TUf (eiθ) is invertible.

The H∞ multidisk problem. We now take the block diagonal entries inΓ(eiθ, f ) to be of the form (4.1). It is then convenient to picture both f andϕ as m ×m matrices with entries fk` and ϕk`, k, ` = 1, . . . , m, instead ofcolumn vectors of height m2 × 1, just as in the Nehari case. Thus, upon writing

Γ = Γ(·, Z) = diag{(K1 − Z)T (K1 − Z), . . . , (Kv − Z)T (Kv − Z)}

with Z = [zk`], k, ` = 1, . . . ,m, it is readily checked that

∂Γ∂zk`

= −diag{(K1 − Z)TEk`, . . . , (Kv − Z)TEk`}(5.3)

and∂2Γ

∂zrs∂zk`= diag{EsrEk`, . . . , Esr Ek`} = EsrEk`,(5.4)

where

(5.5) Ek` = diag{Ek`, . . . , Ek`}

has v identical block diagonal entries. These formulas lead easily to the followingresults.


Lemma 5.2. The performance function Γ for theH∞ multidisk problem is plurisub-harmonic. In fact

m∑r ,s,k,`=1

uTrs∂2Γ

∂zrs∂zkùk` =

∥∥∥ m∑k,`=1

Ekùk`∥∥∥2

for every choice of vectors uk` ∈ Cmv , k, ` = 1, . . . ,m.

Lemma 5.3. Let dµ = diag{dµ1, . . . , dµv} with nonnegative block matrixmeasures dµp of sizem×m. Then, in the H∞ multidisk case,

Vf (eiθ, dµ) = 0 if and only ifv∑p=1{Kp(eiθ)− f(eiθ)}dµp = 0 and {γIm − Γp(eiθ, f (eiθ))}dµp = 0

for p = 1, . . . , v.

Proof. In view of formula (1.15), the first N aliasm2 entries in the constraintVf (eiθ, dµ) = 0 yields the sequence of formulas

0 = tr{∂Γ∂zrs

dµ}=

v∑p=1

tr{∂Γp∂zrs

dµp}= −

v∑p=1

tr{(Kp − f)TErs dµp}

= −v∑p=1

tr{Ers dµp(Kp − f)T} = −v∑p=1

{dµp(Kp − f)T}sr

for s, r = 1, . . . , m. But this leads easily to the first condition in the forwardimplication. The second condition is selfevident, as is the converse. ❐

Proof of Theorem 1.6. Our first objective is to verify that Theorem 1.1 is ap-plicable. The main effort is to translate condition (4). But that is done in thepreceding lemma.

Next, invoking Theorem 1.1 and Lemma 1.3, we see that conditions (a) and(c) are obvious. To prove (b), note that the second optimality condition in Theo-rem 1.1 or Theorem 5.1 implies that

(5.6) − tr(GarsGT ) = tr{ v∑p=1

GpTGp(Kp − f)TErs

}∈ eiθH2

for all r = 1, . . . , m and s = 1, . . . , m. Thus tr[βErs] ∈ eiθH2, which isequivalent to β ∈ eiθH2

m×m. ❐


The null space of σprimal. Now we turn to an analysis of the null space of T ′for the H∞ multidisk problem. Recall from Theorem 2.3 that the key to under-standing the null space of T ′ is an understanding of the operators σprimal andσdual.

Lemma 5.4 (Primal). In the H∞ multidisk case, assume that Kp, Gp for p = 1,. . . , v, and f are continuous on T. Then a matrix B ∈ Cm×m is in the null space ofσprimal at the point eiθ if and only if

Gp(eiθ)(Kp(eiθ)− f(eiθ))B = 0 and Gp(eiθ)BT = 0 for p = 1, . . . , v.

Thus,µpnull(θ) = dim{B ∈ Cm×m | σprimal(B) = 0 at eiθ}.

Proof. For the multidisk problem,

σprimal(B) =

−diag{G1(K1 − f)TB, . . . , Gv(Kv − f)TB}

v∑p=1

GpTGpBT

.Thus, σprimal(B) = 0 if and only if Gp(Kp − f)TB = 0 for p = 1, . . . , v and∑vp=1Gp

TGpBT = 0. The second equation is equivalent to GpBT = 0 for p =1, . . . , v. This completes the proof of the first assertion. The second followsimmediately from the definition of µpnull(θ). ❐

The null space of σdual.

Lemma 5.5. In the H∞ multidisk case, let Bp ∈ Ckp×kp for p = 1, . . . , v.Then the null space of σdual is equal to zero providing that Bp = Okp×kp , p = 1, . . . ,v, is the only solution of the equation

(5.7)v∑p=1

{Kp(eiθ)− f(eiθ)}Gp(eiθ)TBpGp(eiθ) = 0,

that is, µdnull(θ) = 0 for all θ.

Proof. Recall that σdual is regular provided the null space of σdual is 0 at eachpoint eiθ ∈ T. This is an elaboration of the calculation for the Nehari case. It isan easy consequence of the definition of σdual and formula (5.3). ❐

The U condition for the multidisk problem. In view of formula (5.2) and thecalculations for the Nehari problem, it is readily seen that

UfUTf =v∑p=1

ΠT diag{(Kp − f)(Kp − f)T , . . . , (Kp − f)(Kp − f)T}Π,


where the block diagonal matrix inside the curly brackets on the right hasm iden-tical m × m blocks and Π is an m2 × m2 permutation matrix. Thus,Uf (eiθ)Uf (eiθ)T is invertible if and only if Kp(eiθ) − f(eiθ) is invertible forp = 1, . . . , v. However, if v > 1, then Uf (eiθ)TUf (eiθ) is never invertible.

Proof of Theorem 1.9. The first four assumptions of this theorem guaran-tee that the assumptions imposed in Theorem 1.6 are met. Therefore, all theconclusions of the latter are in force. In view of Lemma 5.2 and the presumedsmoothness we have PSCON. Then, with the aid of assumption (5), we obtainthe first two items of SCOM. The third follows from the construction in Section3. In particular, since Ψ(eiθ) is continuous with constant rank, the Moore-Penroseinverse Ψ [−1](eiθ) is also a continuous function of θ on the unit circle (cf [17]).Therefore, ρ is continuous and strictly positive definite on the circle. This guar-antees the existence of the factorization ρ−1 = QTQ with Q outer. However, inorder to obtain a continuous factor Q, we need a little more:

By a general theorem that has been obtained in the work of J. Plemelj, N.I.Mushelisvili and N.P. Vekua, a sufficient condition for the continuity of Q is theHolder continuity of ρ on the circle. Therefore, since γI−Γ(eiθ, f (eiθ)) is Holdercontinuous by assumption, it remains only to check that the Moore-Penrose in-verse Ψ [−1](eiθ) of Ψ(eiθ) is Holder continuous. To this end, let C be a simpleclosed contour in the open right half plane that encircles the nonzero spectrumσ ′(Ψ(eiθ)) of the positive semidefinite matrix Ψ(eiθ) for all θ ∈ [0,2π] and issuch that

(5.8) |λ− µ| ≥ δ > 0

for every choice of λ ∈ C and µ ∈ σ ′(eiθ). The presumed continuity and con-stant rank of Ψ(eiθ) guarantees the existence of such a contour. Then it is readilychecked that Ψ [−1](eiθ) = 1

2πi

∫Cλ−1(λI − Ψ(eiθ))−1 dλ

and hence that

ddθ

Ψ [−1](eiθ) = 12πi

∫Cλ−1(λI − Ψ(eiθ))−1Ψ ′(eiθ)(λI − Ψ(eiθ))−1 dλ.

Therefore, since

(5.9) ‖(λI − Ψ(eiθ))−1‖ ≤ δ−1

for λ ∈ C, thanks to (5.8), the last formula leads easily to the bound∥∥∥∥ ddθΨ [−1](eiθ)∥∥∥∥ ≤ 1

2πδ−2‖Ψ ′(eiθ)‖∫

C|dλ||λ| ,


which in turn guarantees the Holder continuity of Ψ [−1]. The fact that H(eiθ)has constant rank follows easily from the construction in Section 3.

Now we return to the main thread of the proof. Hypotheses (6) and (7) statethat σprimal and σdual are regular. Thus, by Theorem 2.3, T ′ is Fredholm of indexzero. On the other hand, if either (6) or (7) fails, then, by Proposition 3.8, thesymbol M is not invertible for all θ and hence the operator T ′ is not Fredholm. �

6. SUPPLEMENTARY PROOFS

We begin this section with a proof of the optimality theorem on which this paperis based. The proof is a refinement of the proof of Theorem 2 of [12] (which alsoappears as Theorem 17.1.1 of [10]). Then we turn to the GTG factorization of Ψand establish Lemma 1.3, which serves to rephrase condition (a) of Theorem 1.1in terms of G. Subsequently, we prove a few specializations of Theorem 1.1 to themulti performance case and then, finally, justify Proposition 1.14.

Proof of Theorem 1.1. Equation (20.34) on page 217 of [10] implies that ifγ, f is a local solution of the MOPT problem with γIm − Γ(eiθ, f (eiθ)) ≥ 0,then

(6.1) tr

{∂Γ∂z`

( · , f )(dλ)τ}= eiθϕ`

dθ2π, ` = 1, . . . , N,

whereϕ` belongs to the Hardy space H1 and dλ is anm×m nonnegative matrixvalued measure on the circle such that

(6.2)∫ 2π

0tr{dλ} = 1.

Consider the Radon-Lebesgue-Nikodym decomposition

(6.3) (dλ)τ = Ψ dθ2π

+ dµ,

where Ψ is an m ×m mvf that is summable on the circle, and dµ is an m ×mmatrix valued measure whose entries are singular with respect to the Lebesguemeasure. Note that since dλ is nonnegative, so are (dλ)τ , Ψ and dµ. Substituting(6.3) into (6.1) and matching absolutely continuous and singular measures, yieldsthe relations

tr

{∂Γ∂z`

( · , f )Ψ} = eiθϕ`(6.4)

and

tr

{∂Γ∂z`

( · , f )dµ}= 0.(6.5)


Next, by [10, (20.37)] (corrected–by transposing one of the factors–and) adaptedto the present setting, we have

(6.6)∫ 2π

0tr{[γIm − Γ(eiθ, f (eiθ))](dλ)τ} = 0

and hence, upon invoking the decomposition (6.3) and taking advantage of thefact that γIm − Γ( · , f ), Ψ and dµ are all positive semidefinite, we obtain

(6.7)∫ 2π

0tr{[γIm − Γ(eiθ, f (eiθ))]Ψ(eiθ)}dθ = 0

and

(6.8)∫ 2π

0tr{[γIm − Γ(eiθ, f (eiθ))]}dµ = 0.

But (6.5) and (6.8) imply that Vf (eiθ, dµ) = 0 and hence, by assumption (4),dµ = 0. On the other hand, formula (6.7) implies that

tr{[γIm − Γ(eiθ, f (eiθ))]Ψ(eiθ)} = 0

a.e. on the circle. Therefore, since both of the factors in the product are positivesemidefinite, the full matrix

[γIm − Γ(eiθ, f (eiθ))]Ψ(eiθ) = 0

a.e. on the circle. This is conclusion (1.19a). Condition (1.19c) drops out easilyfrom formulas (6.2) and (6.3) and the vanishing of dµ.

Finally, (6.4) implies that

tr

{e−iθ

∂Γ∂zj

( · , f )Ψ} ∈ H1 ∩ L2, j = 1, . . . , N.

Therefore, by the Smirnov maximum principle,

tr

{e−iθ

∂Γ∂zj

( · , f )Ψ} ∈ H2, j = 1, . . . , N.

But this is easily seen to be equivalent to condition (1.19b) ❐

Proof of Lemma 1.3. If (γIm − Γ( · , f ))GTG = 0, then

(γIm − Γ( · , f ))GTG(γIm − Γ( · , f )) = 0,


which implies that G(γIm − Γ( · , f )) = 0. Therefore,

(1) ⇒ (2) ⇒ PH2+[G(γ − Γ( · , f ))] = 0.

Conversely, if this last condition holds, then G(γIm − Γ( · , f )) = F , with F ∈H2⊥+ . Thus, GTG(γIm − Γ( · , f )) = GTF and

12π

∫tr{GTG(γI − Γ( · , f ))}dθ = 1

2π

∫tr{GTF}dθ = 〈F,G〉 = 0,

since F ∈ (H2+)⊥ and G ∈ H2+. But the trace of the product of nonnegativematrices is nonnegative. Since the trace of the product integrates to zero, it mustbe zero for almost all θ. Now the product of nonnegative matrices has trace zeroif and only if the product itself is zero. Therefore

GTG(γIm − Γ( · , f )) = 0.

This proves that (1)⇒(3) and serves to complete the proof. ❐

Now we turn to multi-performance MOPT. In the setting of the multi per-formance MOPT that was introduced in Section 5, it is readily checked that theconditions in (1.19) can be reexpressed in terms of the positive semidefinite diag-onal blocks Ψp ∈ L1

m×m of Ψ and the performance functions Γp, p = 1, . . . , v, asfollows:

There exists a set of positive semidefinite mvf ’s Ψp ∈ L1m×m such that

(6.9)

Ψ1(γI − Γ 1) = · · · = Ψv(γI − Γv) = 0,

tr

{Ψ1 ∂Γ 1

∂zj

}+ · · · + tr

{Ψv ∂Γv∂zj

}∈ eiθH1,

v∑p=1

∫tr(Ψp)dθ = 2π,

Proof of Theorem 5.1. The proof is an immediate consequence of Lemma 1.3and Theorem 1.1, specialized to the H∞ multidisk case. Formula (6.9) translatesthe conclusions of that theorem to the present setting. ❐

Proof of Proposition 1.14. In order to prove Proposition 1.14 we need alemma.

Lemma 6.1. Let Lp and RpT be m × kp matrices of rankkp, for p = 1, . . . ,v and suppose that at least one of the sets {L1, . . . , Lv}, {R1T , . . . , RvT} has linearlyindependent ranges. Then:


(1) The only matrices Bp ∈ Ckp×kp , p = 1, . . . , v, which satisfy the condition

(6.10)v∑p=1

LpBpRp = 0

are the matrices Bp = 0 for p = 1, . . . , v;(2) If

∑vp=1 kp =m, then the only matrix C ∈ Cm×m that meets the conditions

RpCT = 0 and LpTC = 0 for p = 1, . . . , v,

is the matrix C = 0.

We remark that the formulation of part (2) of this lemma is a little deceptive.At first glance it appears that the conditions RpCT = 0 and LpT C = 0 are reen-forcing each other. In fact they are invoked independently, according to which ofthe matrices R or L (that are defined below in the proof ) is invertible.

Proof.Proof of (1). Let

L =[L1 · · · Lv

], RT =

[R1T · · · RpT

]and B = diag{B1, . . . , Bv}.

Then the condition (6.10) is the same as to say that LBR = 0. Now, if the set Lp,p = 1, . . . , v, has linearly independent ranges, then L is left invertible. ThereforeBR = 0, or, what is the same, BpRp = 0, for p = 1, . . . , v. But this in turnimplies that Bp = 0, since the presumed maximal rank condition implies that theRp are all right invertible. This completes the proof of (1), when the Lp, p = 1,. . . , v have linearly independent ranges.

If the RpT , p = 1, . . . , v, have linearly independent ranges, then R is rightinvertible and hence the condition LBR = 0 leads to LB = 0, i.e., LpBp = 0 forp = 1, . . . , v, which again implies that Bp = 0 for all p.

Proof of (2). If the Lp, p = 1, . . . , v, have linearly independent ranges, thenthe assumption

∑vp=1 rankLp =m implies that L is an invertible matrix, and the

condition LpT C = 0, p = 1, . . . , v, implies that LTC = 0, which clearly forcesC = 0. On the other hand, if the RpT have linearly independent ranges, then theassumption

∑vp=1 rankRpT = m implies that RT is an invertible matrix, and the

condition RpCT = 0, p = 1, . . . , v implies that RCT = 0. Therefore, C = 0 inthis case also. ❐

Proof of Proposition 1.14. Since Ψp(eiθ) ≥ 0, the constraints on B in Defini-tion 1.7 are also valid if Ψp is replaced by (Ψp)1/2. They imply that at each pointeiθ ∈ T, B maps the orthogonal complement of

range{Ψ1(eiθ)} + · · · + range{Ψv(eiθ)} = range{Ψ1(eiθ)+ · · · + Ψv(eiθ)}


into the orthogonal complement of

range{(K1 − f)Ψ1(K1 − f)T + · · · + (Kv − f)Ψv(Kv − f)T}(eiθ).These spaces have dimension µD(θ) and µR(θ), respectively, which serves to provethe first formula in the proposition.

The independence of µpnull(θ) from θ is true because rank Ψp(eiθ) is inde-pendent of θ for almost all θ, since Ψp has the analytic factor Gp.

Next, general position implies that

µD(θ) =m−v∑p=1

rankΨp(eiθ)and

µR(θ) =m−v∑p=1

rank(Kp(eiθ)− f(eiθ))Ψp(eiθ)(Kp(eiθ)− f(eiθ))T=m−

v∑p=1

rankΨp(eiθ).Thus general position implies the second formula.

That µpnull(θ) = 0 = µdnull(θ) follows immediately from part (2) and part(1) of Lemma 6.1, respectively. ❐

REFERENCES

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[2] A. BOTTCHER & B. SILBERMANN, Analysis of Toeplitz Operators, Springer Verlag, Berlin,1990.

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Regional Conference Series in Mathematics, American Mathematical Society, Providence, 1987.[9] J.W. Helton & O. Merino, Conditions for Optimality over H∞, SIAM J. Cont. Opt. 31 (1993).


[10] , Classical Control using H∞ Methods: Theory, Optimization and Design, SIAM, Philadel-phia, 1998.

[11] J.W. HELTON, O. MERINO & T. WALKER, Algorithms for Optimizing over Analytic Func-tions, Indiana Univ. Math. J. 42 (1993), 839-874.

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problems, SIAM Review, 19 (1977) 634-662.[18] L. VANDENBERGHE & S. BOYD, Semidefinite Programming, SIAM Review, 38 (1996), 49-95.[19] S. WRIGHT, Primal Dual Optimization, SIAM, 1998.[20] Y. ZHANG, R.A. TAPIA & J.E. DENNIS, On the super linear and quadratic primal dual interior

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HARRY DYM:The Weizmann InstituteRehovot 76100, Israel.E-MAIL: [email protected]. WILLIAM HELTON:University of California at San DiegoLa Jolla, CA 92093, U. S. A.E-MAIL: [email protected] MERINO:University of Rhode IslandKingston, RI 02881, U. S. A.E-MAIL: [email protected]

KEY WORDS AND PHRASES: H∞ optimization, H∞ control, optimization over analytic functions,semidefinite programming, primal-dual optimization.Received : September 28th, 2001.

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