Programme Gaspard MONGE pour l’Optimisation et la ... · Programme Gaspard MONGE pour l’Optimisation et la Recherche Op erationnelle (PGMO) "Nothing in the world takes place without

Programme Gaspard MONGE pour l’Optimisationet la Recherche Operationnelle (PGMO)

”Nothing in the world takes place without optimization, and there is nodoubt that all aspects of the world that have a rational basis can be

explained by optimization methods”

L.Euler (1744)

JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 20

Programme Gaspard MONGE pour l’Optimisationet la Recherche Operationnelle (PGMO)

”Nothing in the world takes place without optimization, and there is nodoubt that all aspects of the world that have a rational basis can be

explained by optimization methods”

L.Euler (1744)


Some anniversaries ...

Henri Poincare

... died in 1912.


Some anniversaries ...

Henri Poincare

... died in 1912.


Alan Turing Turing Memorial

... born in 1912.


Leonid Kantorovich

... born in 1912.


Paul Sabatier, Nobel prize winner in 1912

Paul Sabatier Fermat


1962-1963: Birth of modern convex analysis and optimization.





A variational approach of the rank function.

Jean-Baptiste HIRIART-URRUTYand

Hai Yen LE

Institut de Mathematiques de ToulouseUniversite Paul Sabatier

Septembre 2012


Contents

1 The rank in linear algebra or matricial calculus

2 The rank from the topological and semi-algebraic viewpoint

3 Best approximation in terms of rank

4 Global minimization of the rank

5 Relaxed form of the rank function

6 Surrogates of the rank function

7 Regularization- approximation of the rank function

8 The generalized subdifferentials of the rank function

9 Further notions related to the rank


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The rank in linear algebra or matricial calculus

Contents












Let p := min(m, n), let

rank : Mm,n(R) −→ {0, 1, . . . , p}A 7−→ rank A.

The properties of the rank function are well-known in the context of linearalgebra or matricial calculus. Any book on these subjects presents them;there even are compilations of them. We recall here the most basic ones:

1 rank A = rank AT ; rank A = rank (AAT ) = rank (ATA).

2 If the product AB can be done,

rank (AB) ≤ min(rank A, rank B) (Sylvester inequality).

3 rank A = 0 if and only if A = 0; rank (cA) = rank A for c 6= 0.

4 |rank A− rank B| ≤ rank (A + B) ≤ rank A + rank B.



Let p := min(m, n), let

rank : Mm,n(R) −→ {0, 1, . . . , p}A 7−→ rank A.

The properties of the rank function are well-known in the context of linearalgebra or matricial calculus. Any book on these subjects presents them;there even are compilations of them. We recall here the most basic ones:

1 rank A = rank AT ; rank A = rank (AAT ) = rank (ATA).

2 If the product AB can be done,

rank (AB) ≤ min(rank A, rank B) (Sylvester inequality).

3 rank A = 0 if and only if A = 0; rank (cA) = rank A for c 6= 0.

4 |rank A− rank B| ≤ rank (A + B) ≤ rank A + rank B.



Theorem

1 Let A ∈Mn(R) be non-null diagonalizable, with all real eigenvalues.Then

rank A ≥ (tr A)2

tr (A2),

where tr B denotes the trace of the matrix B.

2 Let A ∈ Sn(R) be non-null and positive definite. Then

tr A

λmax(A)≤ rank A ≤ tr A

λmin(A),

where λmax(A) (resp. λmin(A)) stands for the maximal (resp.minimal) eigenvalue of A.



Theorem

1 Let A ∈Mn(R) be non-null diagonalizable, with all real eigenvalues.Then

rank A ≥ (tr A)2

tr (A2),

where tr B denotes the trace of the matrix B.

2 Let A ∈ Sn(R) be non-null and positive definite. Then

tr A

λmax(A)≤ rank A ≤ tr A

λmin(A),

where λmax(A) (resp. λmin(A)) stands for the maximal (resp.minimal) eigenvalue of A.


The rank from the topological and semi-algebraic viewpoint

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The only (useful) topological property of the rank function is that it islower-semicontinuous: if Aν → A in Mm,n(R), then

lim infν→+∞

rank Aν ≥ rank A.

For k ∈ {0, 1, . . . , p}, consider now the following two subsets of Mm,n(R):

Sk := {A ∈Mm,n(R)| rank A ≤ k},

Σk := {A ∈Mm,n(R)| rank A = k}.

Theorem

1 Σp is an open dense subset of Sp =Mm,n(R).

2 If k < p, the interior of Σk is empty and its closure is Sk .




lim infν→+∞





Theorem






lim infν→+∞





Theorem





Jean-Baptiste Hiriart-Urruty

Bases, outils et principes pour l'analyse variationnelle


Bases, tools and principles for variational analysis

Mathematics

L’étude mathématique des problèmes d’optimisation, ou de ceux dits variationnels de manière générale (c’est-à-dire, « toute situation où il y a quelque chose à minimiser sous des contraintes »), requiert en préalable qu’on en maîtrise les bases, les outils fondamentaux et quelques principes. Le présent ouvrage est un cours répondant en partie à cette demande, il est principalement destiné à des étudiants de Master en formation, et restreint à l’essentiel. Sont abordés successivement : La semicontinuité inférieure, les topologies faibles, les résultats fondamentaux d’ existence en optimisation ; Les conditions d’optimalité approchée ; Des développements sur la projection sur un convexe fermé, notamment sur un cône convexe fermé ; L’analyse convexe dans son rôle opératoire ; Quelques schémas de dualisation dans des problèmes d’optimisation non convexe structurés ; Une introduction aux sous-différentiels généralisés de fonctions non différentiables.

The mathematical study of optimization problems, or of the so-called variational ones (that is to say, “every situation where there is something to be minimized under constraints”), requires first having a solid understanding of bases, fundamental tools and principles. The present book consists of lecture notes that partly meet these requirements; it is mainly intended for students at the Master level, and focuses on the essentials. The following points are addressed: Lower semi-continuity, weak topologies, and fundamental existence results in optimization; Conditions for approximate optimality; Developments in the projection on a closed convex set, more specifically on a closed convex cone: Convex analysis in its operational role; Some duality schemes for specially structured non-convex optimization problems; and An introduction to the generalized sub-differentials of non-smooth functions.

ISBN 978-3-642-30734-8

Hiriart-U

rruty

Mathématiques et Applications 70



M&A70


Mathématiques et Applications 70


Best approximation in terms of rank

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The best approximation problem that we consider in this section is asfollows: Given A ∈Mm,n(R) of rank r ,

(Ak)

{Minimize ‖A−M‖M ∈ Sk

The problem of course depends on the choice of ‖.‖; it is solved when ‖.‖is either the Frobenius-Schur norm or the spectral norm.



Theorem (Eckart and Young)

Let A ∈Mm,n(R) and A = UΣAV be a SVD of A. Choose ‖.‖ as either‖.‖F or ‖.‖sp. Then

Ak := UDkV ,

(where Dk is obtained from D by keeping σ1, . . . , σk and putting 0 in theplace of σk+1, . . . , σr ) is a solution of the best approximation problem(Ak). For the Frobenius-Schur norm case, Ak is the unique solution in(Ak) when σk > σk+1.The optimal values in (Ak) are as following:

minM∈Sk

‖A−M‖F =

√√√√ r∑i=k+1

σ2i ;

minM∈Sk

‖A−M‖sp = σk+1.


Global minimization of the rank

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Global minimization

(P)

{Minimize rank(A)

A ∈ C

Example for C. Aij given for (i , j) ∈ K ⊂ {1, . . . ,m} × {1, . . . , n}.

Theorem (”Esclarecimiento de Valparaiso”)

Every admissible point in (P) is a local minimizer.

JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -1


Global minimization

(P)

{Minimize rank(A)

A ∈ C

Example for C. Aij given for (i , j) ∈ K ⊂ {1, . . . ,m} × {1, . . . , n}.

Theorem (”Esclarecimiento de Valparaiso”)

Every admissible point in (P) is a local minimizer.


Relaxed form of the rank function












For R > 0, we define

A ∈Mm,n(R) 7−→ rankR(A) :=

{rank of A if ‖A‖sp ≤ R;

+∞ otherwise.

Theorem (Fazel, 2002)

We have:

A ∈Mm,n(R) 7−→ co (rankR)(A) :=

{1R ‖A‖∗ if ‖A‖sp ≤ R;+∞ otherwise.



For R > 0, we define

A ∈Mm,n(R) 7−→ rankR(A) :=

{rank of A if ‖A‖sp ≤ R;

+∞ otherwise.

Theorem (Fazel, 2002)

We have:

A ∈Mm,n(R) 7−→ co (rankR)(A) :=

{1R ‖A‖∗ if ‖A‖sp ≤ R;+∞ otherwise.



Proof 1(Fazel)

Calculate the Legendre-Fenchel conjugate φ∗ of φ := rankR .

Then, compute the biconjugate φ∗∗ of φ.



Proof 1(Fazel)

Calculate the Legendre-Fenchel conjugate φ∗ of φ := rankR .

Then, compute the biconjugate φ∗∗ of φ.



Proof 2

Here, one begins by convexifying the restricted counting function (on Rp),and then one applies fine results by A.Lewis on how to get properties onrankR from properties on cR .

Theorem

We have

x ∈ Rp 7−→ co(cR)(x) :=

{1R ‖x‖1 if ‖x‖∞ ≤ R;+∞ otherwise.

(a) The counting function (b) The l1 norm



Proof 2

Here, one begins by convexifying the restricted counting function (on Rp),and then one applies fine results by A.Lewis on how to get properties onrankR from properties on cR .

Theorem

We have

x ∈ Rp 7−→ co(cR)(x) :=

{1R ‖x‖1 if ‖x‖∞ ≤ R;+∞ otherwise.

(c) The counting function (d) The l1 norm



Proof 3

Here we look at the problem with ”geometrical glasses”: instead ofconvexifying functions directly, we firstly convexify (appropriate) sets ofmatrices and, then, get convex hulls (or quasiconvex hulls) of functions asby-products.For k ∈ {0, 1, . . . , p} and R ≥ 0, let

SRk := Sk ∩ {A ∈Mm,n(R)| ‖A‖sp ≤ R}

= {A ∈Mm,n(R)| rank A ≤ k and ‖A‖sp ≤ R}.

Theorem

We have

co SRk = {A ∈Mm,n(R)| ‖A‖sp ≤ R and ‖A‖∗ ≤ Rk}.

Ref. J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of bounded rank.

Applications to quasiconvexification and convexification of the rank function,

Optimization Letters 6(5): 841-849 (2012).



Proof 3




Theorem

We have







Proof 3




Theorem

We have







Proof 3




Theorem

We have







Proof 3

Theorem

We have:

A ∈Mm,n(R) 7−→ (rankR)q(A) =

{d 1R ‖A‖∗e if ‖A‖sp ≤ R,+∞ otherwise,

where dae stands for the smallest integer which is larger than a.


Surrogates of the rank function

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A proposal of a continuous surrogate of the rank is as follows:

A ∈Mm,n(R) 7−→ gsf (A) :=

{‖A‖2∗

‖A‖2F

if A 6= 0,

0 if A = 0.

Theorem (Flores, Ph.D Thesis 2011)

We have:

(G1) gsf (cA) = gsf (A) for c 6= 0.

(G2) 1 ≤ gsf (A) ≤ rank A for A 6= 0.

(G3) If all the non-zero singular values of A are equal, thengsf (A) = rank A.

(G4) rank A = 1 if and only if gsf (A) = 1.



A proposal of a continuous surrogate of the rank is as follows:

A ∈Mm,n(R) 7−→ gsf (A) :=

{‖A‖2∗

‖A‖2F

if A 6= 0,

0 if A = 0.

Theorem (Flores, Ph.D Thesis 2011)

We have:

(G1) gsf (cA) = gsf (A) for c 6= 0.

(G2) 1 ≤ gsf (A) ≤ rank A for A 6= 0.

(G3) If all the non-zero singular values of A are equal, thengsf (A) = rank A.

(G4) rank A = 1 if and only if gsf (A) = 1.



Theorem (Lokam, 2001)

For any non-null B ∈Mm,n(R), define

A ∈Mm,n(R 7−→ gB(A) :=

|〈〈A,B〉〉|‖A‖sp‖B‖sp

if A 6= 0,

0 if A = 0.

ThengB(A) ≤ rank A for all A ∈Mm,n(R).


Regularization- approximation of the rank function

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Smoothed versions

The underlying idea is as follows: Since

rank A = c[σ1(A), . . . , σp(A)]

=

p∑i=1

θ[σi (A)],

where θ(x) = 1 if x 6= 0, θ(0) = 0 (θ is the counting function on R), wehave to design some smooth approximation of the θ function.



Smoothed versions

A first example was proposed by Hiriart-Urruty (2009), it is as following:For ε > 0, let θε be defined as

x ∈ R 7−→ θε(x) := 1− e−x2/ε.

The resulting approximation of the rank function is

A ∈Mm,n(R) 7−→ Rε(A) :=

p∑i=1

[1− e−σ2i (A)/ε].

An alternate expression of the Rε function is

Rε(A) = p − tr(e−ATA/ε).



Smoothed versions


x ∈ R 7−→ θε(x) := 1− e−x2/ε.


A ∈Mm,n(R) 7−→ Rε(A) :=

p∑i=1

[1− e−σ2i (A)/ε].





Smoothed versions


x ∈ R 7−→ θε(x) := 1− e−x2/ε.


A ∈Mm,n(R) 7−→ Rε(A) :=

p∑i=1

[1− e−σ2i (A)/ε].





Smoothed versions

Theorem

We have

(i) Rε(A) ≤ rank A for all ε > 0.

(ii) The sequence of functions (Rε)ε>0 increases when ε decreases, andRε(A)→ rank A for all A when ε→ 0.

(iii) If A 6= 0 and r = rank A,

rank A− Rε(A) ≤ ε

r∑i=1

1

σ2i (A)

,

≤ ε2r∑

i=1

1

σ4i (A)

.



Smoothed versions

Theorem

We have





r∑i=1

1

σ2i (A)

,

≤ ε2r∑

i=1

1

σ4i (A)

.



Smoothed versions

Theorem

We have





r∑i=1

1

σ2i (A)

,

≤ ε2r∑

i=1

1

σ4i (A)

.



Smoothed versions

Another proposal for approximating the rank function, a quite recent one,is due to Zhao. It consists of using, for qll ε > 0, the following evenapproximation of the θ function:

x ∈ R 7−→ τε(x) :=x2

x2 + ε.


A ∈Mm,n(R) 7−→ Zε(A) :=

p∑i=1

σ2i (A)

σ2i (A) + ε

.

Alternate expressions of the Zε function are:

Zε(A) = tr[A(ATA + εIn)−1AT ]

= n − εtr(ATA + εIn)−1.

Ref. Y.-B.Zhao, An approximation theory of matrix rank minimization and its

application to quadratic equations, Linear Algebra and Its Applications (2012), 77-93.



Smoothed versions


x ∈ R 7−→ τε(x) :=x2

x2 + ε.


A ∈Mm,n(R) 7−→ Zε(A) :=

p∑i=1

σ2i (A)

σ2i (A) + ε

.








Smoothed versions


x ∈ R 7−→ τε(x) :=x2

x2 + ε.


A ∈Mm,n(R) 7−→ Zε(A) :=

p∑i=1

σ2i (A)

σ2i (A) + ε

.








Smoothed versions

Theorem (Zhao, 2012)

We have

(i) Zε(A) ≤ rank A for all ε > 0.

(ii) The sequence of functions (Zε)ε>0 increases when ε decreases andZε(A)→ rank A for all A when ε→ 0.


rank A− Zε(A) =r∑

i=1

ε

σ2i (A) + ε

≤ εr∑

i=1

1

σ2i (A)

.



Smoothed versions


We have





i=1

ε

σ2i (A) + ε

≤ εr∑

i=1

1

σ2i (A)

.



Smoothed versions


We have





i=1

ε

σ2i (A) + ε

≤ εr∑

i=1

1

σ2i (A)

.



Moreau-Yosida approximation (or Moreau envelope)

By definition,

(rankR)λ(A) = infB ∈ Mm,n(R)‖B‖sp≤R

{rank B +

1

2λ‖A− B‖2

F

}

The limiting case, i.e. with R = +∞, is

(rank)λ(A) = infB∈Mm,n(R)

{rank B +

1

2λ‖A− B‖2

F

}




Theorem

We have, for all A ∈Mm,n(R) with rank r ≥ 1:

(rank)λ(A) =1

2λ‖A‖2

F −1

2λ

r∑i=1

[σ2i (A)− 2λ]+.

One element in Proxλ(rank)(A), is provided by B := UΣBV , where

• U and V are orthogonal matrices such that A = UΣAV , withΣA = diagm,n[σi (A), . . . , σr (A), 0, . . . , 0] (a singular valuedecomposition of A with σ1(A) ≥ · · · ≥ σr (A) > 0;)

•

σi (B) =

σi (A) if σi (A) >

√2λ

0 or σi (A) if σi (A) =√

2λ

0 if σi (A) <√

2λ.




Theorem

We have, for all A ∈Mm,n(R) with rank r ≥ 1:

(rank)λ(A) =1

2λ‖A‖2

F −1

2λ

r∑i=1

[σ2i (A)− 2λ]+.

One element in Proxλ(rank)(A), is provided by B := UΣBV , where

• U and V are orthogonal matrices such that A = UΣAV , withΣA = diagm,n[σi (A), . . . , σr (A), 0, . . . , 0] (a singular valuedecomposition of A with σ1(A) ≥ · · · ≥ σr (A) > 0;)

•

σi (B) =

σi (A) if σi (A) >

√2λ

0 or σi (A) if σi (A) =√

2λ

0 if σi (A) <√

2λ.



Moreau-Yosida approximations (or Moreau’s envelope)

Theorem

We have, for all A ∈Mm,n(R) of rank r ≥ 1:

(rankR)λ(A) = 12λ‖A‖

2F −

12λ

∑ri=1{σ2

i (A)− [(σi (A)− R)+]2 − 2λ}+.

One element in Proxλ(rankR)(A), is provided by B := UΣBV withΣB = diagm,n[σ1(B), . . . , σp(B)], where

• If√

2λ ≥ R, σi (B) :=

R if σi (A) > 2λ+λ2

2λ ,

0 or R if σi (A) = 2λ+λ2

2λ ,

0 if σi (A) < 2λ+λ2

2λ .

• If√

2λ < R, σi (B) :=

R if σi (A) > R,

σi (A) if√

2λ < σi (A) ≤ R,

0 or σi (A) if√

2λ = σi (A),

0 if σi (A) <√

2λ.



Moreau-Yosida approximations (or Moreau’s envelope)

Theorem

We have, for all A ∈Mm,n(R) of rank r ≥ 1:

(rankR)λ(A) = 12λ‖A‖

2F −

12λ

∑ri=1{σ2

i (A)− [(σi (A)− R)+]2 − 2λ}+.

One element in Proxλ(rankR)(A), is provided by B := UΣBV withΣB = diagm,n[σ1(B), . . . , σp(B)], where

• If√

2λ ≥ R, σi (B) :=

R if σi (A) > 2λ+λ2

2λ ,

0 or R if σi (A) = 2λ+λ2

2λ ,

0 if σi (A) < 2λ+λ2

2λ .

• If√

2λ < R, σi (B) :=

R if σi (A) > R,

σi (A) if√

2λ < σi (A) ≤ R,

0 or σi (A) if√

2λ = σi (A),

0 if σi (A) <√

2λ.



Moreau-Yosida approximation



Moreau-Yosida approximation


The generalized subdifferentials of the rank function

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Which one?All the generalized subdifferentials of the rank function coincide. We note∂(rank) for the common subdifferential. For A ∈Mm,n(R), ∂(rank)(A) isconstructed as follows:

Consider the matrices U ∈ O(m) and V ∈ O(n) such that

U.Diagm,n(σ(A)).V = A

(in other words, we collect all the orthogonal matrices U and V whichgive a singular value decomposition of A).

Consider the ”diagonal” matrices Diagm,n(x∗), where x∗ ∈ Rp is suchthat x∗i = 0 for all i = 1, . . . , r (recall that r = rank A).

Then, collect all the matrices of the form UDiagm,n(x∗)V .




In a single formula,

∂(rank)(A)= {UDiag(x∗)V | U ∈ O(m),V ∈ O(n) such that UΣAV = A

and x∗i = 0 for all i = 1, . . . , r}.




Alternate expression of ∂(rank)(A)

∂(rank)(A) = N(AT )⊗ N(A)

={∑

i ,j aijαiβTj | (αi ) is a basis of N(AT )

(βj) is a basis of N(A)} .

withN(AT ) = {u ∈ Rm| ATu = 0}

N(A) = {v ∈ Rn| Av = 0}.

Dimension of ∂(rank)(A)

∂(rank)(A) is a vector space of dimension (m− r)(n− r) with r = rank A.




Alternate expression of ∂(rank)(A)

∂(rank)(A) = N(AT )⊗ N(A)

={∑

i ,j aijαiβTj | (αi ) is a basis of N(AT )

(βj) is a basis of N(A)} .

withN(AT ) = {u ∈ Rm| ATu = 0}

N(A) = {v ∈ Rn| Av = 0}.

Dimension of ∂(rank)(A)

∂(rank)(A) is a vector space of dimension (m− r)(n− r) with r = rank A.



Illustration

For m = n = 2, we have

If A = 0 then ∂(rank)(0) =M2,2(R).

If rank A = 2 then ∂(rank)(A) = {0}.If rank A = 1 then ∂(rank)(A) is a vector space of dimension 1.


Further notions related to the rank

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The spark of a matrix

Definition

Given A ∈Mm,n(R), the spark of A is the smallest positive integer k suchthat there exists a set of k columns of A which are linearly dependent.

If A is of full column rank, we should adopt +∞ as for the spark of A.

If one column of A is a zero-column: then spark A = 1.

If A does not contain any zero-column and is not of full column rank,

2 ≤ sparkA ≤ rank A + 1.



The spark of a matrix

Definition

Given A ∈Mm,n(R), the spark of A is the smallest positive integer k suchthat there exists a set of k columns of A which are linearly dependent.

If A is of full column rank, we should adopt +∞ as for the spark of A.

If one column of A is a zero-column: then spark A = 1.

If A does not contain any zero-column and is not of full column rank,

2 ≤ sparkA ≤ rank A + 1.



The matrix rigidity

Definition

The rigidity of A ∈Mm,n(R), denoted by RA(k) is the smallest number ofentries that need to be changed in order to reduce the rank of A below k .

A variational formulation of RA(k) can easily be proposed as a rankminimization problem:

RA(k) = minM∈A+Sk

c(M),

where c(M) denotes the number of nonzero entries in the matrix M andSk = {S ∈Mn(R)| rank S ≤ k}



The matrix rigidity

Definition

The rigidity of A ∈Mm,n(R), denoted by RA(k) is the smallest number ofentries that need to be changed in order to reduce the rank of A below k .

A variational formulation of RA(k) can easily be proposed as a rankminimization problem:

RA(k) = minM∈A+Sk

c(M),

where c(M) denotes the number of nonzero entries in the matrix M andSk = {S ∈Mn(R)| rank S ≤ k}



The cp-rank of matrix

A matrix M ∈ Sn(R) is said to be copositive if

〈Mx , x〉 ≥ 0 for all x ∈ Rn+.

A matrix A ∈ Sn(R) is then called completely positive if

〈〈A,M〉〉 ≥ 0 for all copositive matrices M.

Definition

The smallest number of columns of a matrix B in a factorizationA = BBT of a completely positive matrix A is called the cp-rank of A anddenoted as cp-rank A.








Definition









Definition





For the zero matrix (which indeed lies in the cone CPn(R)), we adoptby convention that its cp-rank is 0.

For matrices A ∈ Sn(R) which are not in CPn(R), we posecp-rank A = +∞.

If one wishes to compare the rank and the cp-rank of a matrix, one gets atthe following inequality:

rank A ≤ cp-rank A,

for all completely positive matrices, hence for all symmetric matrices.




For the zero matrix (which indeed lies in the cone CPn(R)), we adoptby convention that its cp-rank is 0.

For matrices A ∈ Sn(R) which are not in CPn(R), we posecp-rank A = +∞.

If one wishes to compare the rank and the cp-rank of a matrix, one gets atthe following inequality:

rank A ≤ cp-rank A,

for all completely positive matrices, hence for all symmetric matrices.



The cp-rank of a matrix

Theorem

(i) If A and B are completely positive,

cp-rank (A + B) ≤ cp-rank A + cp-rank B.

(ii) If A is completely positive and c > 0,

cp-rank (cA) = cp-rank A.

(iii) The cp-rank is a lower-semicontinuous function: If (Aν)ν is asequence of completely positive matrices converging to A, then A iscompletely positive and

lim infν→+∞

cp-rank Aν ≥ cp-rank A.



Upper bounds for the cp-rank

Theorem (Barioli and Berman, 2003)

(i) For every completely positive matrix of rank k ≥ 2,

cp-rank A ≤ k(k + 1)

2− 1.

(ii) For every integer k ≥ 2, there exists a completely positive matrix A

whose rank is k and whose cp-rank is k(k+1)2 − 1.



Upper bounds for the cp-rank

The DJL Conjecture (1994)

If A is an n × n completely positive matrix, n ≥ 4, then

cp-rank A ≤ n2

4.



The convex relaxed form

For R > 0, consider

A ∈ Sn(R) 7→ cp-rankR(A) :=

{cp-rank A if A is CP and ‖A‖∗ ≤ R;

+∞ otherwise.

Theorem

We have

A ∈ Sn(R) 7→ co(cp-rankR)(A) :=

{1R ‖A‖∗ if A is CP and ‖A‖∗ ≤ R;+∞ otherwise.



References

J.-B.Hiriart-Urruty, When only global optimization matters, J. of GlobalOptimization (DOI: 10.1007/s10898-011-9826-7) (2012).

J.-B.Hiriart-Urruty and J.Malick, A fresh variational analysis look at theworld of the positive semidefinite matrices, J. of Optimization Theory andApplications, Vol 153(3) (2012), 551-577.

J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of boundedrank: applications to the quasiconvexification and convexification of therank function, Optimization Letters 6(5): 841-849 (2012).

H.Y.Le, Confexifying the Counting Function on Rp for Convexifying theRank Function on Mm,n(R), J. of Convex Analysis Vol 19(2) (2012).

H.Y.Le, The generalized subdifferentials of the rank function, Optimization

Letters (2012), DOI: 10.1007/s11590-012-0456-x.




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