Exact Differentiable Exterior Penalty for Linear Programming

Olvi MangasarianUW Madison & UCSD La Jolla

Edward WildUW Madison

April 21, 2023

Preliminaries

• Exterior penalty functions in linear and nonlinear programming

– Exact (penalty parameter remains finite)

• Nondifferentiable

– Asymptotic (penalty parameter approaches infinity)

• Differentiable

• Are there exact exterior penalty functions that are differentiable?

– Yes for linear programs• Which is the topic of this talk

Outline

• Sufficient exactness condition for dual exterior penalty function

• Exact primal solution computation from inexact dual exterior penalty function

• Independence of dual penalty function on penalty parameter

• Generalized Newton algorithm & its convergence

• DLE: Direct Linear Equation algorithm & its convergence

• Computational results

• Conclusion & outlook

The Primal & Dual Linear Programs

Primal linear program

Dual linear program

The Dual Exterior Penalty Problem

Divide by 2 and let:

Penalty problem becomes:

Exact Primal Solution Computation

Any solution of the dual penalty problem:

generates an exact solution y of the primal LP:

for sufficiently large but finite as follows:

In addition this solution minimizes:

over the solution set of the primal LP.Ref: Journal of Machine Learning Research 2006

Optimality Condition for Dual Exterior Penalty Problem & Exact Primal LP Solution

A nasc for solving the dual penalty problem:

is:

where P2 Rm£ m is a diagonal matrix of ones and zeros defined as follows:

Solving for u gives:

which gives the following exact primal solution,

Sufficient Condition for Penalty Parameter

• Note that iny = B0((BB0+P)\b) + (B0((BB0+ P) \ (Bd)) - d)

y depends on only through– The implicit dependence of P on u– The explicit dependence on above

• Thus, if is sufficiently large to ensure y is an exact solution of the linear program, then– P (i.e., the active constraint set) does not change with

increasing – B0((BB0+ P) \ (Bd)) - d = 0

Assumed to hold

Ensured computationally

Generalized Newton Algorithm

• Solve the unconstrained problem f(u) = -b0u + ½(||B0u - d||2 + ||(-u)+||2) using a generalized Newton method

• Ordinary Newton method requires gradient and Hessian to compute the Newton direction -(r2f(u))-1rf(u), but f is not twice differentiable

• Instead of ordinary Hessian, we use the generalized Hessian, ∂2f(u) and the generalized Newton direction (∂2f(u))-1rf(u)– rf(u) = -b + B(B0u - d) - (-u)+

– ∂2f(u) = BB0 + diag(sign((-u)+))

Generalized Newton Algorithm (JMLR 2006)minimize f(u) = -b0 u + ½(||B0u - d||2 + ||(-u)+||2)

1) ui + 1 = ui + i ti

• ti = i(∂2f(ui))-1rf(ui) (generalized Newton direction)

• i = max {1, ½, ¼, …} s.t. f(ui) - f(ui + i ti) ¸ -i ¼ rf(ui)0 ti (Armijo stepsize)

2) Stop if ||rf(ui)|| · tol & ||B0((BB0+Pi) \ (Bd)) - d|| · tol

• Pi = diag(sign((-ui)+))

3) If i = imax then ! 10, imax ! 2 ¢ imax

4) i ! i + 1 and go to (1)

Generalized Newton Algorithm Convergence

• Assume tol = 0• Assume B0((BB0+ P) \ (Bd)) - d = 0 implies that is large

enough that an exact solution to the primal is obtained• Then either

– The Generalized Newton Algorithm terminates at ui such that y = B0ui - d is an exact solution to the primal, or

– For any accumulation point ū of the sequence of iterates {ui}, y = B0ū -d is an exact solution to the primal

• Exactness condition is incorporated as a termination criterion

Direct Linear Equation Algorithm

• f(u) = -b0u + ½(||B0u - d||2 + ||(-u)+||2)

• rf(u) = -b + B(B0u - d) - (-u)+ = -b + B(B0u - d) + Pu

• rf(u) = 0 , u = (BB0 + P)-1(Bd + b)• Successively solve rf(u) = 0 for updated

values of the diagonal matrix P = diag(sign((-u)+))

Direct Linear Equation Algorithmminimize f(u) = -b0u + ½(||B0u - d||2 + ||(-u)+||2)

1) Pi = diag(sign((-ui)+))2) ui+1 = (BB0 + Pi) \ (b + Bd)

3) ui+1 ! ui + i (ui+1 - ui)

• i is the Armijo stepsize

4) Stop if ||ui+1 - ui|| · tol & ||B0((BB0+Pi) \ (Bd)) - d|| · tol

5) If i = imax then ! 10, imax ! 2 ¢ imax

6) i ! i + 1 and go to (1)

Direct Linear Equation Algorithm Convergence

• Assume tol = 0• Assume B0((BB0+ P) \ (Bd)) - d = 0 implies that is

large enough that an exact solution to the primal is obtained, and that each matrix in the sequence {BB0+ Pi} is nonsingular

• Then either– The Direct Linear Equation Algorithm terminates at ui such

that y = B0ui - d is an exact solution to the primal, or– For any accumulation point ū of the sequence of iterates {ui},

y = B0ū -d is an exact solution to the primal• Exactness condition is incorporated as a termination

criterion

Solving Primal LPs with More Constraints than Variables

• Difficulty: factoring BB0 • Solution: get exact solution to the dual which requires

factoring a smaller matrix the size of B0B• Given an exact solution of the dual, find the exact solution

of the primal by solving

where B1 and B2 correspond to u1 > 0 and u2 = 0– Requires factoring matrices only of size B0B

Primal exterior penalty problem

The Primal & Dual Linear Programs

Dual linear program

Primal linear program

For sufficiently large , u = (-By + b)+ is an exact solution of the dual linear program

Furthermore, this solution minimizes ||u||2 over the solution set of the dual linear program

Optimality Condition for the Primal Exterior Penalty Problem & Exact Dual LP Solution

A nasc for solving the primal penalty problem:

is:

where Q2 R` £ ` is a diagonal matrix of ones and zeros defined as follows:

Solving for y gives: y = (B0QB) \ ( B0Qb - d)

which gives the following exact dual solution, u = (-By + b)+,

u = (B((B0QB) \ d) - (B((B0QB) \ (B0 Qb)) - b))+

Sufficient Condition for Penalty Parameter

• Note that in

u = (B((B0QB) \ d) - (B((B0QB) \ (B0 Qb)) - b))+

u depends on only through– Q, which depends on through y and – The explicit dependence on above

• Thus, is sufficiently large to ensure u is an exact solution of the linear program if– Q does not change with increasing – diag(sign(u))(B((B0 QB) \ (B0 Qb)) - b) = 0

The subgradient with respect to

Computational Details

• Cholesky factorization used for both methods– Ensure factorizability by adding a small

multiple of the identity matrix– For example, BB0 + P + I for some small – Other approaches left to future work

• Start with = 100 for both methods– Newton method: occasionally increased to 1000– Direct method: not increased in our examples

Computational Results

• When B0((BB0+P) \ (Bd)) - d = 0, optimal solution obtained– Tested on randomly generated linear programs– We know the optimal objective values– This condition is used as a stopping criterion– Relative difference from the true objective value and

maximum constraint violation less then 1e-3, and often smaller than 1e-6

• B0((BB0+ P) \ (Bd)) - d = 0 satisfied efficiently– Our algorithms are compared against the commercial

LP package CPLEX 9.0 (simplex and barrier methods)– Our algorithms are implemented using MATLAB 7.3

Running Time Versus Linear Program SizeProblems with the Same Number of Variables and Constraints

Number of variables (= number of constraints)

Ave

rage

sec

onds

to s

olut

ion

Average Seconds to Solve 10 Random Linear Programs with 100 Variables and

Increasing Numbers of Constraints

Constraints CPLEX Newton LP DLE

1,000 0.1 0.1 0.1

10,000 0.3 1.0 0.5

100,000 3.0 13.0 5.5

1,000,000 44.8 173.5 70.9

Variables CPLEX Newton LP DLE

1,000 0.02 0.04 0.03

10,000 0.05 0.20 0.90

100,000 0.84 3.22 0.91

1,000,000 17.9 29.1 9.3

Average Seconds to Solve 10 Random Linear Programs with 100 Constraints and Increasing

Numbers of Variables

Conclusion

• Presented sufficient conditions for obtaining an exact solution to a primal linear program from a classical dual exterior penalty function

• Precise termination condition given for– Newton algorithm for linear programming (JMLR

2006)– Direct method based on solving the optimality

condition of the convex penalty function

• Algorithms efficiently obtain optimal solutions using the precise termination condition

Future Work

• Deal with larger linear programs

• Application to real-world linear programs

• Direct methods for other optimization problems, e.g. linear complementarity problems

• Further improvements to performance and robustness

Links to Talk & Papers

• http://www.cs.wisc.edu/~olvi

• http://www.cs.wisc.edu/~wildt