Exact Differentiable Exterior Penalty for Linear Programming

1. Exact Differentiable Exterior Penalty for Linear Programming February 12, 2008 Olvi Mangasarian UW Madison & UCSD La Jolla Edward Wild UW Madison TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

2. Dedication • This talk is dedicated to the memory of Herb Keller 1925-2008 • World renowned scholar and numerical analyst • Author of several texts, research monographs, and more than 140 research papers. • Directed the dissertations of 25 PhD students • Oldest bicyclist to negotiate Torrey Pines incline • Splendid and enjoyable colleague and office mate

3. Preliminaries • Exterior penalty functions convert constrained problems to optimization unconstrained optimization problems • Two types of penalty functions • Nondifferentiable penalty functions • Exact: Penalty parameter remains finite • Differentiable penalty functions • Asymptotic: Penalty parameter approaches infinity • Are there exact exterior penalty functions that are differentiable? • Yes for linear programs • Which is the topic of this talk

4. Outline • Sufficient primal LP solution exactness condition based on differentiable dual exterior penalty function • Exact primal LP 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

5. The Primal & Dual Linear Programs Primal linear program Dual linear program

6. The Dual Exterior Penalty Problem where, (-u)+=max{-u,0} Divide by 2 and let: Penalty problem becomes:

7. Exact Primal Solution Computation Any solution of the dual penalty problem: generates anexactsolution 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: OLM,Journal of Machine Learning Research 2006, 1517-1530

8. Optimality Condition for Dual Exterior Penalty Problem & Exact Primal LP Solution A nasc for solving the dual penalty problem: 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,

9. Sufficient Exactness Condition for Penalty Parameter  • Note that in y = 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

10. 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)+))

11. Generalized Newton Algorithm (JMLR 2006)minimize f(u) = -b0 u + ½(||B0u - d||2 + ||(-u)+||2) • 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) • Stop if ||rf(ui)|| · tol & ||B0((BB0+Pi) \ (Bd)) - d|| · tol • Pi = diag(sign((-ui)+)) • If i = imax then ! 10, imax ! 2 ¢ imax • i ! i + 1 and go to (1)

12. 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

13. 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)+))

14. Direct Linear Equation Algorithmminimize f(u) = -b0u + ½(||B0u - d||2 + ||(-u)+||2) • Pi = diag(sign((-ui)+)) • ui+1 = (BB0 + Pi) \ (b + Bd) • ui+1! ui + i (ui+1 - ui) • i is the Armijo stepsize • Stop if ||ui+1 - ui|| · tol & ||B0((BB0+Pi) \ (Bd)) - d|| · tol • If i = imax then ! 10, imax ! 2 ¢ imax • i ! i + 1 and go to (1)

15. 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

16. 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

17. More-Constraints-than-Variables Case Dual LP Primal LP Primal exterior penalty problem

18. Exact Solution of Dual LP from Primal LP Penalty For sufficiently large but finite penalty parameter , a solution y of the primal LP penalty function gives an exact solution u = (-By +  b)+ to the dual LP. Furthermore, this exact dual solution minimizes ||u||2 over the solution set of the dual linear program. From this exact dual solution u an exact solution of the primal LP is found by the unconstrained minimization problem: where B1 and B2 correspond to u1>0 and u2=0 respectively.

19. 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

20. 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

21. Running Time Versus Linear Program SizeProblems with the Same Number of Variables and Constraints Average seconds to solution Number of variables (= number of constraints)

22. Average Seconds to Solve 10 Random Linear Programs with 100 Variables and Increasing Numbers of Constraints

23. Average Seconds to Solve 10 Random Linear Programs with 100 Constraints and Increasing Numbers of Variables

24. 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 differentiable convex penalty function • Algorithms efficiently obtain optimal solutions using the precise termination condition

25. 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

26. Links to Talk & Papers • http://www.cs.wisc.edu/~olvi • http://www.cs.wisc.edu/~wildt

27. 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))+

28. 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 