Crash-Starting the Simplex Method

Transcription

1 Crash-Starting the Simplex Method Ivet Galabova Julian Hall School of Mathematics, University of Edinburgh Optimization Methods and Software December 2017 Ivet Galabova, Julian Hall Crash-Starting Simplex 1/20

2 Overview A linear programming problem is x b Better crash min s.t. c T x Ax = b x 0 x c x Feasibility crash f x 0 Crash x k Crossover x b Simplex x Ivet Galabova, Julian Hall Crash-Starting Simplex 2/20

3 Motivation A good starting basis is essential for the solution of large problems The Idiot crash is a heuristic aiming to improve feasibility Implemented by John Forrest in CLP Seeking primal feasibility prior to primal simplex x 0 Crash x k Crossover x b Simplex x Ivet Galabova, Julian Hall Crash-Starting Simplex 3/20

4 The Idiot crash No documentation but the code John Forrest: so bad that I called it the Idiot algorithm At each iteration minimize a weighted average of objective and infeasibilities Looks like quadratic penalty function approach but... Extra term in function minimized Looks like Augmented Lagrangian (AL) but... Approximate minimization Parameter adjustment Other details Ivet Galabova, Julian Hall Crash-Starting Simplex 4/20

5 Augmented Lagrangian In iteration k x k+1 = arg min x L A = c T x λ T r(x) + µ 2 r(x)t r(x), Residual vector r(x) = b Ax Parameters µ and λ λ i corresponds to the i th row λ converge to the optimal Lagrange multipliers Adjust parameters At start After some iterations s After some iterations l µ 0 = 2 µ l+1 := 10µ l λ 0 = 0 λ s+1 := λ s µr(x s ) Ivet Galabova, Julian Hall Crash-Starting Simplex 5/20

6 Alternating variables approach L A = c T x λ T r(x) + µ 2 r(x)t r(x) Singular Hessian Computationally expensive Alternating variables approach Approximate solution Cheap min x h(x) repeated 100 times to improve accuracy min h(x) x 1 min h(x) x 2 min x n... h(x) Ivet Galabova, Julian Hall Crash-Starting Simplex 6/20

7 The Idiot Crash The Idiot Crash structure Initialize x 0, µ 0, λ 0 = 0 For I 1..nMajor For i 1..nMinor j {1..n} end Choose new µ or λ end min h(x) = c T x λ T r(x) + µ x j 2 r(x)t r(x) Every few major iterations µ is increased Ivet Galabova, Julian Hall Crash-Starting Simplex 7/20

8 Parameters At the end of each major iteration the values of λ or µ are re-adjusted: Start with µ 0, λ 0 = 0, nminor = 2 At the end of each major iteration: Until 10% reduction in primal infeasibility set µ k+1 = 10µ k Set nminor = 105 The first major iteration with a new µ, λ is unchanged The next few major iterations set λ k+1 i = µr i (x k ) Increase µ and repeat λ converges to 0 Ivet Galabova, Julian Hall Crash-Starting Simplex 8/20

9 Idiot nature Not Augmented Lagrangian During initial major iterations µ is smaller so λ ensures reducing primal infeasibility has enough weight min h(x) = c T x λ T r(x) + µ x j 2 r(x)t r(x), Cycles through all columns Random starting point Alternating variables method Early termination of a major iteration is possible Condition on moving average of expected progress Ivet Galabova, Julian Hall Crash-Starting Simplex 9/20

10 Relevant work An Augmented Lagrangian Method for Linear Programming Beale, Hattersley, James (1985) The LP dual active set algorithm Hager (1998) An approximate, efficient LP solver for LP rounding Sridhar, Wright et al (2013) Sparse linear programming via primal and dual augmented coordinate descent Yen, Zhong et al (2015) Ivet Galabova, Julian Hall Crash-Starting Simplex 10/20

11 Performance Generally useless Single variable change limitation Solves Quadratic Assignment Problems (QAPs) Combinatorial optimization problem, special case of facility location Well known for being hard to solve The Idiot crash reduces linearisation solution time significantly Many zero Lagrange multipliers Ivet Galabova, Julian Hall Crash-Starting Simplex 11/20

12 Quadratic Assignment Problems QAP12 Solution time (s) QAP15 Solution time (s) idiot crossover simplex idiot crossover simplex nmajor nmajor Ivet Galabova, Julian Hall Crash-Starting Simplex 12/20

13 Accuracy Residual error 10 4 The Idiot crash on QAPs Residual error 10 1 An approximate, efficient LP solver for LP rounding Sridhar, Wright et al (2013) Residual error 10 3 Sparse linear programming via primal and dual augmented coordinate descent Yen, Zhong et al (2015) Ivet Galabova, Julian Hall Crash-Starting Simplex 13/20

14 Change in objective value Iteration Penalty Idiot AL Penalty method vs Idiot vs Augmented Lagrangian objective value Ivet Galabova, Julian Hall Crash-Starting Simplex 14/20

15 Quadratic Assignment Problems QAP12 Solution time (s) QAP15 Solution time (s) idiot crossover simplex idiot crossover simplex nmajor nmajor Ivet Galabova, Julian Hall Crash-Starting Simplex 15/20

16 Idiot alone min s.t. f P = c T x Ax = b x 0 (P) max f D = b T y s.t. A T y c (D) Idiot applied to (P) Primal feasible Objective f P f Near-optimal, but not provably Idiot applied to (D) Dual feasible Objective f D f Ivet Galabova, Julian Hall Crash-Starting Simplex 16/20

17 Branch & bound Branch & bound tree LP problems at each node Lower bounds on f Allow for pruning Ivet Galabova, Julian Hall Crash-Starting Simplex 17/20

18 Branch & bound Branch & bound tree LP problems at each node Lower bounds on f Allow for pruning Ivet Galabova, Julian Hall Crash-Starting Simplex 18/20

19 Observations Time (s) Problem LB UB f Bounds Idiot+Crossover Simplex qap qap qap QAPs require integer solution to linearization Fast bounds on f accelerate branch & bound algorithms Ivet Galabova, Julian Hall Crash-Starting Simplex 19/20

20 Conclusion Finding a good starting basis beneficial but difficult The Idiot crash is somewhere between a quadratic penalty function method and an Augmented Lagrangian method Limitations Sensitive to parameters Can have disastrous effects Single variable change Effective on some problems Scope for accelerating B&B Ivet Galabova, Julian Hall Crash-Starting Simplex 20/20