Recommended
Recommended
More Related Content
Similar to 2009 : Solving linear optimization problems with MOSEK
Similar to 2009 : Solving linear optimization problems with MOSEK (20)
2009 : Solving linear optimization problems with MOSEK
- 1. Solving Linear Optimization Problems with MOSEK. Bo Jensen MOSEK ApS, Fruebjergvej 3, Box 16, 2100 Copenhagen, Denmark. Email: bo.jensen@MOSEK.com INFORMS Annual San Diego Tuesday Oct 13, 13:30 - 15:00, 2009 http://www.mosek.com
- 2. Introduction 2 / 26
- 3. What is MOSEK Introduction Aim of talk s A software package for solving large-scale optimization Topic problems. Introduction of the optimizers s Solves linear, conic, and nonlinear convex problems. Introduction of the optimizers Mixed integer optimizer. (Continued) s Used to solve problems with up to millions of constraints Computational test setup and variables. Improvements in the s Stand-alone as well as embedded : presolve Improvements in the x C simplex optimizers x Java Improvements in the interior point x MATLAB optimizer x Python Conclusions x .NET 3 / 26
- 4. What is MOSEK (Continued) Introduction Aim of talk s Third party interfaces : Topic Introduction of the x AIMMS optimizers x COIN-OR Introduction of the optimizers x Frontline Systems (Continued) x GAMS Computational test setup x Microsoft Solver Foundation Improvements in the presolve x OptimJ Improvements in the s Version 1 released 1999. simplex optimizers Improvements in the s Version 6 released October 2009 (Next week). interior point optimizer For further information see www.MOSEK.com. Conclusions Prices public available directly on our website ! 4 / 26
- 5. Aim of talk Introduction Aim of talk s Highlight recent key improvements in the linear Topic optimizers. Introduction of the optimizers s Show computational results on a larger set of benchmark Introduction of the optimizers problems. (Continued) Computational test setup Improvements in the presolve Improvements in the simplex optimizers Improvements in the interior point optimizer Conclusions 5 / 26
- 6. Topic Introduction Aim of talk s The problem Topic (P ) min cT x Introduction of the optimizers st Ax = b, Introduction of the optimizers x ≥ 0. (Continued) Computational test s The linear optimizers. setup Improvements in the x Interior-point based. presolve x Primal and dual simplex based. Improvements in the simplex optimizers x Primal network flow simplex. Improvements in the interior point s What are the recent improvements? optimizer Conclusions 6 / 26
- 7. Introduction of the optimizers Introduction Aim of talk s Interior point. Topic Introduction of the x Based on the homogeneous and self-dual model. optimizers x Basis identification. Introduction of the optimizers x Highly tuned implementation. (Continued) x Can exploit multiple processors. Computational test setup s Primal and Dual simplex. Improvements in the presolve x Exploits hyper sparsity (LU, solves and loops). Improvements in the simplex optimizers x Multiple pricing options : Dantzig, partial, devex, Improvements in the exact steepest edge, approximate steepest edge and interior point optimizer hybrid pricing. Conclusions x Crashes an advanced initial basis. x Aggressive anti-degeneracy schemes. 7 / 26
- 8. Introduction of the optimizers (Continued) Introduction Aim of talk s Network optimizer. Topic Introduction of the x Specialized primal simplex implementation. optimizers x Exploits the tree structure of the basis and other Introduction of the optimizers tricks. (Continued) x Can extract embedded network structure. Computational test setup Improvements in the presolve The network optimizer has not been changed since mosek 5 Improvements in the and will be left out of benchmark. simplex optimizers Improvements in the interior point optimizer Conclusions 8 / 26
- 9. Computational test setup 9 / 26
- 10. Computational test setup Introduction All computational results are run on the same machine : Computational test setup Computational test s Intel Xeon 2.27 GHz 16 Cores. setup s 16 GB RAM. How do we measure improvements ? s Running Linux. Improvements in the presolve Selected about 1000 problems from our clients and public Improvements in the ressources simplex optimizers Improvements in the interior point optimizer Conclusions 10 / 26
- 11. How do we measure improvement ? Introduction s Maximum solution time allowed 10000 sec, timeouts are Computational test setup excluded. Computational test setup s Group problems by solution times : How do we measure improvements ? x Small problems => best solution time ≤ 6 sek. Improvements in the x Medium problems => 6 sek < best solution time ≤ presolve Improvements in the 60 sek. simplex optimizers x Large problems => best solution time > 60 Improvements in the interior point s Sum of solution time Bad benchmarking criteria!. optimizer s Geometric mean of ratios of optimizer time i.e Conclusions MOSEK6/MOSEK5. s Number of wins. 11 / 26
- 12. Improvements in the presolve 12 / 26
- 13. Presolve overview Introduction s MOSEK has had an extensive presolve since version 1. Computational test setup See [AND:95]. Improvements in the s It was only employed in cold start cases. presolve Presolve overview s Since MOSEK 5 we can also presolve hotstart. Presolving hotstarts s Presolving hotstarts has been improved in MOSEK 6. Presolving hotstarts Numerical examples Improvements in the simplex optimizers Improvements in the interior point optimizer Conclusions 13 / 26
- 14. Presolving hotstarts Introduction s Assume we have a ”good” (optimal?) primal x∗ or dual Computational test setup y ∗ solution to P . Improvements in the But now the problem has been modified by the user into : presolve Presolve overview Presolving hotstarts ˆ (P ) min cT x ˆ Presolving hotstarts Numerical examples st Ax = ˆ ˆ b, Improvements in the x ≥ 0. simplex optimizers Improvements in the interior point optimizer Conclusions 14 / 26
- 15. Presolving hotstarts Introduction s Assume we have a ”good” (optimal?) primal x∗ or dual Computational test setup y ∗ solution to P . Improvements in the But now the problem has been modified by the user into : presolve Presolve overview Presolving hotstarts ˆ (P ) min cT x ˆ Presolving hotstarts Numerical examples st Ax = ˆ ˆ b, Improvements in the x ≥ 0. simplex optimizers Improvements in the interior point optimizer s Challenges : Conclusions x The problem may have been modified in any way. x Some presolve techniques destroy primal and dual feasibility. s Ex : Assume x∗ is primal feasible and x∗ k = uk , but dual presolve says x∗ k = lk ? 14 / 26
- 16. Presolving hotstarts (Continued) Introduction s The MOSEK presolve strategy: Computational test setup x Aimed at ”hard” hotstarts where presolve really Improvements in the presolve makes a difference. Presolve overview Presolving hotstarts Presolving hotstarts Numerical examples Improvements in the simplex optimizers Improvements in the interior point optimizer Conclusions 15 / 26
- 17. Presolving hotstarts (Continued) Introduction s The MOSEK presolve strategy: Computational test setup x Aimed at ”hard” hotstarts where presolve really Improvements in the presolve makes a difference. Presolve overview x Preserve either primal (primal simplex) or dual Presolving hotstarts Presolving hotstarts feasibility (dual simplex). Numerical examples Improvements in the simplex optimizers Improvements in the interior point optimizer Conclusions 15 / 26
- 18. Presolving hotstarts (Continued) Introduction s The MOSEK presolve strategy: Computational test setup x Aimed at ”hard” hotstarts where presolve really Improvements in the presolve makes a difference. Presolve overview x Preserve either primal (primal simplex) or dual Presolving hotstarts Presolving hotstarts feasibility (dual simplex). Numerical examples x Also try to keep the basis intact. Improvements in the simplex optimizers Improvements in the interior point optimizer Conclusions 15 / 26
- 19. Presolving hotstarts (Continued) Introduction s The MOSEK presolve strategy: Computational test setup x Aimed at ”hard” hotstarts where presolve really Improvements in the presolve makes a difference. Presolve overview x Preserve either primal (primal simplex) or dual Presolving hotstarts Presolving hotstarts feasibility (dual simplex). Numerical examples x Also try to keep the basis intact. Improvements in the simplex optimizers x If your problem takes few iterations in reoptimizing, Improvements in the then you should switch off presolve. interior point optimizer Conclusions 15 / 26
- 20. Numerical examples Introduction Selected 82 hotstart instances Computational test setup Warning biased problem selection ahead! Improvements in the presolve Presolve overview Presolving hotstarts Presolving hotstarts Numerical examples small medium Improvements in the 6-nopre 6 5 6-nopre 6 5 simplex optimizers Num. 74 74 74 8 8 8 Improvements in the interior point Firsts 9 51 18 1 6 1 optimizer Total time 1058.49 170.90 323.95 602.72 98.20 618.54 Conclusions G. avg. 5.79 1.77 2.39 41.68 11.21 44.37 G. avg. r 2.42 0.74 0.94 0.25 Table 1: Presolve hotstart effect on client problems (Dual sim- plex). 16 / 26
- 21. Improvements in the simplex optimizers 17 / 26
- 22. Key areas simplex optimizers Introduction s Speed and stability has in general been greatly improved. Computational test setup Improvements in the presolve Improvements in the simplex optimizers Computational results-primal simplex Computational results-dual simplex Improvements in the interior point optimizer Conclusions 18 / 26
- 23. Key areas simplex optimizers Introduction s Speed and stability has in general been greatly improved. Computational test setup s Larger parts of the code has been rewritten. Improvements in the presolve Improvements in the simplex optimizers Computational results-primal simplex Computational results-dual simplex Improvements in the interior point optimizer Conclusions 18 / 26
- 24. Key areas simplex optimizers Introduction s Speed and stability has in general been greatly improved. Computational test setup s Larger parts of the code has been rewritten. Improvements in the s Parameter tuning (i.e running many test with different presolve internal settings). Improvements in the simplex optimizers Computational results-primal simplex Computational results-dual simplex Improvements in the interior point optimizer Conclusions 18 / 26
- 25. Key areas simplex optimizers Introduction s Speed and stability has in general been greatly improved. Computational test setup s Larger parts of the code has been rewritten. Improvements in the s Parameter tuning (i.e running many test with different presolve internal settings). Improvements in the simplex optimizers s Hybrid pricing in primal simplex has been completely Computational results-primal rewritten. simplex Computational x Improved the automatic switch between partial and results-dual simplex Improvements in the approximate steepest-edge pricing in primal simplex. interior point x Restricted pricing (i.e optimizing over a subset of optimizer Conclusions columns) is now used more intensively. 18 / 26
- 26. Key areas simplex optimizers Introduction s Speed and stability has in general been greatly improved. Computational test setup s Larger parts of the code has been rewritten. Improvements in the s Parameter tuning (i.e running many test with different presolve internal settings). Improvements in the simplex optimizers s Hybrid pricing in primal simplex has been completely Computational results-primal rewritten. simplex Computational x Improved the automatic switch between partial and results-dual simplex Improvements in the approximate steepest-edge pricing in primal simplex. interior point x Restricted pricing (i.e optimizing over a subset of optimizer Conclusions columns) is now used more intensively. s Better crash module in dual simplex. 18 / 26
- 27. Key areas simplex optimizers Introduction s Speed and stability has in general been greatly improved. Computational test setup s Larger parts of the code has been rewritten. Improvements in the s Parameter tuning (i.e running many test with different presolve internal settings). Improvements in the simplex optimizers s Hybrid pricing in primal simplex has been completely Computational results-primal rewritten. simplex Computational x Improved the automatic switch between partial and results-dual simplex Improvements in the approximate steepest-edge pricing in primal simplex. interior point x Restricted pricing (i.e optimizing over a subset of optimizer Conclusions columns) is now used more intensively. s Better crash module in dual simplex. s The automatic refactor frequency in the LU module has been improved. 18 / 26
- 28. Computational results-primal simplex Introduction Computational test setup Improvements in the presolve Improvements in the simplex optimizers Computational results-primal simplex Computational results-dual simplex Improvements in the interior point optimizer Conclusions s Small problems are misleading, since time measurement in MOSEK 5 and 6 are different !. s At least 25% speed up on average. 19 / 26
- 29. Computational results-dual simplex Introduction Computational test setup Improvements in the presolve Improvements in the simplex optimizers Computational results-primal simplex Computational results-dual simplex Improvements in the interior point optimizer Conclusions s Small problems are misleading, since time measurement in MOSEK 5 and 6 are different !. s At least 25% speed up on average. 20 / 26
- 30. Improvements in the interior point optimizer 21 / 26
- 31. Key areas interior point optimizers Introduction s Improved speed in general. Computational test setup s Improved numerical stability on hard problems. Improvements in the presolve Improvements in the simplex optimizers Improvements in the interior point optimizer Computational results-interior point Conclusions 22 / 26
- 32. Computational results-interior point Introduction Computational test setup Improvements in the presolve Improvements in the simplex optimizers Improvements in the interior point optimizer Computational results-interior point Conclusions s The testset is too easy for interior point ! s Interior point was 3-4 times faster than dual simplex ! s About 5-10% speed up (more for hard problems). 23 / 26
- 33. Conclusions 24 / 26
- 34. Conclusions Introduction s Much improved presolve performance for hotstarts. Computational test setup s All optimizers more numerical stable. Improvements in the s Primal simplex optimizer 25% faster on average. presolve s Dual simplex optimizer 25% faster on average. Improvements in the simplex optimizers s Interior point optimizer 5-10% faster on average. Improvements in the interior point optimizer Conclusions Conclusions References 25 / 26
- 35. References Introduction Computational test [AND:95] E. Andersen and K. Andersen, ”Presolve in linear setup programming”, Mathematical Programming, Volume 71 , Improvements in the presolve 1995 Improvements in the simplex optimizers Improvements in the interior point optimizer Conclusions Conclusions References 26 / 26