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2007 : Solving Linear Problems with MOSEK (Seattle 2007)
- 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 Meeting Seattle Nov. 7, 2007 ∗ http://www.mosek.com Erling D. Andersen
- 2. Introduction 2 / 26
- 3. Topics Introduction Topics s The problem: The linear optimizer (P ) min cT x The simplex st Ax = b, optimizers Computational x ≥ 0. results Conclusions s The linear optimizers. x Interior-point optimizer (Not main focus in this talk). x Simplex optimizer. s What is the recent improvements? s What is the (relative) performance? 3 / 26
- 4. The linear optimizer Introduction Topics The general flow : The linear optimizer s Presolve. The simplex optimizers s Form the reduced primal or dual. Computational s Scale (optimizer specific). results s Optimize (interior-point or simplex). Conclusions s Basis identification (interior-point only). s Undo scaling and dualizing. s Postsolve. 4 / 26
- 5. The simplex optimizers 5 / 26
- 6. What makes a good simplex optimizer ? Introduction s Exploit sparsity (i.e. LU and FTRAN and BTRAN The simplex optimizers routines). What makes a good simplex optimizer ? s Exploit problem dependent structure. MOSEK simplex-overview s Choose right path (i.e. good pricing strategy). Exploiting sparsity s Long steps (i.e. avoid degeneracy). aggressively Primal (dual) s Numerical stability (i.e. reliable and consistent results). Degeneracy Dual bound flipping s Fast hotstarts (i.e. MIP and other hotstart applications). idea used more aggressively s Other tricks. Numerical stability Network optimizer Computational results Conclusions 6 / 26
- 7. MOSEK simplex-overview Introduction s Primal and dual simplex optimizer. The simplex optimizers What makes a good x Efficient cold start and warm start. simplex optimizer ? x Crashes an initial basis. MOSEK simplex-overview x Multiple pricing options: Exploiting sparsity aggressively s Full (Dantzig). Primal (dual) Degeneracy s Partial. Dual bound flipping idea used more s Approximate/exact steepest edge. aggressively Numerical stability s Hybrid. Network optimizer x Degeneration handling. Computational results s Revised simplex algorithm + many enhancements. Conclusions s Many enhancements still possible!. 7 / 26
- 8. Exploiting sparsity aggressively Introduction s Simplex algs. require solution of the linear equation The simplex optimizers systems What makes a good simplex optimizer ? Bf = A:j and B T g = ei . MOSEK simplex-overview in each iteration. Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 8 / 26
- 9. Exploiting sparsity aggressively Introduction s Simplex algs. require solution of the linear equation The simplex optimizers systems What makes a good simplex optimizer ? Bf = A:j and B T g = ei . MOSEK simplex-overview in each iteration. Exploiting sparsity aggressively s Assume a sparse LU factorization of the basis Primal (dual) Degeneracy Dual bound flipping B = LU. idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 8 / 26
- 10. Exploiting sparsity aggressively Introduction s Simplex algs. require solution of the linear equation The simplex optimizers systems What makes a good simplex optimizer ? Bf = A:j and B T g = ei . MOSEK simplex-overview in each iteration. Exploiting sparsity aggressively s Assume a sparse LU factorization of the basis Primal (dual) Degeneracy Dual bound flipping B = LU. idea used more aggressively Numerical stability s f can be computed as follow. Solve Network optimizer Computational results ¯ Lf = A:j Conclusions and then ¯ Uf = f. 8 / 26
- 11. Exploiting sparsity aggressively Introduction s Simplex algs. require solution of the linear equation The simplex optimizers systems What makes a good simplex optimizer ? Bf = A:j and B T g = ei . MOSEK simplex-overview in each iteration. Exploiting sparsity aggressively s Assume a sparse LU factorization of the basis Primal (dual) Degeneracy Dual bound flipping B = LU. idea used more aggressively Numerical stability s f can be computed as follow. Solve Network optimizer Computational results ¯ Lf = A:j Conclusions and then ¯ Uf = f. s Simple implementation requires O(nz(L) + nz(U )) flops. 8 / 26
- 12. Exploiting sparsity aggressively (continued) Introduction s Consider the simple example: The simplex optimizers ¯ What makes a good 1 f1 0 simplex optimizer ? MOSEK 0 1 ¯ f2 = x simplex-overview x 0 1 ¯ f3 0 Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 9 / 26
- 13. Exploiting sparsity aggressively (continued) Introduction s Consider the simple example: The simplex optimizers ¯ What makes a good 1 f1 0 simplex optimizer ? MOSEK 0 1 ¯ f2 = x simplex-overview x 0 1 ¯ f3 0 Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping s Clearly sparsity in the RHS can be exploited! (done idea used more aggressively extensively in MOSEK). Numerical stability Network optimizer Computational results Conclusions 9 / 26
- 14. Exploiting sparsity aggressively (continued) Introduction s Consider the simple example: The simplex optimizers ¯ What makes a good 1 f1 0 simplex optimizer ? MOSEK 0 1 ¯ f2 = x simplex-overview x 0 1 ¯ f3 0 Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping s Clearly sparsity in the RHS can be exploited! (done idea used more aggressively extensively in MOSEK). Numerical stability s Gilbert and Peierls [GIL:88] demonstrate how to solve the Network optimizer triangular system in O(minimal number of flops). Computational results Conclusions 9 / 26
- 15. Exploiting sparsity aggressively (continued) Introduction s Consider the simple example: The simplex optimizers ¯ What makes a good 1 f1 0 simplex optimizer ? MOSEK 0 1 ¯ f2 = x simplex-overview x 0 1 ¯ f3 0 Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping s Clearly sparsity in the RHS can be exploited! (done idea used more aggressively extensively in MOSEK). Numerical stability s Gilbert and Peierls [GIL:88] demonstrate how to solve the Network optimizer triangular system in O(minimal number of flops). Computational results s Aim: Solves with L and U and updates to the LU should Conclusions run in O(minimal number of flops) and not in O(m) for instance. 9 / 26
- 16. Exploiting sparsity aggressively (continued) Introduction s Consider the simple example: The simplex optimizers ¯ What makes a good 1 f1 0 simplex optimizer ? MOSEK 0 1 ¯ f2 = x simplex-overview x 0 1 ¯ f3 0 Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping s Clearly sparsity in the RHS can be exploited! (done idea used more aggressively extensively in MOSEK). Numerical stability s Gilbert and Peierls [GIL:88] demonstrate how to solve the Network optimizer triangular system in O(minimal number of flops). Computational results s Aim: Solves with L and U and updates to the LU should Conclusions run in O(minimal number of flops) and not in O(m) for instance. s Drawback: Both L and U must be stored row and column wise because solves with LT and U T are required too. 9 / 26
- 17. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? MOSEK simplex-overview Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 10 / 26
- 18. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? s Primal step size δp : MOSEK simplex-overview lB ≤ xB − δp B −1 aq ≤ uB Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 10 / 26
- 19. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? s Primal step size δp : MOSEK simplex-overview lB ≤ xB − δp B −1 aq ≤ uB Exploiting sparsity aggressively s Basic variables on a bound may imply a zero primal step. Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 10 / 26
- 20. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? s Primal step size δp : MOSEK simplex-overview lB ≤ xB − δp B −1 aq ≤ uB Exploiting sparsity aggressively s Basic variables on a bound may imply a zero primal step. Primal (dual) Degeneracy s Dual step size δd : Dual bound flipping cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL idea used more aggressively cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU Numerical stability Network optimizer Computational results Conclusions 10 / 26
- 21. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? s Primal step size δp : MOSEK simplex-overview lB ≤ xB − δp B −1 aq ≤ uB Exploiting sparsity aggressively s Basic variables on a bound may imply a zero primal step. Primal (dual) Degeneracy s Dual step size δd : Dual bound flipping cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL idea used more aggressively cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU Numerical stability Network optimizer s Non basic variables with zero reduced cost may imply a Computational results zero dual step. Conclusions Degeneration posses both a theoretical and a practical problem for the simplex optimizer ! 10 / 26
- 22. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? s Primal step size δp : MOSEK simplex-overview lB ≤ xB − δp B −1 aq ≤ uB Exploiting sparsity aggressively s Basic variables on a bound may imply a zero primal step. Primal (dual) Degeneracy s Dual step size δd : Dual bound flipping cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL idea used more aggressively cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU Numerical stability Network optimizer s Non basic variables with zero reduced cost may imply a Computational results zero dual step. Conclusions Degeneration posses both a theoretical and a practical problem for the simplex optimizer ! What is our options ? 10 / 26
- 23. Primal (dual) Degeneracy Introduction The simplex optimizer may take very small or zero step sizes, The simplex optimizers why ? What makes a good simplex optimizer ? s Primal step size δp : MOSEK simplex-overview lB ≤ xB − δp B −1 aq ≤ uB Exploiting sparsity aggressively s Basic variables on a bound may imply a zero primal step. Primal (dual) Degeneracy s Dual step size δd : Dual bound flipping cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL idea used more aggressively cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU Numerical stability Network optimizer s Non basic variables with zero reduced cost may imply a Computational results zero dual step. Conclusions Degeneration posses both a theoretical and a practical problem for the simplex optimizer ! What is our options ? One approach is to perturb lj and uj (cj ). 10 / 26
- 24. Primal (dual) Degeneracy (continued) Introduction MOSEK 5 has been improved on degenerated problems: The simplex optimizers What makes a good s Better and more aggressive perturbation scheme. simplex optimizer ? MOSEK simplex-overview Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 11 / 26
- 25. Primal (dual) Degeneracy (continued) Introduction MOSEK 5 has been improved on degenerated problems: The simplex optimizers What makes a good s Better and more aggressive perturbation scheme. simplex optimizer ? s Sparsity issues important (very tricky). MOSEK simplex-overview Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 11 / 26
- 26. Primal (dual) Degeneracy (continued) Introduction MOSEK 5 has been improved on degenerated problems: The simplex optimizers What makes a good s Better and more aggressive perturbation scheme. simplex optimizer ? s Sparsity issues important (very tricky). MOSEK simplex-overview s Clean up perturbations with dual (primal) simplex. Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 11 / 26
- 27. Primal (dual) Degeneracy (continued) Introduction MOSEK 5 has been improved on degenerated problems: The simplex optimizers What makes a good s Better and more aggressive perturbation scheme. simplex optimizer ? s Sparsity issues important (very tricky). MOSEK simplex-overview s Clean up perturbations with dual (primal) simplex. Exploiting sparsity aggressively s Many examples where ”tailed” solves are substantial Primal (dual) Degeneracy reduced. Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 11 / 26
- 28. Primal (dual) Degeneracy (continued) Introduction MOSEK 5 has been improved on degenerated problems: The simplex optimizers What makes a good s Better and more aggressive perturbation scheme. simplex optimizer ? s Sparsity issues important (very tricky). MOSEK simplex-overview s Clean up perturbations with dual (primal) simplex. Exploiting sparsity aggressively s Many examples where ”tailed” solves are substantial Primal (dual) Degeneracy reduced. Dual bound flipping s Still room for improvement. idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 11 / 26
- 29. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 12 / 26
- 30. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping s This involves flipping nonbasic variables to opposite idea used more aggressively bound to remain dual feasible and cost one extra solve. Numerical stability Network optimizer Computational results Conclusions 12 / 26
- 31. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping s This involves flipping nonbasic variables to opposite idea used more aggressively bound to remain dual feasible and cost one extra solve. Numerical stability Network optimizer s Longer dual steplengths. Computational results Conclusions 12 / 26
- 32. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping s This involves flipping nonbasic variables to opposite idea used more aggressively bound to remain dual feasible and cost one extra solve. Numerical stability Network optimizer s Longer dual steplengths. Computational s Reduces degeneracy. results Conclusions 12 / 26
- 33. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping s This involves flipping nonbasic variables to opposite idea used more aggressively bound to remain dual feasible and cost one extra solve. Numerical stability Network optimizer s Longer dual steplengths. Computational s Reduces degeneracy. results s Less iterations. Conclusions 12 / 26
- 34. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping s This involves flipping nonbasic variables to opposite idea used more aggressively bound to remain dual feasible and cost one extra solve. Numerical stability Network optimizer s Longer dual steplengths. Computational s Reduces degeneracy. results s Less iterations. Conclusions s More flexibility in pivot choice (i.e. potentially more stable). 12 / 26
- 35. Dual bound flipping idea used more aggressively Introduction The simplex Dual step size δd : optimizers cj − y T Aj − (+)δd (ei B −1 N )j ≥ 0 ∀j ∈ NL What makes a good simplex optimizer ? MOSEK cj − y T Aj − (+)δd (ei B −1 N )j ≤ 0 ∀j ∈ NU simplex-overview Exploiting sparsity A ranged variable i.e. −∞ < lj < xj < uj < ∞ may not be aggressively binding in dual min-ratio if profitable. Primal (dual) Degeneracy Dual bound flipping s This involves flipping nonbasic variables to opposite idea used more aggressively bound to remain dual feasible and cost one extra solve. Numerical stability Network optimizer s Longer dual steplengths. Computational s Reduces degeneracy. results s Less iterations. Conclusions s More flexibility in pivot choice (i.e. potentially more stable). s Improves sparsity of the basis when degenerated! (i.e. if xB i becomes feasible no basis exchange is needed). 12 / 26
- 36. Dual bound flipping idea used more aggressively (continued) Introduction Bound flipping examples: The simplex optimizers What makes a good Iter Time simplex optimizer ? MOSEK Problem Rows Cols NB WB NB WB simplex-overview Exploiting sparsity osa-60 10280 232966 6938 5111 58.12 8.84 aggressively Primal (dual) world 34506 32734 54566 32606 218.81 50.03 Degeneracy pds-40 66844 212859 34274 26599 96.51 18.48 Dual bound flipping idea used more ken-18 105127 154699 151203 51452 258.18 13.92 aggressively Numerical stability client 27216 20567 80555 63660 208.40 84.09 Network optimizer Computational WB = MOSEK 5 Dual simplex with bound flips results Conclusions NB = MOSEK 5 Dual simplex with no bound flips 13 / 26
- 37. Numerical stability Introduction s Improving numerical stability. The simplex optimizers What makes a good x Moved LU update before updating solution. simplex optimizer ? MOSEK simplex-overview Exploiting sparsity aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 14 / 26
- 38. Numerical stability Introduction s Improving numerical stability. The simplex optimizers What makes a good x Moved LU update before updating solution. simplex optimizer ? MOSEK s Saves one solve with L in ei T B −1 [GOL:77]. simplex-overview Exploiting sparsity s More stable approach. aggressively Primal (dual) Degeneracy Dual bound flipping idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 14 / 26
- 39. Numerical stability Introduction s Improving numerical stability. The simplex optimizers What makes a good x Moved LU update before updating solution. simplex optimizer ? MOSEK s Saves one solve with L in ei T B −1 [GOL:77]. simplex-overview Exploiting sparsity s More stable approach. aggressively Primal (dual) x Better handling of singularities (sing. variables are Degeneracy Dual bound flipping temporary fixed). idea used more aggressively Numerical stability Network optimizer Computational results Conclusions 14 / 26
- 40. Numerical stability Introduction s Improving numerical stability. The simplex optimizers What makes a good x Moved LU update before updating solution. simplex optimizer ? MOSEK s Saves one solve with L in ei T B −1 [GOL:77]. simplex-overview Exploiting sparsity s More stable approach. aggressively Primal (dual) x Better handling of singularities (sing. variables are Degeneracy Dual bound flipping temporary fixed). idea used more aggressively x Switch to safe mode if deemed unstable. Numerical stability Network optimizer Computational results Conclusions 14 / 26
- 41. Network optimizer Introduction MOSEK 5 features a network simplex optimizer. The simplex optimizers What makes a good s Solves pure network flow problems (i.e. LP’s with two simplex optimizer ? non-zeros in each column either 1 or -1). MOSEK simplex-overview s Can extract embedded network structure in a model (i.e. Exploiting sparsity aggressively network with side constraints). Primal (dual) Degeneracy s Using standard interface, only one parameter has to be Dual bound flipping set. idea used more aggressively s Huge problems can be solved in limited time, for instance Numerical stability Network optimizer a problem with 8 million variables can be solved in less Computational than 200 seconds. results Conclusions 15 / 26
- 42. Computational results 16 / 26
- 43. Test setup Introduction s 577 problems (mixed size). The simplex optimizers s A Dual Core server with 4GB RAM running Windows Computational 2003 (Intel CPU). results Test setup s A Quad Core server with 8GB RAM running Windows Network Vs. Standard simplex 2003 (Intel CPU). Primal Simplex s See [HM:07] for a benchmark comparing Mosek with Dual Simplex Numerical difficult other solvers. problems-primal simplex Numerical difficult All results presented in one table is obtained using one of the problems-dual simplex two computers only. Conclusions 17 / 26
- 44. Network Vs. Standard simplex Introduction The simplex optimizers small medium Computational netw psim dsim netw psim dsim results Num. 30 30 30 43 43 43 Test setup Network Vs. Firsts 30 0 1 43 0 0 Standard simplex Total time 13.7 114.8 27.8 589.9 10676.6 3015.2 Primal Simplex Dual Simplex G. avg. 0.39 2.42 0.70 6.30 91.74 19.70 Numerical difficult problems-primal simplex Numerical difficult large problems-dual simplex netw psim dsim Conclusions Num. 2 2 2 Firsts 2 0 0 Total time 366.3 2905.8 968.9 G. avg. 182.98 1115.71 468.76 Table 1: Performance of the network flow, primal simplex and dual simplex optimizer on pure network problems. 18 / 26
- 45. Primal Simplex Introduction The simplex optimizers small medium large Computational 5 4 5 4 5 4 results Num. 399 399 148 148 30 30 Test setup Network Vs. Firsts 329 245 91 62 22 11 Standard simplex Total time 100.4 101.7 2425.3 8962.3 29905.2 39333.2 Primal Simplex Dual Simplex G. avg. 0.06 0.07 7.49 9.24 591.39 746.01 Numerical difficult problems-primal simplex Numerical difficult Table 2: Performance of the version 4 and version 5 primal problems-dual simplex simplex optimizer Conclusions 19 / 26
- 46. Dual Simplex Introduction The simplex optimizers small medium large Computational 5 4 5 4 5 4 results Num. 412 412 150 150 21 21 Test setup Network Vs. Firsts 198 286 133 22 18 5 Standard simplex Total time 84.8 106.4 1852.9 7611.3 23678.9 38994.3 Primal Simplex Dual Simplex G. avg. 0.10 0.08 4.65 8.70 544.44 1065.24 Numerical difficult problems-primal simplex Numerical difficult Table 3: Performance of the version 4 and version 5 dual sim- problems-dual simplex plex optimizer Conclusions 20 / 26
- 47. Numerical difficult problems-primal simplex Introduction The simplex optimizers small medium large Computational 5 4 5 4 5 4 results Num. 9 9 19 19 2 2 Test setup Network Vs. Firsts 5 5 13 6 2 0 Standard simplex Total time 2.7 2.8 235.9 319.6 1297.7 1503.3 Primal Simplex Dual Simplex G. avg. 0.19 0.18 7.19 9.54 413.26 464.04 Numerical difficult Fails 0 0 0 3 0 3 problems-primal simplex Numerical difficult problems-dual Table 4: Performance of the version 4 and 5 of the primal sim- simplex Conclusions plex optimizer on numerical difficult problems. 21 / 26
- 48. Numerical difficult problems-dual simplex Introduction The simplex optimizers small medium large Computational 5 4 5 4 5 4 results Num. 11 11 19 19 4 4 Test setup Network Vs. Firsts 7 6 13 6 4 0 Standard simplex Total time 3.9 6.6 3198.3 345.9 4736.3 12820.5 Primal Simplex Dual Simplex G. avg. 0.24 0.31 8.44 9.67 802.24 2525.35 Numerical difficult Fails 0 0 0 1 0 1 problems-primal simplex Numerical difficult problems-dual Table 5: Performance of the version 4 and 5 dual simplex opti- simplex Conclusions mizer on numerical difficult problems. 22 / 26
- 49. Conclusions 23 / 26
- 50. Conclusions Introduction s Simplex: The simplex optimizers x MOSEK 5 substantial faster than MOSEK 4. Computational results x MOSEK 5 more stable than MOSEK 4. Conclusions x Dual simplex faster than primal. Conclusions A number open issues exists References 24 / 26
- 51. A number open issues exists Introduction s Simplex: The simplex optimizers x Degeneracy (non-perturbation method might be Computational results needed in extreme cases). Conclusions x Improve primal pricing. Conclusions A number open x Better crashing on special problems. issues exists x Choose more sparse path. References 25 / 26
- 52. References Introduction The simplex [HM:07] H.Mittelmann http://plato.la.asu.edu/bench.html optimizers Computational [GIL:88] J. R. Gilbert and T. Peierls, ”Sparse partial pivoting in time results proportional to arithmetic operations”, SIAM J. Sci. Statist. Conclusions Comput., 9, 1988, pp. 862–874. Conclusions A number open issues exists [GOL:77] D. Goldfarb, ”On the Bartels-Golub decomposition for References linear programming bases,” Mathematical. Programming, 13, 1977, pp 272-279 [KOS:02] E. Kostina, ”The Long Step Rule in the Bounded-Variable Dual Simplex Method: Numerical Experiments”, Mathematical Methods of Operations Research, 55 2002, I. 3. [MAR:03] Maros I, ”A Generalized Dual Phase-2 Simplex Algorithm”, European Journal of Operational Research, 149, 2003, pp. 1–16 26 / 26