An improved primal simplex algorithm and column generation for degenerate linear programs Abdelmoutalib Metrane ´ Ecole Polytechnique and GERAD Montreal Canada Column Generation 2008
Introduction Improved Primal Simplex Column generation for degenerate linear programs Joint work with Issmail Elhallaoui, Post-doc, Polytechnique, Montreal Guy Desaulniers, Professor, Polytechnique, Montreal Fran¸ cois Soumis, Professor, Polytechnique, Montreal Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
Introduction Improved Primal Simplex Column generation for degenerate linear programs Introduction 1 Improved Primal Simplex 2 The reduced problem The complementarity problem IPS Algorithm Numerical results Conclusion Column generation for degenerate linear programs 3 Aggregated columns Algorithm Numerical results Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
Introduction Improved Primal Simplex Column generation for degenerate linear programs Linear programming LP = c ⊤ x ( LP ) z min (1) x s.t. Ax = b (2) x ≥ 0 (3) where x ∈ R n , c ∈ R n , b ∈ R m , and A ∈ R m × n Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
Introduction Improved Primal Simplex Column generation for degenerate linear programs Degeneracy in linear programming Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
Introduction Improved Primal Simplex Column generation for degenerate linear programs Degeneracy in linear programming Instance cp 5 cp 6 cp 7 Number of variables 10 15 21 Number of constraints 16 32 64 Number of non-zero elements/column 8 16 32 Number of extreme points 26 158 544 Number of LPF bases 126 48,414 6,450,702 Avg nb of LPF bases/extreme point 4.8 306.4 11,857.9 Table 1: Results of the lrs2 algorithm for three set partitioning instances Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
Introduction Improved Primal Simplex Column generation for degenerate linear programs Set partitioning case: Dynamic constraint aggregation, Multi-phase dynamic constraint aggregation Elhallaoui, I., A. Metrane, F. Soumis, and G. Desaulniers (2005). 1 Multi-phase Dynamic Constraint Aggregation for Set Partitioning Type Problems. Under minor revisions in Math Programming . I. Elhallaoui, G. Desaulniers, A. Metrane, F. Soumis: Bi- Dynamic 2 Constraint Aggregation and Subproblem Reduction. Computer and Operation Research 35 (2008) 1713-1724. Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Introduction 1 Improved Primal Simplex 2 The reduced problem The complementarity problem IPS Algorithm Numerical results Conclusion Column generation for degenerate linear programs 3 Aggregated columns Algorithm Numerical results Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Definition Let B = { A k / k ∈ K ⊂ { 1 ... n } } . A column D is said to be compatible with B if there exists λ such that � D = λ k A k k ∈ K Definition A reduced problem w. r. t to a set B is a problem where only columns compatible with B are considered and the redundant constraints are removed. Reduced Problem RP B For a degenerate basic solution x , let B = { A i / x i > 0 } = { A 1 , . . . , A d / d < m } . Denote by RP B the reduced problem w.r.t to a set B. Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion LP Incompatible b B B Compatible columns = Redundant b NB constraints are removed Figure 1: Reduced Problem Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Reduced Problem = b B B Compatible columns Instead of working with LP, it is better to work with RP B because this problem is smaller than LP. Figure 2: Reduced Problem Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion After solving the RP B Find one or more columns such that if we add these columns to the reduced problem, the optimal value decreases. Questions: Do these columns exist? How can we find these columns? Is it easy to find them? Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Motivation We know that a solution x is an optimal solution of LP if and only if there exists a dual solution π to LP such that c j := c j − π ⊤ A j ¯ = 0 , ∀ j ∈ { 1 ... d } (4) c j := c j − π ⊤ A j ¯ ≥ 0 , ∀ j ∈ { d + 1 ... n } (5) Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Motivation We know that a solution x is an optimal solution of LP if and only if there exists a dual solution π to LP such that c j := c j − π ⊤ A j ¯ = 0 , ∀ j ∈ { 1 ... d } (4) c j := c j − π ⊤ A j ¯ ≥ 0 , ∀ j ∈ { d + 1 ... n } (5) Complementarity problem max (6) s s , π c j − π ⊤ A j = 0 , s.t. ∀ j ∈ { 1 ... d } (7) c j − π ⊤ A j ≥ s , ∀ j ∈ I = { d + 1 ... n } (8) Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Duality of complementarity problem CP � ( CP B ) B = min v j ¯ (9) z c j v j ∈ I s.t. Mv = 0 (10) e ⊤ v = 1 (11) v ≥ 0 . (12) Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Duality of complementarity problem CP � ( CP B ) B = min v j ¯ (9) z c j v j ∈ I s.t. Mv = 0 (10) e ⊤ v = 1 (11) v ≥ 0 . (12) Let x B optimal solution of RP B Proposition B ≥ 0 , then ( x ∗ If z CP B , 0) is an optimal solution to LP. B < 0 , then ( x ∗ If z CP B , 0) is not an optimal solution to LP. Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Duality of complementarity problem CP � ( CP B ) B = min v j ¯ (9) z c j v j ∈ I s.t. Mv = 0 (10) e ⊤ v = 1 (11) v ≥ 0 . (12) Let x B optimal solution of RP B Proposition B ≥ 0 , then ( x ∗ If z CP B , 0) is an optimal solution to LP. B < 0 , then ( x ∗ If z CP B , 0) is not an optimal solution to LP. − → Add � � ⇒ the optimal value of RP B decreases. A j / v ∗ j > 0 Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
The reduced problem Introduction The complementarity problem Improved Primal Simplex IPS Algorithm Column generation for degenerate linear programs Numerical results Conclusion Find an initial basis B Construct and solve RP B Construct and solve CP Add A to RP B j B such that v > 0 * j Stop: Optimal solution z CP < 0 ? No Yes B Figure 3: The improved primal simplex algorithm Abdelmoutalib Metrane An improved primal simplex algorithm and column generation for degenerate
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