Simplex method(s) for solving LPs in standard form

Transcription

1 Simplex method: outline I The Simplex Method is a family of algorithms for solving LPs in standard form (and their duals) I Goal: identify an optimal basis, as in Definition 3.3 I Versions we will consider: I Primal I Dual (along with LP sensitivity analysis) I Parametric primal-dual IOE 610: LP II, Fall 2013 Simplex method for solving LPs Page 98 Simplex method(s) for solving LPs in standard form I Idea behind the Simplex method(s) for LPs in standard form: I Start at a basis B I Move to an adjacent basis I Repeat until I an optimal basis (Definition 3.3) is reached, or I lack of optimal solution is proven I Questions: I What initial basis do we choose? I How do we move between adjacent bases? I How can we guarantee that the process terminates with a correct conclusion? I How long will it take? IOE 610: LP II, Fall 2013 Simplex method for solving LPs Page 99

2 Primal Simplex Method I Assumptions: I The primal LP is feasible I AbasisassociatedwithaBFSisavailable I The Primal Simplex Algorithm I Start at a BFS x and associated basis B I Calculate c T = c T c T B B 1 A I If c 0, stop current basis is optimal. Otherwise, pick some j 2 N with c j < 0. I Move to an adjacent feasible basis in which Aj is a basic column. I Repeat until an optimal basis is reached, or unboundedness (of the LP) is proven. IOE 610: LP II, Fall 2013 Primal Simplex Method Page 100 Feasible directions and basic directions (LP) min c T x s.t. x 2 P; P = {x : Ax = b, x 0} Definition 3.1 Let x 2 P. d 2< n is a feasible direction at x if 9 > 0: x + d 2 P. I Special case: x is a BFS with basis B; j non-basic index I d is the jth basic direction at x if d i =0, i 2 N, i 6= j, d j =1, d B =(d B(1),...,d B(m) ) T = I Note: I If x is non-degenerate, d is a feasible direction I If x is degenerate, d may not be a feasible direction! I E ect on the cost function of moving in jth basic direction: jth reduced cost B 1 A j c j = c T d = c T B d B + c j = c j c T B B 1 A j IOE 610: LP II, Fall 2013 Primal Simplex Method Page 101

3 An iteration of the (primal) simplex method: a pivot 1. Let x =(x B ; x N )=(B 1 b; 0) beabfs,andb the associated basis (assume for now x is non-degenerate) 2. Compute c j = c j c T B B 1 A j 8j 2 N. If c 0 terminate; optimal solution found. Else, choose j : c j < Let d be the j basic direction at x: d i =0, i 2 N, i 6= j, d j =1, d B =(d B(1),...,d B(m) ) T = If d B 0, terminate; LP is unbounded. 4. Else, let? = min i:d B(i) <0 d B(i), l = argmin i:d B(i) <0 d B(i) 5. Let l be as above,? = x B(l) d B(l).Formanewbasismatrixby replacing A B(l) with A j (x l leaves, and x j enters, the basis). The values of the new basic variables: y j =?, y B(i) = +? d B(i), i 6= l IOE 610: LP II, Fall 2013 Primal Simplex Method Page 102 B 1 A j Sanity checks Theorem 3.2 (a) The columns A B(i), i 6= l and A j are linearly independent and therefore, the matrix constructed in step 5 is a basis. (b) The vector y = x +? d is a basic feasible solution associated with the above matrix. Theorem 3.3 Assume that the feasible set is nonempty and that every basic feasible solution is nondegenerate. Then the simplex method terminates after a finite number of iterations. At termination, there are the following two possibilities: (a) We have an optimal basis B and an associated basic feasible solution which is optimal. (b) We have found a vector d satisfying Ad = 0, d 0, c T d < 0, and the problem is unbounded. IOE 610: LP II, Fall 2013 Primal Simplex Method Page 103

4 What if there are degenerate BFSs? I If current BFS x is degenerate, then it is possible that? = 0. I It is still possible, however, to perform a change of basis, although we will stay at the same solution, and see no improvement in the objective. I It is possible that a series of iterations as above will cause us to loop through a series of bases, returning to the original one: cycling. I Can be avoided by appropriate pivoting rules. IOE 610: LP II, Fall 2013 Primal Simplex Method Page 104 Pivot selection How to choose j and l (if there are ties) in steps 2 and 5 of the algorithm? Some possible pivoting rules (out of many others): I Choose j = argmin c j fastest rate of decrease of the cost. I Choose j = argmax? j c j largest cost decrease. I Choose the smallest j with c j < 0 less computation. If the smallest subscript rule is also used to break ties in choosing l, no cycling occurs in degenerate problems I Thus, with appropriate pivoting rules, simplex method will find an optimal basis matrix in any standard form LP with finite optimal value! I Lexicographic pivoting rule (See Section 3.4) another anti cycling pivoting rule. IOE 610: LP II, Fall 2013 Primal Simplex Method Page 105

5 Un-answered (for now) questions I How do we find a starting feasible basis, or find out/prove that the LP is infeasible? I What is the computational e ciency of the (Primal) Simplex Method: I Time to execute 1 iteration? I Number of iterations required? I Let s postpone these for now, and explore I I other properties of optimal bases. and other variants of the simplex method. IOE 610: LP II, Fall 2013 Primal Simplex Method Page 106 Sensitivity analysis: changes in c (P) min c T x (D) max b T p s.t. Ax = b s.t. p T A apple c T x 0 I Suppose optimal basis B is known, but c becomes c + t c, for some given c 2< n I For what values of t is B still optimal? I xb = B 1 b 0 does not change I Need: (cj + t c j ) (c B + t c B ) T B 1 A j 0 for all j 2 N I Let c j = c j c T B B 1 A j I B stays dual-feasible (and hence optimal) for max j: c j >0 c j c j apple t apple min j: c j <0 c j c j I If change in c is such that B is no longer dual-feasible, re-solve with Primal Simplex (starting at current basis B) IOE 610: LP II, Fall 2013 Sensitivity analysis Page 107

6 Sensitivity analysis: changes in b I Suppose optimal basis B is known, but b becomes b + t b, for some given b 2< m I For what values of t is B still optimal? I s N = c N c T B B 1 A N = c N 0 does not change I Need: B 1 (b + t b) 0 I Let x B = B 1 b I B stays primal-feasible (and hence optimal) for max i: >0 apple t apple min i: <0 I If change in b is such that B is no longer primal-feasible, re-solve? I Note: current basis B is dual-feasible, but not primal feasible... IOE 610: LP II, Fall 2013 Sensitivity analysis Page 108 Primal vs. Dual simplex I Primal Simplex: I Starts with a primal-feasible basis I Goes from a basis to an (adjacent) basis I Strives to achieve dual feasibility (hence selection of entering variable) I Maintains primal feasibility throughout (hence selection of exiting variable) I Dual Simplex? I Starts with a dual-feasible basis I Goes from a basis to an (adjacent) basis I Strives to achieve primal feasibility I Maintains dual feasibility throughout I Development of Dual Simplex algorithm: I Given a dual-feasible basis, I Determine which variable should exit to work towards primal feasibility I Then determine which variable should enter to maintain dual feasibility IOE 610: LP II, Fall 2013 Dual Simplex algorithm Page 109

7 An iteration of the dual simplex method: a pivot 1. Let (p; s) =(p; s B ; s N )=((c T B B 1 ) T ; 0; c N )withs N 0 be adualbfs,andb the associated basis 2. Compute x B = B 1 b.ifx B 0 terminate; solution found. Else, choose l : x B(l) < 0. (x B(l) leaves the basis) 3. Let v be the lth row of B 1 A: If v 4. Else, let v T = e T l B 1 A 0, terminate;(primal)lpisinfeasible. j = argmin i:v i <0 c i v i (x j enters the basis) 5. Let l and j be as above. Form a new basis matrix by replacing A B(l) with A j. IOE 610: LP II, Fall 2013 Dual Simplex algorithm Page 110 Analysis of the dual simplex pivot Entering variable: x j ;exitingvariable:x B(l) I New basic (p new ) T = p T + t? p = c T B B 1 + c j e T l B 1 I For i in the (new) basis, c i new = 0 by construction I For i nonbasic (after the basis change), c new i = c i (p new ) T A i = c i c j e T l B 1 A i = c i c j v i I For i = B(l): c new B(l) = c B(l) I For i 6= B(l): c i new c = c j i I New objective value: c j v B(l) =0 c j 1 0 v i 0 by choice of j (p new ) T b = c T B B 1 b+ c j e T l B 1 b = c T B v B 1 b+ c j x j v B(l) c T B B 1 b j I If c j > 0, then the pivot is non-degenerate and new dual BFS has a better (dual) objective function value I If all pivots are non-degenerate, bases are never repeated, proving convergence I Usual anti-cycling pivoting rules can be used if needed IOE 610: LP II, Fall 2013 Dual Simplex algorithm Page 111

8 Sensitivity analysis summary Given a primal and dual feasible basis B, I If c changes to c + t dual-feasible I If b changes to b + t primal-feasible c, find range of t for which B remains b, find range of t for which B remains I If both changes happen simultaneously, combined analyses finds range of t for which B remains optimal IOE 610: LP II, Fall 2013 Dual Simplex algorithm Page 112 Simplex method summary so far X Described: I Primal simplex method, to use when a starting primal-feasible B is available I Dual simplex method, to use when a starting dual-feasible B is available X Argued that both versions of simplex terminate after a finite number of iterations I Either finds an optimal basis, I or proves that the problem is unbounded Easy argument with no degeneracy; references to anti-cycling pivoting rules with degeneracy I Still unanswered: I How many iterations? I How much time per iteration? I How do we get a primal- or dual- feasible basis to start with? I or re-design simplex method to be able to start at any basis! IOE 610: LP II, Fall 2013 Dual Simplex algorithm Page 113