The Ascendance of the Dual Simplex Method: A Geometric View
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1 The Ascendance of the Dual Simplex Method: A Geometric View Robert Fourer 4er@ampl.com AMPL Optimization Inc AMPL U.S.-Mexico Workshop on Optimization and Its Applications Huatulco 8-12 January
2 The Ascendance of the Dual Simplex Method: A Geometric View First described in the 1950s, the dual simplex evolved in the 1990s to become the method most often used in solving linear programs. Factors in the ascendance of the dual simplex method include Don Goldfarb s proposal for a steepest-edge variant, and an improved understanding of the bounded-variable extension. The ways that these come together to produce a highly effective algorithm are still not widely appreciated, however. This talk employs a geometric approach to the dual simplex method to provide a unified and straightforward description of the factors that work in its favor. 2
3 Motivation 3
4 Primal Linear Program Minimize Subject to, 0 Basic variables, nonbasic variables Coefficient columns of basic variables form a nonsingular matrix Basic solution 0, Feasible basic solution 0 4
5 Dual Linear Program Maximize Subject to Binding constraints, nonbinding constraints Coefficient rows of binding constraints form a nonsingular matrix Vertex solution Feasible vertex solution 5
6 Dual Linear Program Maximize Subject to, 0 Binding constraints, nonbinding constraints Coefficient rows of binding constraints form a nonsingular matrix Vertex solution 0, Feasible vertex solution 0 6
7 Primal Simplex Method Given feasible basic solution and corresponding basis matrix Choose a nonbasic variable to enter solve select : 0 Choose a basic variable to leave solve select : Θ min / Update Θ Θ for all 7
8 Dual Simplex Method Given feasible vertex solution and corresponding basis matrix Choose a binding constraint to leave solve select : 0 Choose a nonbinding constraint to enter solve select : Φ / min / Update Φ Φ for all 8
9 Inner Products with : Column-Wise Work of is the same for any For each nonzero in column of, add to sum Need all in an -vector Primal simplex Select one 0 ( ) for inner products, but often Dual simplex Compute min / ( ) Always inner products 9
10 Inner Products with : Row-Wise Store by row as well as by column Accumulate inner products together Work of depends on sparsity of For each nonzero, for each nonzero in row of, add to sum for Faster in dual simplex tends to be especially sparse Faster in primal if you update them all / for all 10
11 Primal Steepest Edge 11
12 Primal Steepest Edge Simplex step If enters, solution changes by Entering variable increases to Basic variables change by (where ) Objective is reduced by Steepness of step / reduction of objective per unit change in solution Main steepest-edge computations Choose largest / over all with 0 Update for Update for... 12
13 Primal Steepest Edge Updating / known after updating not known except for, but is known 2 Hard part is Solve Then compute as for each One extra solve and extra inner products 13
14 Dual Steepest Edge 14
15 Dual Steepest Edge 15
16 Dual Steepest Edge 16
17 Dual Steepest Edge Simplex step If constraint is relaxed, Slack variable increases to Variables change by (where ) Objective is reduced by Steepness of step / reduction of objective per unit change in solution Main steepest-edge computations Choose largest / over all with 0 Update for Update for... 17
18 Dual Steepest Edge Updating ( / known after updating not known except for, but is known 2 Hard part is Solve Then is for each One extra solve but no extra inner products 18
19 Bounded Variables 19
20 Bounded Variables Generalize 0to l State simplex methods for l, finite Extend to allow some l and/or Check that l 0, reduces to previous case Further improve the dual simplex method Take longer steps Adapt to degeneracy 20
21 Primal LP with Bounded Variables Minimize Subject to, l Basic variables, nonbasic variables Coefficient columns of basic variables form a nonsingular matrix Basic solution l for, for l Feasible basic solution l, 21
22 Dual LP with Bounded Variables 22
23 Dual LP with Bounded Variables 23
24 Dual LP with Bounded Variables 24
25 Dual LP with Bounded Variables Maximize Subject to : l What is : l?? Sum of concave piecewise-linear functions :l Slope of for 0 Slope of l for 0 Example for 0 l : :l l 25
26 Dual LP with Bounded Variables Maximize Subject to : l Binding constraints, nonbinding constraints Coefficient rows of binding constraints form a nonsingular matrix Vertex solution 0, for 0 :l l for 0 Always feasible! 26
27 Primal Simplex, Bounded Variables Given feasible basic solution and corresponding basis matrix Choose a nonbasic variable to enter solve select : 0 or select : 0 Choose a basic variable to leave solve ( ) or ( ) select Θ min Θ,Θ,Θ : : Θ l min l : Θ min : Θ l Update... 27
28 Dual Simplex, Bounded Variables Given feasible vertex solution and corresponding basis matrix Choose a binding constraint to leave solve l select : l or Choose a nonbinding constraint to enter solve (if l ) or (if ) select Φ min Φ,Φ ) : : Φ / min / : Φ / min / Update... :l l 28
29 In Principle, Bounds Can Be Infinite Some l and/or?? A basis may be infeasible 0but l 0but Minimize sum of infeasible variables to get feasible Replace each l by 1 Replace each by 1 Replace all finite bounds by 0 All l 0 and?? Then 0 Primal and dual algorithms reduce to their simpler forms 29
30 In Practice, Bounds Tend to be Finite Decisions are bounded Variables are bounded Slacks on inequality constraints are bounded Integer-valued decisions have small values Many are zero-one! Standard presolve routines compute bounds Compute bounds where none given by user Tighten bounds in multiple passes 30
31 Long Steps Standard iteration increases to Φ (if l ) or decreases to Φ (if ) moves to 0 Suppose increases/decreases further... moves past 0, but solution remains feasible Rate of improvement in objective degrades by l If increasing, objective slope changes from to l If decreasing, objective slope changes from l to If rate still positive, can continue until next, reaches 0 :l l 31
32 Long Step Iteration Replace single ratio test by a loop Set initial objective improvement rate to l (if l ) or (if ) Form set of all ratios that may be reached: :, 0 :, 0 Repeat for increasing Let l Let until 0 Equivalent to weighted selection In theory, faster than sorting In practice, a small part of iteration cost 32
33 Re-Optimization for MIPs Change bounds for some fractional, Increase l to, resulting in l Decrease to, resulting in Either way, binding constraint can leave Continue with dual simplex steps until optimal again Fix some fractional binary, Increase l to 1, resulting in l Decrease to l 0, resulting in Either way, binding constraint can leave Since now l, using long steps the constraint will never become binding again 33
34 Degeneracy Choosing a binding constraint to relax For 0,place For 0,place For 0??? Guess if you think is likely to increase Guess if you think is likely to decrease... can t be sure though until you choose solve l select : l or :l l 34
35 Degeneracy Choosing a nonbinding constraint to add Set initial objective improvement rate to l (if l ) or (if ) Collect all ratios that may be reached: :, 0 :, 0 Some of these ratios may be zero! For, = 0 and 0 For, = 0 and 0 Repeat for every 0 Let l Let while 0 If 0, iteration is degenerate If still 0, continue with nondegenerate iteration :l l 35
36 Degeneracy: Benefit of Long Steps For some 0, you guess wrong, but decreases, but increases As a result, you are overly optimistic about the objective improvement rate r Long-step ratio test corrects for your wrong guesses If the corrected 0then you can take a nondegenerate step after all None of this changes the optimality condition l for all 36
37 Ascendance of the Dual Simplex Fast inner products Dual steepest edge Bounded-variable extension Feasibility for finite bounds Long steps More nondegenerate steps 37
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