Constraint Branching and Disjunctive Cuts for Mixed Integer Programs

Transcription

1 Constraint Branching and Disunctive Cuts for Mixed Integer Programs Constraint Branching and Disunctive Cuts for Mixed Integer Programs Michael Perregaard Dash Optimization

2 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 2 Small Example min z s.t. 2x x, x + 2 2x, z Z + z 99 Optimal LP solution: z = Optimal MIP solution: z = Consider pure branch-and-bound. Will alternately branch on fractional x or x 2. Reuires exhaustive search of (x, x 2 ) = (,49.5), (.5,49), (,48.5),, (49.5, ) solutions to search. times more with new x 3. Alternatively, branch on x + x2 49 x + x

3 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 3 Branching from Disunctive Cuts Branching is imposing a disunction valid for all (feasible) integer solutions, but not the current LP solution. Disunctive cuts are derived from some base disunction and often a strengthening argument. Gomory s Mixed Integer cuts. Lift-and-Proect cuts. Reduce and Split cuts of Andersen, Cornuéols and Li (23). The strengthening of the cut can be transformed into a strengthening of the base disunction. Use the strengthened base disunction for branching.

4 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 4 Basic Mixed Integer Program We consider solving: min s. t. cx Ax x x = b Z for Solve using branch-and-bound. I Standard branching selects a single fractional variable x and imposes disunction x x x x Can we find a better disunction?

5 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 5 Disunctive Normal Form Example For constraint Q k = D x d = where e.g. x and x 2 are fractional, we can create a disunction x x x2 x + x2

6 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 6 where Example Q (, ) + d I I d Z Split Disunctions ( di xi d di xi d + for Q. If x, x 2 and x 3 are fractional binaries, we can consider the disunction ) ( x x ) ( x2 x2 ) ( x3 x3 ) Leads to 2 3 = 8 branches.

7 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 7 Basic Disunctive (Intersection) Cut Given disunction (in nonbasic space) where with d >, then Q dn xn d α N x N α d = max Q d is a valid ineuality that cuts off the LP solution =. x N

8 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 8 Strengthening Disunctions (Balas, Jeroslaw 98) Let f be the largest value for which d x f is valid for (MIP). Set h = ( d f ). Let m Z satisfies with Σ α α Q for I, Q, be any set of integers that m for all I. Then = = is a valid ineuality for (MIP) max{( d Q max{ d Q α N N x N + d N m } h I ) d } for for N \ I

9 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 9 Strengthening Disunctions [continued] Instead of strengthening cut, as in modify the disunction directly, as in () Basic disunctive cut from () identical to strengthened cut. I N d d I d h m d Q Q \ for } max{ for } ) max{( = α + = α I N I N I I I Q d x d x m h d \ \ ) ( + +

10 Constraint Branching and Disunctive Cuts for Mixed Integer Programs Strengthening Conunctions Given valid disunction for (MIP) Q ( di x d di x d + ) Let m Z for I, Q, be any set of integers. Then Q ( + ) d m x d ( d + m ) x d ) ( I I I I I I + is a valid disunction for (MIP) since integer. m I xi must be Gomory s Mixed Integer cuts and Lift-and-Proect cuts strengthens in nonbasics. Andersen, Cornuéols, Li cuts iteratively strengthens individual basics and all nonbasics.

11 Constraint Branching and Disunctive Cuts for Mixed Integer Programs General Branching Alternatives Ryan-Foster for Set Packing and Set Partitioning. B.A. Foster and D.M. Ryan (98). Specifically designed for Set Partitioning constraints: Basis Reduction H.W. Lenstra (983) Polynomial algorithm for solving integer programs for fixed number of variables. General Branching of Mehrotra, Owen (2) Tests each variable using LP reoptimization to determine best coefficient.

12 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 2 Even more Alternatives General Branching of Karamanov, Cornuéols (Monday) Branches on Gomory cut related disunction. Column Basis Reduction of Pataki (Thursday) Generalized Branching Methods of Mehrotra (Friday)

13 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 3 Small Examples obective

14 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 4 General branching + Branch on any linear disunction. General vs. - branching - Adds new constraints matrix size grows. - More difficult to get implications. - More basic integers less reduced cost tightening.? Heaps of choices - branching - Branch on - disunctions only + Changes bounds matrix size unchanged. + Easy to get implications (bound propagation). + Branched variables will be non-basic allows reduced cost tightening.? Easy to find best choice.

15 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 5 Evaluating a Disunction Work in space of nonbasic variables A B x B + A N x Measure the uality of a disunction N = b B B through that of the implied disunctive cut αx, with x B Q dn xn d α = max Q = { d } d A b A A N x N

16 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 6 Evaluating a Cut Andersen, Cornuéols, Li (23) suggests minimizing the L 2 -norm of cut coefficients for continuous variables. What about scaling and cost? Consider reduced costs. Cost to satisfy the cut by increasing non-basic variable x is at least c α. Make cut expensive to satisfy maximize c minimize α c. Since c α N x N can be zero, we estimate a cut by g( α) = α N + δ c α, or c N

17 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 7 Improving a Disunction - Nonbasics Express disunction in nonbasics x N Q Strengthened cut coefficients in nonbasics are α α Find optimal dbxb + dnxn d Q dn xn d = = max{( d Q max{ d Q m + d m } h ) d for for N I N \ for each independently easy. } I Note: For simple split disunction xk xk x optimal gives Gomory s Mixed Integer cut. m x k k

18 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 8 Improving a Disunction - Basics Same strengthening applies to basic variable x k Q mk h xk + dn xn d Use the row i of the simplex tableau in which x k is basic: x k + aix = bi N to re-express the disunction in nonbasics: Q ( dn mk hain) xn d mk hai Problem: Find optimal discrete amount m k h to add simplex tableau row i (without basic x k ) to each term of the disunction.

19 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 9 Procedure. Convert Xpress selected branching variable x k into a simple disunction 2. Apply Gomory-esue strengthening to coefficients of non-basics in D. 3. Are there more basic, integer variables to use for strengthening? If not, stop. 4. Select basic, integer variable x i. Calculate optimal continuous coefficient m i in D. Update D with the better of or. Repeat from 2. m i mi

20 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 2 Miplib 3 Test Set 65 instances, 5 with general integers Miplib instances, 5 with general integers H. Mittelmann s test set 63 instances, 6 with general integers 46 uniue instances, 3 with general integers.

21 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 2 Instances with General Integers Name Rows Columns Binaries Integers Int.Gap Name Rows Columns Binaries Integers Int.Gap arki msc98-ip e9 atlanta-ip e9 mzzv bell3a mzzv42z bell neos blend neos flugpl neos gen neos gesa neos gesa2_o noswot gesa net gesa3_o net_o gt roll manna rout momentum timtab momentum timtab Instances not suited for general integer branching.

22 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 22 Computational Settings Implemented in C using Xpress 25B optimizer library. Uses Xpress callbacks to override default branches with new constraint branches. No in-tree cutting. No heuristics. Best-first search. Run on a dual processor Opteron 246 system (2GHz, 4GB RAM, Linux OS).

23 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 23 Instance arki atlanta-ip bell3a bell5 blend2 dsbmip flugpl gesa2_o gesa2 gesa3_o gesa3 gt2 msc98-ip No Strengthening Time Nodes (Bound) (758565) (972878) Nonbasic Strengthening (82.88) Simple Strengthening Time 5 2 Nodes (Bound) (758295) (82.92) (972878) Instance mzzv mzzv42z neos neos6 neos2 neos7 net_o net roll3 timtab timtab2 Not finished in 8 seconds gen, manna8: solved on root (excluded). noswot: can t raise bound (excluded). atlanta-ip, dsbmip, msc98-ip, mzzv, mzzv42z, neos: very few branches on integers rout No Strengthening Time Nodes (Bound) (-2728) (434) (-46) (79934) (2453) (64457) (69572) Simple Strengthening Time Nodes (Bound) (-2728) (432) (-468) (73934) 35 3 (2456) (47) (57727) (6629)

24 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 24 Instance arki atlanta-ip bell3a bell5 blend2 dsbmip flugpl gesa2_o gesa2 gesa3_o gesa3 gt2 msc98-ip No Strengthening Time Nodes (Bound) (758565) (82.88) (972878) Full Strengthening Full Strengthening Time Nodes (Bound) (75852) (82.89) (972878) Instance mzzv mzzv42z neos neos6 neos2 neos7 net_o net roll3 rout timtab timtab2 No Strengthening Time Nodes (Bound) (-2728) (434) (-46) (79934) (2453) (64457) (69572) Simple Strengthening Time Nodes (Bound) (-2728) (432) (-468) (7378) (2459) (53) (552455) (64262) Not finished in 8 seconds. bell3a, bell5: Half the number of nodes of Nonbasic Strengthening. flugpl: reduced from 329 to 3 nodes.

25 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 25 Nonbasic Strengthening #Better #Worse Branching on Binaries No Strengthening Comparing results from Nonbasic Strengthening on all Binary/Integer branches against previous results. 4 8 Strengthening on integer branches 3 7 Full Strengthening #Better #Worse No Strengthening 7 7 Strengthening on integer branches 5 5 Comparing results from Full Strengthening on all Binary/Integer branches against previous results.

26 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 26 Reduced Cost Scaling of Cut Coefficients f = + c δ δ = (no scaling) δ = median reduced cost δ =. median reduced cost. #Best #Worst 2 3 6

27 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 27 Improvement in Cut Estimate Average improvement in cut estimate relative to initial disunction when applying either nonbasic improvement or full improvement atlanta-ip dsbmip neos mzzv42z msc98-ip mzzv neos7 blend2 gesa2_o timtab2 gesa2 timtab gt2 net noswot gesa3 net_o gesa3_o bell5 flugpl rout roll3 arki bell3a neos6 neos2 Nonbasic improvement Full improvement

28 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 28 Basic Improvement Coefficients Average optimal continuous coefficient for basic integer variables, excluding when zero is optimal net_o net mzzv roll3 mzzv42z neos6 atlanta-ip timtab2 blend2 timtab gesa3_o neos2 gt2 msc98-ip gesa3 rout neos gesa2_o gesa2 flugpl arki bell3a dsbmip bell5 neos7

29 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 29 Results on Full Test Set Using full strengthening on both binary and integer branches (37 instances). Strengthening? Both finished, least nodes One finished Both unfinished, best bound No Full 8 2 Instance bell5 bell3a mod8 gt2 flugpl mzzv42z l52lav lseu neos4 neos No strengthening Time Nodes Full strengthening Time Nodes Top with best performance when applying full strengthening

30 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 3 No Improvements L52LAV Full Improvements Nonbasic Improvements 476 Improv. mindeg maxdeg 476 None Nonbas Full Sum of min and max degradation over best 25 nodes.

31 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 3 Client Set Small cutting stock problems with general integers. Strengthening Name Rows Cols Root Obective Best Sol. None Full Full on cycles only d % (686) 7732 % (37) 7732 % (227) d % (62444) 75 % (35) 75 % (248) d % (93588) 3275 % (7) 3275 % (87)

32 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 32 Client Set 2 Lot sizing problems with general integers. Name Rows Cols Root Obective Strengthening None Non-basic Full C % (56) % (45) C % (7) % (2372) C % (3946) % (223) C % (45) % (45)

33 Constraint Branching and Disunctive Cuts for Mixed Integer Programs 33 Future Directions Select initial disunction independently of Xpress. Evaluation of disunctions. Xpress uses e.g. pseudo costs, strongbranch estimates and history values to select a branch candidate. How can this be carried over to general branching? Assimilate ideas from/compare against other general branching schemes. Basis reduction LP guided strengthening of disunction. IMA general branching presentations.. Efficiency (no exploitation of sparsity at the moment). Include most promising scheme in future release of Xpress?.