IPO Investigating Polyhedra by Oracles

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2 IPO Investigating Polyhedra by Oracles Matthias Walter Otto-von-Guericke Universität Magdeburg Joint work with Volker Kaibel (OVGU) Aussois Combinatorial Optimization Workshop 2016

3 PORTA & Polymake Approach max c, x st x Z E + x(δ(v)) = 1 v V Recognized class of facets: x(δ(s)) 1 S V, S odd Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

4 PORTA & Polymake Approach max c, x st x Z E + x(δ(v)) = 1 v V All extr points: Recognized class of facets: x(δ(s)) 1 S V, S odd Enumeration Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

5 PORTA & Polymake Approach max c, x st x Z E + x(δ(v)) = 1 v V All extr points: All equations: ( )x = ( )x = Recognized class of facets: x(δ(s)) 1 S V, S odd Enumeration Convex hull tool (eg, double-description, lrs beneath&beyond, ) All facets: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

6 PORTA & Polymake Approach max c, x st x Z E + x(δ(v)) = 1 v V All extr points: All equations: ( )x = ( )x = Recognized class of facets: x(δ(s)) 1 S V, S odd Enumeration Convex hull tool (eg, double-description, lrs beneath&beyond, ) All facets: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

7 IPO Approach max c, x st x Z E + x(δ(v)) = 1 v V Recognized class of facets: x(δ(s)) 1 S V, S odd Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

8 IPO Approach max c, x st x Z E + x(δ(v)) = 1 v V All equations: ( )x = ( )x = Recognized class of facets: x(δ(s)) 1 S V, S odd Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

9 IPO Approach max c, x st x Z E + x(δ(v)) = 1 v V All equations: ( )x = ( )x = Recognized class of facets: x(δ(s)) 1 S V, S odd Only some useful facets: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

10 IPO Approach max c, x st x Z E + x(δ(v)) = 1 v V MIP solver All equations: ( )x = ( )x = Recognized class of facets: x(δ(s)) 1 S V, S odd Only some useful facets: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

11 IPO Approach max c, x st x Z E + x(δ(v)) = 1 v V MIP solver All equations: ( )x = ( )x = Recognized class of facets: x(δ(s)) 1 S V, S odd Only some useful facets: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

12 Facets Finding Facets Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

13 Quadratic Matching Polytopes Consider the quadratic matching polytope of order n with one quadratic term: } P n := convhull {(χ(m), y) {0, 1} En +1 : M matching in K n, y = x 1,2x 3,4 Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

14 Quadratic Matching Polytopes Consider the quadratic matching polytope of order n with one quadratic term: } P n := convhull {(χ(m), y) {0, 1} En +1 : M matching in K n, y = x 1,2x 3,4 Hupp, Klein & Liers, 15 obtained a bunch of facets: x(δ(v)) 1 for all v V n x e 0 for all e E n y x 1,2 and y x 3,4 (Note that y x 1,2 + x 3,4 1 is no facet) Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

15 Quadratic Matching Polytopes Consider the quadratic matching polytope of order n with one quadratic term: } P n := convhull {(χ(m), y) {0, 1} En +1 : M matching in K n, y = x 1,2x 3,4 Hupp, Klein & Liers, 15 obtained a bunch of facets: x(δ(v)) 1 for all v V n x e 0 for all e E n y x 1,2 and y x 3,4 x(e[s]) + y S 1 2 for certain odd S (Note that y x 1,2 + x 3,4 1 is no facet) x(e[s]) S 1 2 for certain odd S x(e[s]) + x(e[s \ {1, 2}]) + x 3,4 y S 2 for certain odd S x(e[s]) + x 2,a + x 3,a + x 4,a + y S 2 for certain even S and nodes a x 1,2 + x 1,a + x 2,a + x(e[s]) + x 3,4 + x 3,b + x 4,b y S 2 even S and certain nodes a, b + 1 for certain Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

16 Some are Missing! Excerpt from their paper: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

17 Some are Missing! Excerpt from their paper: Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

18 QMP: An IP Model param n := 6; set V := { 1 to n }; set E := { <u,v> in V*V with u < v }; set F := { <1,2>,<3,4>,<1,5>,<2,5>,<3,6>,<4,6>,<1,3>,<2,4> }; var x[e] binary; var y binary; maximize weights: 10*x[1,2] + 10*x[3,4] + 2*x[1,5] + 2*x[2,5] + 2*x[3,6] + 2*x[4,6] + 4*x[1,3] + 4*x[2,4] -10*y + sum <u,v> in E-F: -1000*x[u,v]; subto degree: forall <w> in V: (sum <u,v> in E with u == w or v == w: x[u,v]) <= 1; subto product1: y <= x[1,2]; subto product2: y <= x[3,4]; subto product3: y >= x[1,2] + x[3,4] - 1; Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

19 QMP: Running IPO % /ipo-facets product-matching-missingzpl Dimension: 16 Found a new facet: x#1#2 + x#1#3 + x#1#4 + x#3#4 - y <= 1, certified by 16 points and 0 rays Found a new facet: x#1#2 + x#1#4 + x#2#4 + x#3#4 - y <= 1, certified by 16 points and 0 rays Found a new facet: x#1#2 + x#1#5 + x#2#5 <= 1, certified by 16 points and 0 rays Found a new facet: x#3#4 + x#3#6 + x#4#6 <= 1, certified by 16 points and 0 rays Found a new facet: x#1#2 + x#1#3 + x#1#4 + x#2#3 + x#2#4 + 2*x#3#4 + x#3#6 + x#4#6 - y <= 2, certified by 16 points and 0 rays Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

20 QMP: Running IPO % /ipo-facets product-matching-missingzpl Dimension: 16 Found a new facet: x#1#2 + x#1#3 + x#1#4 + x#3#4 - y <= 1, certified by 16 points and 0 rays Found a new facet: x#1#2 + x#1#4 + x#2#4 + x#3#4 - y <= 1, certified by 16 points and 0 rays Found a new facet: x#1#2 + x#1#5 + x#2#5 <= 1, certified by 16 points and 0 rays Found a new facet: x#3#4 + x#3#6 + x#4#6 <= 1, certified by 16 points and 0 rays Found a new facet: x#1#2 + x#1#3 + x#1#4 + x#2#3 + x#2#4 + 2*x#3#4 + x#3#6 + x#4#6 - y <= 2, certified by 16 points and 0 rays Last facet is not of the previous types! Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

21 Adjacency of Vertices Checking Adjacency of Vertices Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

22 TSP Polytopes Oracles Oracle: concorde (famous TSP solver) Heuristic: nearest neighbor plus 2-opt, searching once from each node Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

23 TSP Polytopes Oracles Oracle: concorde (famous TSP solver) Heuristic: nearest neighbor plus 2-opt, searching once from each node Results for 10, 000 random tests: Nodes Adjacent Time/pair LP Heuristics Oracles Cache Tours Vertices % 03 s 05 % 01 % 979 % 01 % % 04 s 07 % 01 % 975 % 01 % % 06 s 11 % 01 % 961 % 05 % % 08 s 15 % 01 % 931 % 25 % 1, % 10 s 20 % 02 % 861 % 87 % 5, % 15 s 23 % 02 % 775 % 173 % 15, % 21 s 27 % 02 % 674 % 269 % 33, % 30 s 38 % 03 % 542 % 388 % 66, % 49 s 51 % 03 % 396 % 520 % 125, % 101 s 77 % 03 % 229 % 658 % 232, % 243 s 129 % 02 % 110 % 716 % 406, Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

24 Affine Hull Computing Affine Hulls Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

25 Oracles for Large MIPs Oracles Oracle: SCIP-300-ex Heuristic: SCIP-311 with postprocessing Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

26 Oracles for Large MIPs Oracles Oracle: SCIP-300-ex Heuristic: SCIP-311 with postprocessing Postprocessing of solutions Let I [n] be the set of integral variables 1 For x Q n, obtain x from x by rounding x i for all i I 2 Compute optimal choice for x [n]\i using an exact LP solver, eg, SoPlex Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

27 Constraint Dimensions < 0 % [40 %, 60 %) facets [0 %, 20 %) [60 %, 80 %) equations [20 %, 40 %) [80 %, 100 %) < 0 % [40 %, 60 %) facets [0 %, 20 %) [60 %, 80 %) equations [20 %, 40 %) [80 %, 100 %) Original Instances bell3b bell5 bm23 cracpb1 dcmulti egout flugpl misc01 misc02 misc03 misc05 misc07 mod013 p0033 p0040 p0291 pipex rgn sample2 sentoy stein15 stein27 stein45 stein9 vpm1 Presolved Instances bell3b bell5 bm23 cracpb1 dcmulti egout flugpl misc01 misc02 misc03 misc05 misc07 mod013 p0033 p0040 p0291 pipex rgn sample2 sentoy stein15 stein27 stein45 stein9 vpm Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

28 Summary Get it at Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

29 Summary Get it at Functionality Compute affine hull & dimension Separate point by facet Find facets helpful for objective Check adjacency of vertices Check if point is vertex Oracles & Heuristics SCIP Own code (C++ interface) External code (Python interface): SCIP-ex, concorde, etc Projections & faces of other oracles Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15

30 Summary Get it at Functionality Compute affine hull & dimension Separate point by facet Find facets helpful for objective Check adjacency of vertices Check if point is vertex Oracles & Heuristics SCIP Own code (C++ interface) External code (Python interface): SCIP-ex, concorde, etc Projections & faces of other oracles release party: bar after dinner Matthias Walter IPO Investigating Polyhedra by Oracles Aussois / 15