Cloud Branching. Timo Berthold. joint work with Domenico Salvagnin (Università degli Studi di Padova)
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1 Cloud Branching Timo Berthold Zuse Institute Berlin joint work with Domenico Salvagnin (Università degli Studi di Padova) DFG Research Center MATHEON Mathematics for key technologies 21/May/13, CPAIOR 2013, IBM T. J. Watson Research Center
2 MIP & Branching min s.t. c T x Ax b x Z I 0 R C 0 Mixed Integer Program: linear objective & constraints integer variables continuous variables Branching for MIP: based on LP relaxation fractional variables tries to improve dual bound Timo Berthold: Cloud Branching 2 / 17
3 MIP & Branching min s.t. c T x Ax b x Z I 0 R C 0 Mixed Integer Program: linear objective & constraints integer variables continuous variables Branching for MIP: based on LP relaxation fractional variables tries to improve dual bound x i xi x i xi for i I and xi / Z Timo Berthold: Cloud Branching 2 / 17
4 Branching rules in MIP Most infeasible branching often referred to as a simple, standard rule computationally as bad as random branching! Strong branching [ApplegateEtAl1995] solve LP relaxations for some candidates, choose best effective w.r.t. number of nodes, expensive w.r.t. time Pseudocost branching [BenichouEtAl1971] try to estimate LP values, based on history information effective, cheap, but weak in the beginning can be combined with strong branching Timo Berthold: Cloud Branching 3 / 17
![r.t. time Pseudocost branching [BenichouEtAl1971] try to estimate LP values, based on history information effective, cheap,](/docs-images/48/19055204/images/page_4.jpg "but weak in the beginning can be combined with strong branching Timo Berthold: Cloud Branching 3 / 17")
5 Branching rules in MIP Most infeasible branching often referred to as a simple, standard rule computationally as bad as random branching! Strong branching [ApplegateEtAl1995] solve LP relaxations for some candidates, choose best effective w.r.t. number of nodes, expensive w.r.t. time Pseudocost branching [BenichouEtAl1971] try to estimate LP values, based on history information effective, cheap, but weak in the beginning can be combined with strong branching Timo Berthold: Cloud Branching 3 / 17
![r.t. time Pseudocost branching [BenichouEtAl1971] try to estimate LP values, based on history information effective, cheap,](/docs-images/48/19055204/images/page_5.jpg "but weak in the beginning can be combined with strong branching Timo Berthold: Cloud Branching 3 / 17")
6 Degeneracy naïvely: 1 optimal solution, determined by n constraints Timo Berthold: Cloud Branching 4 / 17
7 Degeneracy naïvely: 1 optimal solution, determined by n constraints at a second thought: 1 optimal solution, k > n tight constraints Timo Berthold: Cloud Branching 4 / 17
8 Degeneracy naïvely: 1 optimal solution, determined by n constraints at a second thought: 1 optimal solution, k > n tight constraints but really: an optimal polyhedron Timo Berthold: Cloud Branching 4 / 17
9 Degeneracy naïvely: 1 optimal solution, determined by n constraints at a second thought: 1 optimal solution, k > n tight constraints but really: an optimal polyhedron Timo Berthold: Cloud Branching 4 / 17
10 Degeneracy naïvely: 1 optimal solution, determined by n constraints at a second thought: 1 optimal solution, k > n tight constraints but really: an optimal polyhedron Goal of this talk: branch on a set (a cloud) of solutions Timo Berthold: Cloud Branching 4 / 17
11 Two questions Goal of this talk: branch on a cloud of solutions 1. How do we get extra optimal solutions? 2. Why should that be a good idea anyway? Timo Berthold: Cloud Branching 5 / 17
12 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) Timo Berthold: Cloud Branching 6 / 17
13 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j x 1 = 0.4 x 2 = 0.55 Timo Berthold: Cloud Branching 6 / 17
![feasibility pump objective (pump&reduce [Achterberg2010]) (x, x)](/docs-images/48/19055204/images/page_13.jpg "= x j x j x 1 = 0.4 x 2 = 0.")
14 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = x 1 + x 2 x 1 = 0.4 x 2 = 0.55 Timo Berthold: Cloud Branching 6 / 17
![feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) =](/docs-images/48/19055204/images/page_14.jpg "x j x j = x 1 + x 2 x 1 = 0.4 x 2 = 0.")
15 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = x 1 + x 2 x 1 [0.4, 0.55] x 2 [0.55, 0.9] Timo Berthold: Cloud Branching 6 / 17
![feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j](/docs-images/48/19055204/images/page_15.jpg "x j = x 1 + x 2 x 1 [0.4, 0.55] x 2 [0.55, 0.")
16 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = +x 1 + x 2 x 1 [0.4, 0.55] x 2 [0.55, 0.9] Timo Berthold: Cloud Branching 6 / 17
![feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j](/docs-images/48/19055204/images/page_16.jpg "x j = +x 1 + x 2 x 1 [0.4, 0.55] x 2 [0.55, 0.")
17 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = +x 1 + x 2 x 1 [0.4, 0.9] x 2 [0.55, 1.0] Timo Berthold: Cloud Branching 6 / 17
![feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j](/docs-images/48/19055204/images/page_17.jpg "x j = +x 1 + x 2 x 1 [0.4, 0.9] x 2 [0.55, 1.")
18 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = +x 1 + x 2 x 1 [0.4, 0.9] x 2 [0.55, 1.0] intervals instead of values Timo Berthold: Cloud Branching 6 / 17
![objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = +x 1 + x 2 x 1 [0.](/docs-images/48/19055204/images/page_18.jpg "4, 0.9] x 2 [0.55, 1.")
19 Generating cloud points 1. How do we get extra optimal solutions? restrict LP to optimal face min/max each variable (OBBT) or feasibility pump objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = +x 1 + x 2 x 1 [0.4, 0.9] x 2 [0.55, 1.0] intervals instead of values stalling is cheap! Timo Berthold: Cloud Branching 6 / 17
![objective (pump&reduce [Achterberg2010]) (x, x) = x j x j = +x 1 + x 2 x 1 [0.4, 0.](/docs-images/48/19055204/images/page_19.jpg "9] x 2 [0.55, 1.0] intervals instead of values stalling is cheap!")
20 Exploiting cloud points (pseudocost) 2. Why should that be a good idea anyway? z = c T x z = c T x j + j x j x j x j x j x j x j pseudocost update ς + j = and ς xj x j = j xj x j pseudocost-based estimation + j = Ψ + j ( x j x j ) and j = Ψ j (x j x j ) Timo Berthold: Cloud Branching 7 / 17
21 Exploiting cloud points (pseudocost) 2. Why should that be a good idea anyway? z = c T x z = c T x j + j xj l x u j xj j j x j x j x j pseudocost update ς + j =... better: ς + xj x j = j xj u j pseudocost-based estimation + j = Ψ + j ( x j x j ) and j = Ψ j (x j x j ) Timo Berthold: Cloud Branching 7 / 17
22 Exploiting cloud points (pseudocost) 2. Why should that be a good idea anyway? z = c T x z = c T x j j + j + j xj l x u j xj j j xj l x u j xj j j pseudocost update ς + j =... better: ς + xj x j = j xj u j pseudocost-based estimation + j = Ψ + j ( x j x j )... better: + j = Ψ + j ( x j u j) Timo Berthold: Cloud Branching 7 / 17
23 Observation Lemma Let x be an optimal solution of the LP relaxation at a given branch-and-bound node and xj l j xj u j xj. Then 1. for fixed and, it holds that ς + j ς + j and ς j ς j, respectively; 2. for fixed Ψ + j and Ψ j, it holds that + j + j and j j, respectively. Timo Berthold: Cloud Branching 8 / 17
24 Exploiting cloud points (strong branch) 2. Why should that be a good idea anyway? Full strong branching: solves 2 #frac. var s many LPs uses product of improvement values as branching score improvement on both sides better than on one Timo Berthold: Cloud Branching 9 / 17
25 Exploiting cloud points (strong branch) 2. Why should that be a good idea anyway? Full strong branching: solves 2 #frac. var s many LPs uses product of improvement values as branching score improvement on both sides better than on one Benefit of cloud intervals: frac. var. gets integral in cloud point one LP spared cloud branching acts as a filter new frac. var. s new candidates (one side known) Timo Berthold: Cloud Branching 9 / 17
26 Cloud strong branching algorithm Idea: Use 3-partition F 2, F 1, F 0 of branching candidates if strict improvements in both directions for F 2, disregard F 1 F 0 if strict improvement in one direction for F 2 F 1, disregard F 0 Timo Berthold: Cloud Branching 10 / 17
27 Cloud strong branching algorithm Idea: Use 3-partition F 2, F 1, F 0 of branching candidates if strict improvements in both directions for F 2, disregard F 1 F 0 if strict improvement in one direction for F 2 F 1, disregard F 0 alternatively: Only use F 2 Timo Berthold: Cloud Branching 10 / 17
28 Cloud strong branching algorithm Idea: Use 3-partition F 2, F 1, F 0 of branching candidates if strict improvements in both directions for F 2, disregard F 1 F 0 if strict improvement in one direction for F 2 F 1, disregard F 0 alternatively: Only use F 2 stop pump&reduce procedure when new cloud point does not imply new integral bound Timo Berthold: Cloud Branching 10 / 17
29 Cloud strong branching algorithm Idea: Use 3-partition F 2, F 1, F 0 of branching candidates if strict improvements in both directions for F 2, disregard F 1 F 0 if strict improvement in one direction for F 2 F 1, disregard F 0 alternatively: Only use F 2 stop pump&reduce procedure when new cloud point does not imply new integral bound Note: In our experiments, we do not use cloud points for anything else (heuristics, cuts) Timo Berthold: Cloud Branching 10 / 17
30 Implementation into SCIP SCIP: Solving Constraint Integer Programs standalone solver / branch-cut-and-price-framework modular structure via plugins free for academic use: very fast non-commercial MIP and MINLP solver time in seconds x 3.94x non-commercial commercial GLPK 4.47 lpsolve CBC SCIP CLP x SCIP SoPlex SCIP Cplex x x 0.77x Xpress x Cplex x 0.17x 0 Gurobi solved (of 87 instances) results by H. Mittelmann (10/Jan/2013) Timo Berthold: Cloud Branching 11 / 17
31 SCIP 3.0 SCIP: Solving Constraint Integer Programs better support of MINLP new presolvers and propagators AMPL and MATLAB interface (beta) first releases of GCG and UG time in seconds solved (of 87) 1.6x 5.2x 4.0x 3.0x 3.0x 2.5x 2.0x 1.3x 1.1x 1.0x sip 1.2 SoPlex 1.2.1a SCIP 0.7 SoPlex SCIP 0.80 SoPlex SCIP 0.90 SoPlex SCIP 1.00 SoPlex SCIP 1.1 SoPlex SCIP 1.2 SoPlex SCIP 2.0 SoPlex SCIP 2.1 SoPlex SCIP 3.0 SoPlex Timo Berthold: Cloud Branching 12 / 17
32 Test sets MMM MIPLIB 3.0, MIPLIB 2003, MIPLIB2010 industry and academics 168 instances from diverse applications huge collection of 350 instances many combinatorial ones mainly collected from NEOS server Timo Berthold: Cloud Branching 13 / 17
33 Computational results cloud statistics Test set %Succ Pts LPs %Sav MMM applicable on some MMM, but many instances only few cloud points on average significants amount of LPs saved (if affected) Timo Berthold: Cloud Branching 14 / 17
34 Computational results strong branch cloud branch Test set Nodes Time (s) Nodes Time (s) MMM little less nodes 5.5% faster on MMM (few affected instances) 30 % faster on COR@L Timo Berthold: Cloud Branching 15 / 17
35 Wrap up Conclusion: Cloud branching... exploits knowledge of alternative relaxation optima can help to improve pseudocost predictions makes full strong branching up to 30% faster Outlook pseudocost, reliability, hybrid branching cloud points from alternative relaxations (MINLP!) nonchimerical + cloud + propagation =? Timo Berthold: Cloud Branching 16 / 17
36 Cloud Branching Timo Berthold Zuse Institute Berlin joint work with Domenico Salvagnin (Università degli Studi di Padova) DFG Research Center MATHEON Mathematics for key technologies 21/May/13, CPAIOR 2013, IBM T. J. Watson Research Center
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