Extending SCIP for solving mixed-integer nonlinear programs

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1 Extending SCIP for solving mixed-integer nonlinear programs Stefan Vigerske Humboldt-Universität zu Berlin DFG Research Center MATHEON Mathematics for key technologies Spring Workshop on Computational Issues in Mixed Integer Nonlinear Programming

2 MINLP project in Berlin: cooperation of Humboldt University (Römisch, V.) and Zuse Institute (Grötschel, Bley, Gleixner, Koch, Pfetsch) started October 2008 work in progress joint work with T. Gellermann (TU Darmstadt), A. Gleixner (ZIB), T. Koch (ZIB), A. Neumaier (Vienna), M. Pfetsch (TU Braunschweig) Introduction Extending SCIP for solving mixed-integer nonlinear programs 2 / 31

3 MINLP project in Berlin: cooperation of Humboldt University (Römisch, V.) and Zuse Institute (Grötschel, Bley, Gleixner, Koch, Pfetsch) started October 2008 work in progress joint work with T. Gellermann (TU Darmstadt), A. Gleixner (ZIB), T. Koch (ZIB), A. Neumaier (Vienna), M. Pfetsch (TU Braunschweig) Introduction Aim: teach Constraint-Integer-Programming solver SCIP how to handle nonlinear constraints LP-based branch-and-cut Extending SCIP for solving mixed-integer nonlinear programs 2 / 31

4 MINLP project in Berlin: cooperation of Humboldt University (Römisch, V.) and Zuse Institute (Grötschel, Bley, Gleixner, Koch, Pfetsch) started October 2008 work in progress joint work with T. Gellermann (TU Darmstadt), A. Gleixner (ZIB), T. Koch (ZIB), A. Neumaier (Vienna), M. Pfetsch (TU Braunschweig) Introduction Aim: teach Constraint-Integer-Programming solver SCIP how to handle nonlinear constraints LP-based branch-and-cut start with quadratic problems extend step-by-step to other types of constraints Extending SCIP for solving mixed-integer nonlinear programs 2 / 31

5 hybrid estim estimate dfs spx qso restart dfs bfs xprs none implics dualfix bound shift msk intto binary Node selector default default LP probing Presolver cpx clp trivial Event Display default rounding rins pscost diving rootsol diving rens Impli cations Dialog simple rounding shifting oneopt fix cnf octane cip lp Tree Cutpool veclen diving ccg mps objpscost diving Reader Heuristic mutation opb zpl Conflict actcons diving local branching ppm sos Pricer rlp coef diving sol int shifting linesearch diving cross over dins cmir clique fixand infer guided diving flow cover Propa gator intdiving Variable feaspump gomory pseudo obj Relaxer fracdiving Separator implied bounds zero half root redcost setppc or relps cost sos1 logicor intobj strong cg random mcf redcost sos2 linear Branch var bound and Constraint Handler knap sack allfull strong pscost full strong mostinf bound xor disjunc. integral in ference leastinf count sols indi cator SCIP is a Branch-Cut-Price framework for Solving Constraint Integer Programs SCIP developed by T. Achterberg, T. Berthold, G. Gamrath, S. Heinz, T. Koch, A. Martin, M. Pfetsch, C. Raack, R. Waniek, M. Winkler, K. Wolter SCIP Extending SCIP for solving mixed-integer nonlinear programs 3 / 31

6 hybrid estim estimate dfs spx qso restart dfs bfs xprs none implics dualfix bound shift msk intto binary Node selector default default LP probing Presolver cpx clp trivial Event Display default rounding rins pscost diving rootsol diving rens Impli cations Dialog simple rounding shifting oneopt fix cnf octane cip lp Tree Cutpool veclen diving ccg mps objpscost diving Reader Heuristic mutation opb zpl Conflict actcons diving local branching ppm sos Pricer rlp coef diving sol int shifting linesearch diving cross over dins cmir clique fixand infer guided diving flow cover Propa gator intdiving Variable feaspump gomory pseudo obj Relaxer fracdiving Separator implied bounds zero half root redcost setppc or relps cost sos1 logicor intobj strong cg random mcf redcost sos2 linear Branch var bound and Constraint Handler knap sack allfull strong pscost full strong mostinf bound xor disjunc. integral in ference leastinf count sols indi cator SCIP is a Branch-Cut-Price framework for Solving Constraint Integer Programs SCIP developed by T. Achterberg, T. Berthold, G. Gamrath, S. Heinz, T. Koch, A. Martin, M. Pfetsch, C. Raack, R. Waniek, M. Winkler, K. Wolter constraint oriented everything is a plugin SCIP Extending SCIP for solving mixed-integer nonlinear programs 3 / 31

7 hybrid estim estimate dfs spx qso restart dfs bfs xprs none implics dualfix bound shift msk intto binary Node selector default default LP probing Presolver cpx clp trivial Event Display default rounding rins pscost diving rootsol diving rens Impli cations Dialog simple rounding shifting oneopt fix cnf octane cip lp Tree Cutpool veclen diving ccg mps objpscost diving Reader Heuristic mutation opb zpl Conflict actcons diving local branching ppm sos Pricer rlp coef diving sol int shifting linesearch diving cross over dins cmir clique fixand infer guided diving flow cover Propa gator intdiving Variable feaspump gomory pseudo obj Relaxer fracdiving Separator implied bounds zero half root redcost setppc or relps cost sos1 logicor intobj strong cg random mcf redcost sos2 linear Branch var bound and Constraint Handler knap sack allfull strong pscost full strong mostinf bound xor disjunc. integral in ference leastinf count sols indi cator SCIP is a Branch-Cut-Price framework for Solving Constraint Integer Programs SCIP developed by T. Achterberg, T. Berthold, G. Gamrath, S. Heinz, T. Koch, A. Martin, M. Pfetsch, C. Raack, R. Waniek, M. Winkler, K. Wolter constraint oriented everything is a plugin can be used as very efficient MIP solver strong preprocessing, branching rules, domain propagators, many heuristics and separators SCIP Extending SCIP for solving mixed-integer nonlinear programs 3 / 31

8 hybrid estim estimate dfs spx qso restart dfs bfs xprs none implics dualfix bound shift msk intto binary Node selector default default LP probing Presolver cpx clp trivial Event Display default rounding rins pscost diving rootsol diving rens Impli cations Dialog simple rounding shifting oneopt fix cnf octane cip lp Tree Cutpool veclen diving ccg mps objpscost diving Reader Heuristic mutation opb zpl Conflict actcons diving local branching ppm sos Pricer rlp coef diving sol int shifting linesearch diving cross over dins cmir clique fixand infer guided diving flow cover Propa gator intdiving Variable feaspump gomory pseudo obj Relaxer fracdiving Separator implied bounds zero half root redcost setppc or relps cost sos1 logicor intobj strong cg random mcf redcost sos2 linear Branch var bound and Constraint Handler knap sack allfull strong pscost full strong mostinf bound xor disjunc. integral in ference leastinf count sols indi cator SCIP is a Branch-Cut-Price framework for Solving Constraint Integer Programs SCIP developed by T. Achterberg, T. Berthold, G. Gamrath, S. Heinz, T. Koch, A. Martin, M. Pfetsch, C. Raack, R. Waniek, M. Winkler, K. Wolter constraint oriented everything is a plugin can be used as very efficient MIP solver strong preprocessing, branching rules, domain propagators, many heuristics and separators free for academic use SCIP Extending SCIP for solving mixed-integer nonlinear programs 3 / 31

9 Overview 1 Introduction 2 Quadratic Constraints 3 Nonlinear Constraints (MINLP) 4 Heuristic 5 Preliminary Numerical Results 6 Latest Work Extending SCIP for solving mixed-integer nonlinear programs 4 / 31

10 Overview 1 Introduction 2 Quadratic Constraints 3 Nonlinear Constraints (MINLP) 4 Heuristic 5 Preliminary Numerical Results 6 Latest Work Extending SCIP for solving mixed-integer nonlinear programs 5 / 31

11 Quadratic Constraints: Presolve Presolve: l x Ax + b x u upgrade to linear constraint if possible, substitute {0, 1} 2 by {0, 1},... Extending SCIP for solving mixed-integer nonlinear programs 6 / 31

12 Quadratic Constraints: Presolve l x Ax + b x u Presolve: upgrade to linear constraint if possible, substitute {0, 1} 2 by {0, 1},... find block structure, i.e., partition (J k ) k of {1,..., n} s.t.: l k x J k A k x Jk + b x u and disaggregate l k z k + b x u, z k = x J k A k x Jk Extending SCIP for solving mixed-integer nonlinear programs 6 / 31

13 Quadratic Constraints: Presolve l x Ax + b x u Presolve: upgrade to linear constraint if possible, substitute {0, 1} 2 by {0, 1},... find block structure, i.e., partition (J k ) k of {1,..., n} s.t.: l k x J k A k x Jk + b x u and disaggregate l k z k + b x u, z k = x J k A k x Jk domain reduction Extending SCIP for solving mixed-integer nonlinear programs 6 / 31

14 Quadratic Constraints: Presolve l x Ax + b x u Presolve: upgrade to linear constraint if possible, substitute {0, 1} 2 by {0, 1},... find block structure, i.e., partition (J k ) k of {1,..., n} s.t.: l k x J k A k x Jk + b x u and disaggregate l k z k + b x u, z k = x J k A k x Jk domain reduction compute min/max eigenvalues for each A k Extending SCIP for solving mixed-integer nonlinear programs 6 / 31

15 Quadratic Constraints: Separation xj k A k x Jk + b x u Separation: try to cut off LP solution by a linear cut k Extending SCIP for solving mixed-integer nonlinear programs 7 / 31

16 Quadratic Constraints: Separation xj k A k x Jk + b x u Separation: try to cut off LP solution by a linear cut linearize if convex (min. eigenvalue of A k 0) k Extending SCIP for solving mixed-integer nonlinear programs 7 / 31

17 Quadratic Constraints: Separation xj k A k x Jk + b x u Separation: try to cut off LP solution by a linear cut linearize if convex (min. eigenvalue of A k 0) k apply McCormick for each bilinear term, x i x j x L i x j + x L j x i x L i x L j x i x j x U i x j + x U j x i x U i x U j secant underestimator for concave terms ( x 2 ) Extending SCIP for solving mixed-integer nonlinear programs 7 / 31

18 Quadratic Constraints: Propagation Univariate case: [Domes and Neumaier 2008] ax 2 + bx [l, u] Extending SCIP for solving mixed-integer nonlinear programs 8 / 31

19 Quadratic Constraints: Propagation Univariate case: [Domes and Neumaier 2008] ax 2 + bx [l, u] forward propagation: compute [ l, ū] := {a x 2 + b x : x [x L, x U ]} [l, u] backward propagation: compute {x : a x 2 + b x [ l, ū]} Extending SCIP for solving mixed-integer nonlinear programs 8 / 31

20 Quadratic Constraints: Propagation Separable case: [Domes and Neumaier 2008] a k xk 2 + b kx k [l, u] k Extending SCIP for solving mixed-integer nonlinear programs 8 / 31

21 Quadratic Constraints: Propagation Separable case: [Domes and Neumaier 2008] a k xk 2 + b kx k [l, u] k forward propagation: compute [l k, u k ] := {a k xk 2 + b k x k : x k [xk L, x k U ]} backward propagation: compute {x k : a k xk 2 + b k x k [l, u] j, u j ]} j k[l Extending SCIP for solving mixed-integer nonlinear programs 8 / 31

22 Quadratic Constraints: Propagation Non-separable case: [Domes and Neumaier 2008] a k,k xk 2 + (b k + a k,j x j )x k [l, u] k j:j k Extending SCIP for solving mixed-integer nonlinear programs 8 / 31

23 Quadratic Constraints: Propagation Non-separable case: [Domes and Neumaier 2008] a k,k xk 2 + (b k + a k,j x j )x k [l, u] k j:j k forward propagation: compute [l k, u k ] := {a k,k xk 2 +(b k+ a k,j [xj L, xj U ]) x k : x k [xk L, x k U ]} j:j k backward propagation: compute {x k : a k,k xk 2 +(b k+ a k,j [xj L, xj U ]) x k [l, u] [l k, u k ]} j:j k j:j k Extending SCIP for solving mixed-integer nonlinear programs 8 / 31

24 Overview 1 Introduction 2 Quadratic Constraints 3 Nonlinear Constraints (MINLP) 4 Heuristic 5 Preliminary Numerical Results 6 Latest Work Extending SCIP for solving mixed-integer nonlinear programs 9 / 31

25 Nonlinear Constraints: Presolve Presolve: Given constraint h(x) [l, u], h C 2 (R n, R) Extending SCIP for solving mixed-integer nonlinear programs 10 / 31

26 Nonlinear Constraints: Presolve Presolve: Given constraint h(x) [l, u], h C 2 (R n, R) upgrade to linear or quadratic constraint, if possible Extending SCIP for solving mixed-integer nonlinear programs 10 / 31

27 Nonlinear Constraints: Presolve Presolve: Given constraint h(x) [l, u], h C 2 (R n, R) upgrade to linear or quadratic constraint, if possible find block structure and quadratic terms: h(x) = b x + k x Q k A k x Qk + r g r (x Nr ) with disjoint subsets Q k and N r of {1,..., n} disaggregate as h(x) = b x + k z k + r z r, z k = x Q k A k x Qk, z r = g r (x Nr ) Extending SCIP for solving mixed-integer nonlinear programs 10 / 31

28 Nonlinear Constraints: Presolve Presolve: Given constraint h(x) [l, u], h C 2 (R n, R) upgrade to linear or quadratic constraint, if possible find block structure and quadratic terms: h(x) = b x + k x Q k A k x Qk + r g r (x Nr ) with disjoint subsets Q k and N r of {1,..., n} disaggregate as h(x) = b x + k z k + r z r, z k = x Q k A k x Qk, z r = g r (x Nr ) Check probable convexity/concavity of each g r : compute sign of eigenvalues of Hessian 2 g r in sample points Extending SCIP for solving mixed-integer nonlinear programs 10 / 31

29 Nonlinear Constraints: Separation Separation (overview): Given constraint g(x) 0, g C 2 (R n, R). Solution ˆx of LP relaxation such that g(ˆx) > 0. Extending SCIP for solving mixed-integer nonlinear programs 11 / 31

30 Nonlinear Constraints: Separation Separation (overview): Given constraint g(x) 0, g C 2 (R n, R). Solution ˆx of LP relaxation such that g(ˆx) > 0. If g( ) is convex, linearize at ˆx: g(ˆx) + g(ˆx)(x ˆx) 0 Extending SCIP for solving mixed-integer nonlinear programs 11 / 31

31 Nonlinear Constraints: Separation Separation (overview): Given constraint g(x) 0, g C 2 (R n, R). Solution ˆx of LP relaxation such that g(ˆx) > 0. If g( ) is convex, linearize at ˆx: g(ˆx) + g(ˆx)(x ˆx) 0 If g( ) is not convex, generate interval gradient cut: g( x) + d L i (x i x i ) + d U i (x i x i ) 0 i: x i =x L i i: x i =x U i Extending SCIP for solving mixed-integer nonlinear programs 11 / 31

32 Nonlinear Constraints: Separation Separation (overview): Given constraint g(x) 0, g C 2 (R n, R). Solution ˆx of LP relaxation such that g(ˆx) > 0. If g( ) is convex, linearize at ˆx: g(ˆx) + g(ˆx)(x ˆx) 0 If g( ) is not convex, generate interval gradient cut: g( x) + d L i (x i x i ) + d U i (x i x i ) 0 i: x i =x L i i: x i =x U i If ˆx is not cut off, generate quadratic cut and add as new quadratic constraint: x Ax + b x + c 0 Extending SCIP for solving mixed-integer nonlinear programs 11 / 31

33 Nonlinear Constraints: Interval Gradient Cuts Given function g(x) and box [x L, x U ]. Compute interval gradient (using CppAD) [d L, d U ] = g([x L, x U ]), i.e., g(x) [d L, d U ] for all x [x L, x U ]. Extending SCIP for solving mixed-integer nonlinear programs 12 / 31

34 Nonlinear Constraints: Interval Gradient Cuts Given function g(x) and box [x L, x U ]. Compute interval gradient (using CppAD) [d L, d U ] = g([x L, x U ]), i.e., g(x) [d L, d U ] for all x [x L, x U ]. Let ˆx [x L, x U ]. Then g(ˆx) + min d [d L,d U ] d (x ˆx) g(x) Extending SCIP for solving mixed-integer nonlinear programs 12 / 31

35 Nonlinear Constraints: Interval Gradient Cuts Given function g(x) and box [x L, x U ]. Compute interval gradient (using CppAD) [d L, d U ] = g([x L, x U ]), i.e., g(x) [d L, d U ] for all x [x L, x U ]. Let ˆx [x L, x U ]. Then g(ˆx) + i:x i ˆx i d L i (x i ˆx i ) + di U i:x i <ˆx i (x i ˆx i ) g(x) Extending SCIP for solving mixed-integer nonlinear programs 12 / 31

36 Nonlinear Constraints: Interval Gradient Cuts Given function g(x) and box [x L, x U ]. Compute interval gradient (using CppAD) [d L, d U ] = g([x L, x U ]), i.e., g(x) [d L, d U ] for all x [x L, x U ]. Let ˆx [x L, x U ]. Then g(ˆx) + i:x i ˆx i d L i (x i ˆx i ) + di U i:x i <ˆx i Simple version (implemented): move ˆx to closest vertex of box (ˆx i {x L i, x U i }) g(ˆx) + d L i (x i ˆx i ) + di U (x i ˆx i ) g(x) (x i ˆx i ) g(x) i:ˆx i =x L i i:ˆx i =x U i Extending SCIP for solving mixed-integer nonlinear programs 12 / 31

37 Nonlinear Constraints: Interval Gradient Cuts Given function g(x) and box [x L, x U ]. Compute interval gradient (using CppAD) [d L, d U ] = g([x L, x U ]), i.e., g(x) [d L, d U ] for all x [x L, x U ]. Let ˆx [x L, x U ]. Then g(ˆx) + i:x i ˆx i d L i (x i ˆx i ) + di U i:x i <ˆx i Simple version (implemented): move ˆx to closest vertex of box (ˆx i {x L i, x U i }) g(ˆx) + d L i (x i ˆx i ) + di U (x i ˆx i ) g(x) (x i ˆx i ) g(x) i:ˆx i =x L i i:ˆx i =x U i fast, exact, weak, improves by branching Extending SCIP for solving mixed-integer nonlinear programs 12 / 31

38 Nonlinear Constraints: Separation by Quadratic Cuts g(x) 0, g C 2 ([x L, x U ], R), ˆx such that g(ˆx) > 0 Extending SCIP for solving mixed-integer nonlinear programs 13 / 31

39 Nonlinear Constraints: Separation by Quadratic Cuts g(x) 0, g C 2 ([x L, x U ], R), ˆx such that g(ˆx) > 0 Aim: Find q(x) = x T Ax + b T x + c that solves min A,b,c x [x L,x U ] g(x) q(x)dx such that q(x) g(x), for all x [x L, x U ], q(ˆx) = g(ˆx). Extending SCIP for solving mixed-integer nonlinear programs 13 / 31

40 Nonlinear Constraints: Separation by Quadratic Cuts g(x) 0, g C 2 ([x L, x U ], R), ˆx such that g(ˆx) > 0 Approach: Compute by solving min A,b,c q(x) = x T Ax + b T x + c g(x) q(x)dx x S such that q(x) g(x), for all x S, for a sample set S [ x L, x U]. Improve sample set adaptively. q(ˆx) = g(ˆx), Extending SCIP for solving mixed-integer nonlinear programs 13 / 31

41 Computation of Quadratic Cuts initial choice: S = vert([x L, x U ]) {random points} f(x) x^ Extending SCIP for solving mixed-integer nonlinear programs 14 / 31

42 Computation of Quadratic Cuts initial choice: S = vert([x L, x U ]) {random points} f(x) q(x) x^ Extending SCIP for solving mixed-integer nonlinear programs 14 / 31

43 Computation of Quadratic Cuts for x S with g(x) = q(x), maximize the error q(x) g(x) x f(x) q(x) x^ x* Extending SCIP for solving mixed-integer nonlinear programs 14 / 31

44 Computation of Quadratic Cuts for x S with g(x) = q(x), maximize the error q(x) g(x) x if q(x ) g(x ) > δ tol, add x to S and recompute q(x) f(x) q(x) x^ x* Extending SCIP for solving mixed-integer nonlinear programs 14 / 31

45 Computation of Quadratic Cuts for x S with g(x) = q(x), maximize the error q(x) g(x) δ max f(x) x^ δ max q(x) Extending SCIP for solving mixed-integer nonlinear programs 14 / 31

46 Computation of Quadratic Cuts for x S with g(x) = q(x), maximize the error q(x) g(x) δ max if δ max < δ tol, lower q(x) by δ max f(x) x^ δ max q(x) Extending SCIP for solving mixed-integer nonlinear programs 14 / 31

47 Separation by Quadratic Cuts (cont.) heuristic: cut is only probably valid due to sampling approach Extending SCIP for solving mixed-integer nonlinear programs 15 / 31

48 Separation by Quadratic Cuts (cont.) heuristic: cut is only probably valid due to sampling approach expensive to compute: solve several NLPs and LPs Extending SCIP for solving mixed-integer nonlinear programs 15 / 31

49 Separation by Quadratic Cuts (cont.) heuristic: cut is only probably valid due to sampling approach expensive to compute: solve several NLPs and LPs tight: derived directly from function values Extending SCIP for solving mixed-integer nonlinear programs 15 / 31

50 Separation by Quadratic Cuts (cont.) heuristic: cut is only probably valid due to sampling approach expensive to compute: solve several NLPs and LPs tight: derived directly from function values works in low dimensions Extending SCIP for solving mixed-integer nonlinear programs 15 / 31

51 Nonlinear Constraints: Domain Propagation Represent algebraic structure of problem in a directed acyclic graph (DAG): nodes: variables, operations, constraints arcs: flow of computation Extending SCIP for solving mixed-integer nonlinear programs 16 / 31

52 Nonlinear Constraints: Domain Propagation Represent algebraic structure of problem in a directed acyclic graph (DAG): nodes: variables, operations, constraints arcs: flow of computation Example: x + 2 xy + 2 y [, 7] x 2 y 2xy + 3 y [0, 2] x, y [1, 16] [1,16] [1,16] x y [,7] [0,2] 3 Extending SCIP for solving mixed-integer nonlinear programs 16 / 31

53 Nonlinear Constraints: Domain Propagation Represent algebraic structure of problem in a directed acyclic graph (DAG): nodes: variables, operations, constraints arcs: flow of computation Example: x + 2 xy + 2 y [, 7] x 2 y 2xy + 3 y [0, 2] x, y [1, 16] Forward propagation: compute bounds on intermediate nodes (top-down) [1,16] [1,16] x y [1,4] [1,256] [1,256] [1,4] 2 [1,16] [1,1024] [5,7] [0,2] Extending SCIP for solving mixed-integer nonlinear programs 16 / 31

54 Nonlinear Constraints: Domain Propagation Represent algebraic structure of problem in a directed acyclic graph (DAG): nodes: variables, operations, constraints arcs: flow of computation Example: x + 2 xy + 2 y [, 7] x 2 y 2xy + 3 y [0, 2] x, y [1, 16] Forward propagation: compute bounds on intermediate nodes (top-down) Backward propagation: reduce bounds by reverse operations (bottom-up) [1,9] [1,16] x y [1,3] [1,256] [1,16] [1,4] 2 [1,4] [1,511] [5,7] [0,2] Extending SCIP for solving mixed-integer nonlinear programs 16 / 31

55 Nonlinear Constraints: Domain Propagation Represent algebraic structure of problem in a directed acyclic graph (DAG): nodes: variables, operations, constraints arcs: flow of computation Example: x + 2 xy + 2 y [, 7] x 2 y 2xy + 3 y [0, 2] x, y [1, 16] [1,9] [1,16] x y [1,3] [1,256] [1,16] [1,4] 2 Forward propagation: compute bounds on intermediate nodes (top-down) Backward propagation: reduce 2 2 bounds by reverse operations + + (bottom-up) [5,7] implemented in Couenne [Belotti et.al.] (open-source) [1,4] [1,511] Extending SCIP for solving mixed-integer nonlinear programs 16 / 31 2 [0,2] 3

56 Overview 1 Introduction 2 Quadratic Constraints 3 Nonlinear Constraints (MINLP) 4 Heuristic 5 Preliminary Numerical Results 6 Latest Work Extending SCIP for solving mixed-integer nonlinear programs 17 / 31

57 Heuristics nothing fancy yet... solve (locally) MINLP with discrete variables fixed NLP min Extending SCIP for solving mixed-integer nonlinear programs 18 / 31

58 Heuristics nothing fancy yet... solve (locally) MINLP with discrete variables fixed NLP get fixation of discrete variables from integral solution of LP relaxation in nodes min Extending SCIP for solving mixed-integer nonlinear programs 18 / 31

59 Heuristics nothing fancy yet... solve (locally) MINLP with discrete variables fixed NLP get fixation of discrete variables from integral solution of LP relaxation in nodes integral solution found by one of SCIPs MIP heuristics min Extending SCIP for solving mixed-integer nonlinear programs 18 / 31

60 Overview 1 Introduction 2 Quadratic Constraints 3 Nonlinear Constraints (MINLP) 4 Heuristic 5 Preliminary Numerical Results 6 Latest Work Extending SCIP for solving mixed-integer nonlinear programs 19 / 31

61 Implementation Implementation using some other software packages: reading problem via GAMS interface Gradients and Hessians via CppAD LP solver: CPLEX 11.2 local search heuristic by GAMS NLP solver CONOPT 3 quadratic estimators by LaGO subproblems (max error g(x) q(x)) by Ipopt domain propagation on DAGs by Couenne eigenvalues via Lapack projects.coin-or.org/lago projects.coin-or.org/ipopt projects.coin-or.org/couenne Extending SCIP for solving mixed-integer nonlinear programs 20 / 31

Implementation Implementation using some other software packages: reading problem via GAMS interface Gradients and Hessians via CppAD LP solver: CPLEX 11.

62 Implementation Implementation using some other software packages: reading problem via GAMS interface Gradients and Hessians via CppAD LP solver: CPLEX 11.2 local search heuristic by GAMS NLP solver CONOPT 3 quadratic estimators by LaGO subproblems (max error g(x) q(x)) by Ipopt domain propagation on DAGs by Couenne eigenvalues via Lapack projects.coin-or.org/lago projects.coin-or.org/ipopt projects.coin-or.org/couenne -lscip.linux.x86.gnu.opt -lobjscip.linux.x86.gnu.opt -llpicpx.linux.x86.gnu.opt -llago -llagointerfaceos -losicpx -losi -lcoinutils -lipopt -lcplex -los -lcouenne -lbonmin -lcbc -lcbcsolver -losiclp -lclp -lcgl -losi -lcoinutils -llagointerfacegams libsmag.a gclib.a libg2d.a clicelib.a libf90pallib.a iolib.a -lpthread -lm -ldl -lz -lreadline -lncurses -lgfortranbegin -lgfortran -lm -lgcc_s Extending SCIP for solving mixed-integer nonlinear programs 20 / 31

63 Testset Model instances: 49 MINLPs from [Belotti et.al. 2008] in average: 200 variables (105 discrete), 151 constraints 16 convex, 18 nonconvex quadratic, 15 nonconvex nonquadratic cecil_13 classical_40_0 classical_40_1 clay0203h clay0204h csched1 csched2 c-schedule4fur7feed du-opt5 du-opt eniplac enpro48pb enpro56pb ex1233 ex1243 ex1244 ex1252 fo7 ibell3a ibienst1 imisc07 iran8x32 lop97icx m6 multistage no7_ar2_1 no7_ar3_1 no7_ar4_1 nous1 nous2 nvs19 nvs23 o7_2 par72 robust_30_0 robust_30_1 shortfall_30_0 shortfall_30_1 space25a space25 stockcycle synheat synheatmod tln12 tln5 tln6 tln7 tls5 tls6 Time limit: 1 hour Gap tolerance: 0.01% Extending SCIP for solving mixed-integer nonlinear programs 21 / 31

64 Results on convex models separation by linearization of constraint functions (1st order Taylor) instance var discr nlnz time[s] gap classical_40_ < 0.01% classical_40_ < 0.01% clay0203h < 0.01% clay0204h < 0.01% du-opt < 0.01% du-opt < 0.01% fo < 0.01% ibell3a < 0.01% m < 0.01% no7_ar2_ < 0.01% no7_ar3_ < 0.01% no7_ar4_ < 0.01% o7_ < 0.01% stockcycle < 0.01% tls % tls % Extending SCIP for solving mixed-integer nonlinear programs 22 / 31

65 Results on nonconvex quadratic models separation by McCormick underestimators instance var discr nlnz time[s] gap ibienst < 0.01% imisc < 0.01% iran8x < 0.01% lop97icx % nous % nous < 0.01% nvs < 0.01% nvs < 0.01% robust_30_ % robust_30_ % shortfall_30_ % shortfall_30_ % space space25a % tln < 0.01% tln % tln % tln % Extending SCIP for solving mixed-integer nonlinear programs 23 / 31

66 Results on nonconvex nonquadratic models separation by interval gradient cuts and quadratic cuts instance var discr nlnz time[s] gap cecil_ < 0.01% csched < 0.01% csched < 0.01% c-sched eniplac < 0.01% enpro48pb < 0.01% enpro56pb < 0.01% ex < 0.01% ex < 0.01% ex < 0.01% ex < 0.01% multistage % par synheat < 0.01% synheatmod % Extending SCIP for solving mixed-integer nonlinear programs 24 / 31

67 Comparison with BARON and LindoGlobal Metric: Solved = gap < 0.01% Extending SCIP for solving mixed-integer nonlinear programs 25 / 31

68 Overview 1 Introduction 2 Quadratic Constraints 3 Nonlinear Constraints (MINLP) 4 Heuristic 5 Preliminary Numerical Results 6 Latest Work Extending SCIP for solving mixed-integer nonlinear programs 26 / 31

69 { X := (x, y) R n R + : Second Order Cone Constraints } n (α i x i ) 2 (βy) 2 i=1 Extending SCIP for solving mixed-integer nonlinear programs 27 / 31

70 X := { (x, y) R n R + : Second Order Cone Constraints } n (α i x i ) 2 (βy) 2 i=1 Ben-Tal and Nemirovski (2001): X can be linearly outer-approximated within arbitrary precision For n = 2 introduce variables a j, b j, j = 0,..., N, and constraints π π a 0 α 1x 1, a j = cos a 2 j+1 j 1 + sin b 2 j+1 j 1, a N βy π π π b 0 α 2 x 2, b j sin a 2 j+1 j 1 + cos b 2 j+1 j 1, b N tan a 2 N+1 N. Extending SCIP for solving mixed-integer nonlinear programs 27 / 31

71 X := { (x, y) R n R + : Second Order Cone Constraints } n (α i x i ) 2 (βy) 2 i=1 Ben-Tal and Nemirovski (2001): X can be linearly outer-approximated within arbitrary precision For n = 2 introduce variables a j, b j, j = 0,..., N, and constraints π π a 0 α 1x 1, a j = cos a 2 j+1 j 1 + sin b 2 j+1 j 1, a N βy π π π b 0 α 2 x 2, b j sin a 2 j+1 j 1 + cos b 2 j+1 j 1, b N tan a 2 N+1 N. N determines approximation error δ(n) = 1 cos ( π 2 N+1 ) 1 = O(4 N ) Extending SCIP for solving mixed-integer nonlinear programs 27 / 31

72 X := { (x, y) R n R + : Second Order Cone Constraints } n (α i x i ) 2 (βy) 2 i=1 Ben-Tal and Nemirovski (2001): X can be linearly outer-approximated within arbitrary precision For n = 2 introduce variables a j, b j, j = 0,..., N, and constraints π π a 0 α 1x 1, a j = cos a 2 j+1 j 1 + sin b 2 j+1 j 1, a N βy π π π b 0 α 2 x 2, b j sin a 2 j+1 j 1 + cos b 2 j+1 j 1, b N tan a 2 N+1 N. 1 N determines approximation error δ(n) = cos ( ) π 1 = O(4 N ) 2 N+1 For n > 2, reformulate as set of SOC constraints: n/2 (α i x i ) 2 z1 2, i=1 n i= n/2 +1 (α i x i ) 2 z 2 2, z z 2 2 (βy) 2 Extending SCIP for solving mixed-integer nonlinear programs 27 / 31

73 Second Order Cone (SOC) Constraints: First Results if quadratic constraint is SOC, add linear outer-approximation variables and constraints, but keep also original constraint n = 30 robust30-0 robust30-1 shortfall30-0 shortfall30-1 Extending SCIP for solving mixed-integer nonlinear programs 28 / 31

74 Second Order Cone (SOC) Constraints: First Results if quadratic constraint is SOC, add linear outer-approximation variables and constraints, but keep also original constraint n = 30 robust30-0 robust30-1 shortfall30-0 shortfall30-1 N = 0 root gap 52% 51% 12% 14% final gap 48% 48% 4% 6% time 1h 1h 1h 1h nodes Extending SCIP for solving mixed-integer nonlinear programs 28 / 31

75 Second Order Cone (SOC) Constraints: First Results if quadratic constraint is SOC, add linear outer-approximation variables and constraints, but keep also original constraint n = 30 robust30-0 robust30-1 shortfall30-0 shortfall30-1 N = 0 root gap 52% 51% 12% 14% final gap 48% 48% 4% 6% time 1h 1h 1h 1h nodes N = 3 root gap 5.2% 2.8% 0.6% 0.8% time 27s 17s 472s 781s nodes Extending SCIP for solving mixed-integer nonlinear programs 28 / 31

76 Second Order Cone (SOC) Constraints: First Results if quadratic constraint is SOC, add linear outer-approximation variables and constraints, but keep also original constraint n = 30 robust30-0 robust30-1 shortfall30-0 shortfall30-1 N = 0 root gap 52% 51% 12% 14% final gap 48% 48% 4% 6% time 1h 1h 1h 1h nodes N = 3 root gap 5.2% 2.8% 0.6% 0.8% time 27s 17s 472s 781s nodes N = 5 root gap 0.2% 0.8% 0% 0.4% time 3s 3s 3s 18s nodes Extending SCIP for solving mixed-integer nonlinear programs 28 / 31

77 Second Order Cone (SOC) Constraints: First Results if quadratic constraint is SOC, add linear outer-approximation variables and constraints, but keep also original constraint n = 30 robust30-0 robust30-1 shortfall30-0 shortfall30-1 N = 0 root gap 52% 51% 12% 14% final gap 48% 48% 4% 6% time 1h 1h 1h 1h nodes N = 3 root gap 5.2% 2.8% 0.6% 0.8% time 27s 17s 472s 781s nodes N = 5 root gap 0.2% 0.8% 0% 0.4% time 3s 3s 3s 18s nodes N = 10 root gap 0% 0.6% 0% 0.3% time 7s 8s 6s 39s nodes Extending SCIP for solving mixed-integer nonlinear programs 28 / 31

78 Second Order Cone (SOC) Constraints: First Results if quadratic constraint is SOC, add linear outer-approximation variables and constraints, but keep also original constraint n = 30 robust30-0 robust30-1 shortfall30-0 shortfall30-1 N = 0 root gap 52% 51% 12% 14% final gap 48% 48% 4% 6% time 1h 1h 1h 1h nodes N = 3 root gap 5.2% 2.8% 0.6% 0.8% time 27s 17s 472s 781s nodes N = 5 root gap 0.2% 0.8% 0% 0.4% time 3s 3s 3s 18s nodes N = 10 root gap 0% 0.6% 0% 0.3% time 7s 8s 6s 39s nodes N = 20 root gap 0% 0.6% 0% 0.3% time 166s 71s 15s 675s nodes Extending SCIP for solving mixed-integer nonlinear programs 28 / 31

79 End Thank you! Extending SCIP for solving mixed-integer nonlinear programs 29 / 31

80 Overview 7 Branching Rule Extending SCIP for solving mixed-integer nonlinear programs 30 / 31

81 Branching Variable Selection Which variable to select for branching? if LP relaxation solution is fractional, choose integer variable (SCIP default: reliability branching on pseudo cost values) Extending SCIP for solving mixed-integer nonlinear programs 31 / 31

82 Branching Variable Selection Which variable to select for branching? if LP relaxation solution is fractional, choose integer variable (SCIP default: reliability branching on pseudo cost values) selection of continuous variable: look at Belotti et.al.: Branching and bounds tightening techniques for non-convex MINLP Extending SCIP for solving mixed-integer nonlinear programs 31 / 31

83 Branching Variable Selection Which variable to select for branching? if LP relaxation solution is fractional, choose integer variable (SCIP default: reliability branching on pseudo cost values) selection of continuous variable: look at Belotti et.al.: Branching and bounds tightening techniques for non-convex MINLP search for pareto-optimal branching rule w.r.t. br-plain performance and ease of implementation Extending SCIP for solving mixed-integer nonlinear programs 31 / 31

84 Branching Variable Selection Which variable to select for branching? if LP relaxation solution is fractional, choose integer variable (SCIP default: reliability branching on pseudo cost values) selection of continuous variable: look at Belotti et.al.: Branching and bounds tightening techniques for non-convex MINLP search for pareto-optimal branching rule w.r.t. br-plain performance and ease of implementation for constraint h i (x) u violated in ˆx, let φ i := (scaled) gap between h i (ˆx) and its best linear underestimator in ˆx Extending SCIP for solving mixed-integer nonlinear programs 31 / 31

85 Branching Variable Selection Which variable to select for branching? if LP relaxation solution is fractional, choose integer variable (SCIP default: reliability branching on pseudo cost values) selection of continuous variable: look at Belotti et.al.: Branching and bounds tightening techniques for non-convex MINLP search for pareto-optimal branching rule w.r.t. br-plain performance and ease of implementation for constraint h i (x) u violated in ˆx, let φ i := (scaled) gap between h i (ˆx) and its best linear underestimator in ˆx select variable x j with maximal variable infeasibility 0.1 x j in h i (x) φ i max φ i x j in h i (x) min x j in h i (x) φ i Extending SCIP for solving mixed-integer nonlinear programs 31 / 31

Extending a CIP framework for solving mixed integer nonlinear programs

Extending a CIP framework for solving mixed integer nonlinear programs Stefan Vigerske joint with T. Berthold, P. Belotti, T. Gellermann, A. Gleixner, S. Heinz, T. Koch, M. Pfetsch DFG Research Center