Solving Mixed-Integer Nonlinear Programs

Transcription

1 Solving Mixed-Integer Nonlinear Programs (with SCIP) Ambros M. Gleixner Zuse Institute Berlin MATHEON Berlin Mathematical School 5th Porto Meeting on Mathematics for Industry, April 10 11, 2014, Porto 1

2 Outline Solving MINLPs (with SCIP) Solving convex MINLPs Solving nonconvex MINLPs Modeling, Reformulation, Presolving Primal Solutions: The Undercover Heuristic 2

3 Outline Solving MINLPs (with SCIP) Solving convex MINLPs Solving nonconvex MINLPs Modeling, Reformulation, Presolving Primal Solutions: The Undercover Heuristic 3

4 What is Mixed-Integer Nonlinear Programming? min c T x for c R n, s. t. g k (x) 0 for k = 1,..., m, g k : [l, u] R C 1, x [l, u], x i Z for i I {1,..., n} g k convex local = global optimality g k nonconvex suboptimal local optima 4

5 Convex MINLP Assumption g 1,..., g m convex NLP-based replace LP by NLP solver branch on integer var.s with fractional NLP value

6 Convex MINLP Assumption g 1,..., g m convex NLP-based replace LP by NLP solver branch on integer var.s with fractional NLP value

7 Convex MINLP Assumption g 1,..., g m convex NLP-based replace LP by NLP solver branch on integer var.s with fractional NLP value

8 Convex MINLP Assumption g 1,..., g m convex NLP-based replace LP by NLP solver branch on integer var.s with fractional NLP value 10 5 LP-based underestimate by gradient cuts g k (ˆx) + g k (ˆx) T (x ˆx)

9 Convex MINLP Assumption g 1,..., g m convex NLP-based replace LP by NLP solver branch on integer var.s with fractional NLP value 10 5 LP-based underestimate by gradient cuts g k (ˆx) + g k (ˆx) T (x ˆx) 0 bound by polyhedral relaxation at MIP/NLP/sub-NLP solutions at node LP solutions

10 Convex MINLP Assumption g 1,..., g m convex NLP-based replace LP by NLP solver branch on integer var.s with fractional NLP value 10 5 LP-based underestimate by gradient cuts g k (ˆx) + g k (ˆx) T (x ˆx) 0 bound by polyhedral relaxation at MIP/NLP/sub-NLP solutions at node LP solutions Many algorithms, many solvers α-ecp [Westerlund and Pettersson], BONMIN [Bonami et al.], DICOPT [Duran and Grossmann], sbb [ARKI Software & Consulting],... [see, e.g., Bonami, Biegler, Conn, Cornuéjols, Grossmann, Laird, Lee, Lodi, Margot, Sawaya, Wächter 2008] 5

11 rdivision Applications, applications, applications industrial engineering: mining with stockpiling constraints manufacturing: sheet metal design chemical industry: design of synthesis processes networks: operation and design of water and gas networks energy production and distribution: plant design, power scheduling biological engineering: cell modeling... vdilution vdegradation 6

12 Applications, applications, applications industrial engineering: mining with stockpiling constraints manufacturing: sheet metal design chemical industry: design of synthesis processes networks: operation and design of water and gas networks energy production and distribution: plant design, power scheduling biological engineering: cell modeling... nonconvex! vdilution vdegradation rdivision 6

13 Applications, applications, applications industrial engineering: mining with stockpiling constraints manufacturing: sheet metal design chemical industry: design of synthesis processes networks: operation and design of water and gas networks energy production and distribution: plant design, power scheduling biological engineering: cell modeling... nonconvex! nonconvex! vdilution vdegradation rdivision 6

14 Applications, applications, applications industrial engineering: mining with stockpiling constraints manufacturing: sheet metal design chemical industry: design of synthesis processes networks: operation and design of water and gas networks energy production and distribution: plant design, power scheduling biological engineering: cell modeling... nonconvex! nonconvex! nonconvex! vdilution vdegradation rdivision 6

15 Applications, applications, applications industrial engineering: mining with stockpiling constraints manufacturing: sheet metal design chemical industry: design of synthesis processes networks: operation and design of water and gas networks energy production and distribution: plant design, power scheduling biological engineering: cell modeling... nonconvex! nonconvex! nonconvex! nonconvex! vdilution vdegradation rdivision 6

16 Applications, applications, applications industrial engineering: mining with stockpiling constraints manufacturing: sheet metal design chemical industry: design of synthesis processes networks: operation and design of water and gas networks energy production and distribution: plant design, power scheduling biological engineering: cell modeling... nonconvex! nonconvex! nonconvex! nonconvex! vdilution vdegradation nonconvex! rdivision 6

17 Open Pit Mine Production Scheduling with Stockpiles Open Pit Mine Scheduling Basic Version Variables: x i, t {0,1} f m i, t [0,1] f i p, t [0,1] block i fully mined by t % of block i mined in t % of block i processed in t Constraints: material flow conservation mining & processing capacities mining precedences Given: Blocks (or aggregates) (rock, metal grade, precedences) Capacities, Costs (Mining, Processing; per rock ton) Metal Prices Decisions: When to mine which block? Which block to process? Goal: max NPV Solution Approach: Time-indexed MILP formulation Andreas Bley, Branch-and-bound techniques for a mine production scheduling problem 7

18 Open Pit Mine Production Scheduling with Stockpiles Open Pit Mine Scheduling With Stockpile Practical extension: Stockpile for interim storage better use of capacities Difficulty: stockpile mixes material Andreas Bley, Branch-and-bound techniques for a mine production scheduling problem 8

19 Open Pit Mine Production Scheduling with Stockpiles Open Pit Mine Scheduling With Stockpile I I f 1,t f 2, t Practical extension: Stockpile for interim storage rock Q t met Q t rock P t met P t better use of capacities Difficulty: stockpile mixes material Aggregated stockpile model f I i, t [0,1] rock met Q, Q t rock t P, P t met t % of block i into stockpiled total rock / metal tons held total rock / metal tons out Mixing constraints: met rock met P rock met rock Q t P (metal fraction out t t t P(grade t Qtout = grade held) rock met rock PQ Q = rock fraction out) t t t Mathematical model: Same rock / metal mix held and taken out of stockpile Nonlinear constraints Andreas Bley, Branch-and-bound techniques for a mine production scheduling problem 9

20 Outline Solving MINLPs (with SCIP) Solving convex MINLPs Solving nonconvex MINLPs Modeling, Reformulation, Presolving Primal Solutions: The Undercover Heuristic 10

21 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 11

22 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 11

23 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

24 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

25 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

26 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

27 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

28 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

29 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

30 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

31 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

32 Nonconvex MINLP Now some g 1,..., g m nonconvex Relaxation gradient cuts invalid linear relaxation of convex hull convexification gap Spatial branch-and-bound branch on int. variables with fractional LP value branch on variables in violated nonlinear constraints [McCormick 76, Smith and Pantelides 99, Tawarmalani and Sahinidis 02, Belotti et al. 09, Vigerske 13,... ] 12

33 Convex Relaxation Convex envelopes largest convex function that underestimates some g j (x) difficult to find in general known for many elementary cases: convex, univariate concave, bilinear,... 13

34 Convex Relaxation Convex envelopes largest convex function that underestimates some g j (x) difficult to find in general known for many elementary cases: convex, univariate concave, bilinear,... Example McCormick underestimators for x 1 x 2 (x 1 l 1 ) (x 2 l 2 ) 0 13

35 Convex Relaxation Convex envelopes largest convex function that underestimates some g j (x) difficult to find in general known for many elementary cases: convex, univariate concave, bilinear,... Example McCormick underestimators for x 1 x 2 (x 1 l 1 ) (x 2 l 2 ) 0 x 1 x 2 l 1 x 2 l 2 x 1 + l 1 l

36 Convex Relaxation Convex envelopes largest convex function that underestimates some g j (x) difficult to find in general known for many elementary cases: convex, univariate concave, bilinear,... Example McCormick underestimators for x 1 x 2 (x 1 l 1 ) (x 2 l 2 ) 0 x 1 x 2 l 1 x 2 l 2 x 1 + l 1 l 2 0 x 1 x 2 l 1 x 2 + l 2 x 1 l 1 l 2 13

37 Convex Relaxation Factorable functions. recursive sum of products of univariate functions). reformulate into simple cases by introducing new variables and equations q g (x) = exp(x12 ) ln(x2 ) x1 [0, 2], x2 [1, 2]

38 Convex Relaxation Factorable functions. recursive sum of products of univariate functions). reformulate into simple cases by introducing new variables and equations q g (x) = exp(x12 ) ln(x2 ) x1 [0, 2], x2 [1, 2] g= y1 6 y1 = y2 y3 4 y2 = exp(y4 ) y3 = ln(x2 ) y4 = x

39 Convex Relaxation Factorable functions. recursive sum of products of univariate functions). reformulate into simple cases by introducing new variables and equations q g (x) = exp(x12 ) ln(x2 ) x1 [0, 2], x2 [1, 2] g= y1 6 y1 = y2 y3 4 y2 = exp(y4 ) y3 = ln(x2 ) y4 = x Tighter relaxations Reformulation-Linearization-Technique, SDP cuts, Disjunctive Programming,... 14

40 General MINLP solving techniques Gradient cuts Underestimators Spatial branching g x g c x Presolving Bound tightening Primal heuristics 15

41 General MINLP solving techniques Gradient cuts Underestimators Spatial branching g x g c x Presolving Bound tightening Primal heuristics 15

42 Outline Solving MINLPs (with SCIP) Solving convex MINLPs Solving nonconvex MINLPs Modeling, Reformulation, Presolving Primal Solutions: The Undercover Heuristic 16

43 Modeling Considerations Provide bounds on variables (as tight as possible) tighter relaxations 17

44 Modeling Considerations Provide bounds on variables (as tight as possible) tighter relaxations Scaling ideally: nonzeros with absolute values in the range [0.01, 100] also intermediate expressions are important: ( exp 1 ) [0, 0.4] for x [10 6 1, 1], but x x [1, 106 ] 17

45 Modeling Considerations Provide bounds on variables (as tight as possible) tighter relaxations Scaling ideally: nonzeros with absolute values in the range [0.01, 100] also intermediate expressions are important: ( exp 1 ) [0, 0.4] for x [10 6 1, 1], but x x [1, 106 ] Prefer Linearity and Convexity x y = 1 nonlinear and nonconvex 17

46 Modeling Considerations Provide bounds on variables (as tight as possible) tighter relaxations Scaling ideally: nonzeros with absolute values in the range [0.01, 100] also intermediate expressions are important: ( exp 1 ) [0, 0.4] for x [10 6 1, 1], but x x [1, 106 ] Prefer Linearity and Convexity x = y linear and thus convex 17

47 Modeling Considerations Provide bounds on variables (as tight as possible) tighter relaxations Scaling ideally: nonzeros with absolute values in the range [0.01, 100] also intermediate expressions are important: ( exp 1 ) [0, 0.4] for x [10 6 1, 1], but x x [1, 106 ] Prefer Linearity and Convexity xy 1 nonconvex 17

48 Modeling Considerations Provide bounds on variables (as tight as possible) tighter relaxations Scaling ideally: nonzeros with absolute values in the range [0.01, 100] also intermediate expressions are important: ( exp 1 ) [0, 0.4] for x [10 6 1, 1], but x x [1, 106 ] Prefer Linearity and Convexity y 1 x convex 17

49 Reformulation of products with binary variables A quadratic term x can be linearly reformulated: N a k y k with x {0, 1} k=1 auxiliary continuous variable w additional linear constraints M L x w M U x, N N a k y k M U (1 x) w a k y k M L (1 x), k=1 k=1 where M L and M U are bounds on N k=1 a ky k. 18

50 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching 19

51 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 in [ 1, 1] [ 1, 1] 19

52 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 (x + y) 2 1 in [ 1, 1] [ 1, 1] 19

53 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 x + y 1 in [ 1, 1] [ 1, 1] feasible region

54 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 (x + y) 2 1 in [ 1, 1] [ 1, 1] using McCormick underestimators: x 2 + 2w + y 2 1 w L y x + L x y L x L y w U y x + U x y U x U y

55 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 (x + y) 2 1 in [ 1, 1] [ 1, 1] using McCormick underestimators: x 2 + 2w + y 2 1 w L y x + L x y L x L y w U y x + U x y U x U y branched into 4 subproblems 19

56 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 (x + y) 2 1 in [ 1, 1] [ 1, 1] using McCormick underestimators: x 2 + 2w + y 2 1 w L y x + L x y L x L y w U y x + U x y U x U y branched into 16 subproblems 19

57 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 (x + y) 2 1 in [ 1, 1] [ 1, 1] using McCormick underestimators: x 2 + 2w + y 2 1 w L y x + L x y L x L y w U y x + U x y U x U y branched into 64 subproblems 19

58 Convexity check for quadratic constraints A quadratic constraint x T Ax + b T x c: convex if A is positive-semidefinite check by computing its minimal eigenvalue with LAPACK if yes: gradient cuts are valid enforcement by separation instead of branching Example x 2 + 2xy + y 2 1 (x + y) 2 1 in [ 1, 1] [ 1, 1] A = ( ) 1 1 positive-semidefinite 1 1 gradient cuts at 4 corners yield exact feasible region

59 Second-order cone upgrade Quadratic constraints of the form N α k xk 2 α N+1 xn k=1 N α k xk 2 α N+1 x N+1 k=1 with α 1,..., α N+1 0, L N+1 0 describe a convex feasible region. Example x 2 + y 2 z 2 0 in [ 1, 1] [ 1, 1] [0, 1] feasible region ice cream cone 20

60 Second-order cone upgrade Quadratic constraints of the form N X 2 αk xk2 αn+1 xn+1 v u N ux 60 t αk xk2 6 αn+1 xn+1 k=1 k=1 with α1,..., αn+1 > 0, LN+1 > 0 describe a convex feasible region. Example x 2 + y 2 z in [ 1, 1] [ 1, 1] [0, 1] using secant underestimator: ( x2 + y2 + w 6 1 z 2 z 2 ) w > (L U) z (U (z Lz ) (Lz )2 Lz ) 20

61 Second-order cone upgrade Quadratic constraints of the form N X 2 αk xk2 αn+1 xn+1 v u N ux 60 t αk xk2 6 αn+1 xn+1 k=1 k=1 with α1,..., αn+1 > 0, LN+1 > 0 describe a convex feasible region. Example x 2 + y 2 z in [ 1, 1] [ 1, 1] [0, 1] using secant underestimator: ( x2 + y2 + w 6 1 z 2 z 2 ) w > (L U) z (U (z Lz ) (Lz )2 Lz ) 20

62 Second-order cone upgrade Quadratic constraints of the form N X 2 αk xk2 αn+1 xn+1 v u N ux 60 t αk xk2 6 αn+1 xn+1 k=1 k=1 with α1,..., αn+1 > 0, LN+1 > 0 describe a convex feasible region. Example x 2 + y 2 z in [ 1, 1] [ 1, 1] [0, 1] using secant underestimator: ( x2 + y2 + w 6 1 z 2 z 2 ) w > (L U) z (U (z Lz ) (Lz )2 Lz ) after branching on z = 0.25, 0.5,

63 Second-order cone upgrade Quadratic constraints of the form N α k xk 2 α N+1 xn k=1 N α k xk 2 α N+1 x N+1 k=1 with α 1,..., α N+1 0, L N+1 0 describe a convex feasible region. Example x 2 + y 2 z 2 0 in [ 1, 1] [ 1, 1] [0, 1] using gradient cuts at 8 corners 20

64 General MINLP solving techniques Gradient cuts Underestimators Spatial branching g x g c x Presolving Bound tightening Primal heuristics 21

65 General MINLP solving techniques Gradient cuts Underestimators Spatial branching g x g c x Presolving Bound tightening Primal heuristics 21

66 Outline Solving MINLPs (with SCIP) Solving convex MINLPs Solving nonconvex MINLPs Modeling, Reformulation, Presolving Primal Solutions: The Undercover Heuristic 22

67 Primal Solutions Feasible LP solutions... Standard MIP heuristics applied to MIP relaxation NLP local search MINLP heuristics. nonlinear feasibility pumps [Bonami et al. 2009, D Ambrosio et al. 2010] min. RENS [Berthold 2013]. Undercover [Berthold and G. 2013] 23

68 The Motivation Large Neighborhood Search: common paradigm in MIP heuristics fix a subset of variables easy subproblem solve MIP: easy = few integralities MINLP: easy = few nonlinearities observation: any MINLP can be reduced to a MIP by fixing (sufficiently many) variables. Experience: Often, few fixings are sufficient! idea: fix variables in minimum cover solution of LP/NLP relaxation as fixing values 24

69 The Structure Definition Let us be given a domain box [L, U] = i [L i, U i ], a function g j : [L, U] R, x g j (x) on [L, U], and a set C N := {1,..., n} of variable indices. We call C a cover of g if and only if for all x [L, U] the set {(x, g j (x)) x [L, U], x k = x k for all k C} is an affine set intersected with [L, U] R. We call C a cover of P if and only if C is a cover for g 1,..., g m. 25

70 Covers of an MINLP Definition Let P be an MINLP with g 1,..., g m twice continuously differentiable on the interior of [L, U]. We call G P = (V P, E P ) the co-occurrence graph of P with node set V P = {1,..., n} and edge set E P = { ij i, j V, k {1,..., m} : 2 x i x j g k (x) 0 }, 26

71 Covers of an MINLP Definition Let P be an MINLP with g 1,..., g m twice continuously differentiable on the interior of [L, U]. We call G P = (V P, E P ) the co-occurrence graph of P with node set V P = {1,..., n} and edge set E P = { ij i, j V, k {1,..., m} : Example 2 x i x j g k (x) 0 }, S s 1 min... s.t. s 1t i a i for all i = 1,... s j t 1 b j for all j = 1,... T t 1 26

72 Covers of an MINLP Definition Let P be an MINLP with g 1,..., g m twice continuously differentiable on the interior of [L, U]. We call G P = (V P, E P ) the co-occurrence graph of P with node set V P = {1,..., n} and edge set E P = { ij i, j V, k {1,..., m} : 2 x i x j g k (x) 0 }, Theorem [Berthold and G. 2010, 2013] C {1,..., n} is a cover of P if and only if it is a vertex cover of the co-occurrence graph G P. 26

73 Covers of an MINLP Definition Let P be an MINLP with g 1,..., g m twice continuously differentiable on the interior of [L, U]. We call G P = (V P, E P ) the co-occurrence graph of P with node set V P = {1,..., n} and edge set E P = { ij i, j V, k {1,..., m} : 2 x i x j g k (x) 0 }, Theorem [Berthold and G. 2010, 2013] C {1,..., n} is a cover of P if and only if it is a vertex cover of the co-occurrence graph G P. Corollary Computing a minimum cover of an MINLP is N P-hard. 26

74 Computing a minimum cover Auxiliary binary variables α k = 1 : x k is fixed in P C(α) := {k α k = 1} is a cover of P if and only if α k = 1 for all loops kk E P, (1) α k + α j 1 for all edges kj E p, k > j. (2) Covering problem { n min α k : (1), (2), α {0, 1} n}. (3) k=1 27

75 Computing a minimum cover Auxiliary binary variables α k = 1 : x k is fixed in P C(α) := {k α k = 1} is a cover of P if and only if α k = 1 for all loops kk E P, (1) α k + α j 1 for all edges kj E p, k > j. (2) Covering problem { n min α k : (1), (2), α {0, 1} n}. (3) k=1 27

76 Computing a minimum cover Auxiliary binary variables α k = 1 : x k is fixed in P C(α) := {k α k = 1} is a cover of P if and only if α k = 1 for all loops kk E P, (1) α k + α j 1 for all edges kj E p, k > j. (2) Covering problem { n min α k : (1), (2), α {0, 1} n}. (3) k=1 27

77 Optimization matters The co-occurence graph of the bilinear program min... s.t. s 1 t i a i for all i = 1,..., s j t 1 b j for all j = 1,..., is S s 1 T t 1 The cover S of complicating variables may be arbitrarily large compared to the minimum cover {s 1, t 1 }. 28

78 A simple example max x 2 + x 3 s.t. x 1 + x 2 + x3 2 4, x 1, x 2, x 3 0, x 1, x 2 Z. Fixing x 3 to any value within its bounds yields a linear subproblem. 29

79 The Undercover Heuristic 1 Input: MINLP P 2 begin 3 compute a solution x of an approximation of P; 4 round x k for all k I; 5 determine a cover C of P; 6 solve the sub-mip of P given by fixing x k = x k for all k C; 30

80 The Undercover Heuristic 1 Input: MINLP P 2 begin 3 compute a solution x of an approximation of P; 4 round x k for all k I; 5 determine a cover C of P; 6 solve the sub-mip of P given by fixing x k = x k for all k C; 30

81 The Undercover Heuristic 1 Input: MINLP P 2 begin 3 compute a solution x of an approximation of P; 4 round x k for all k I; 5 determine a cover C of P; 6 solve the sub-mip of P given by fixing x k = x k for all k C; 30

82 The Undercover Heuristic 1 Input: MINLP P 2 begin 3 compute a solution x of an approximation of P; 4 round x k for all k I; 5 determine a cover C of P; 6 solve the sub-mip of P given by fixing x k = x k for all k C; 30

83 The Undercover Heuristic 1 Input: MINLP P 2 begin 3 compute a solution x of an approximation of P; 4 round x k for all k I; 5 determine a cover C of P; 6 solve the sub-mip of P given by fixing x k = x k for all k C; 30

84 The Undercover Heuristic 1 Input: MINLP P 2 begin 3 compute a solution x of an approximation of P; 4 round x k for all k I; 5 determine a cover C of P; 6 solve the sub-mip of P given by fixing x k = x k for all k C; Remark: MIP heuristics: trade-off fixing many vs. few variables here: eliminate nonlinearities by fixing as few as possible variables minimum cover! 30

85 NLP postprocessing NLP postprocessing All sub-mip solutions are fully feasible for the original MINLP. Still, sub-mip solution x could be improved by NLP local search: fix all integer variables of the original MINLP to their values in x solve the resulting NLP to local optimality 31

86 Fix-and-propagate & Backtracking Fix-and-propagate Do not fix variables in C simultaneously, but sequentially and propagate after each fixing. If xk falls out of bounds then fix to the closest bound (similar to [FischettiSalvagnin09]) recompute the approximation Backtracking If fix-and-propagate deduces infeasibility, apply a one-level backtracking: undo last fixing and try another value 32

87 Avoiding/exploiting Infeasibility If the sub-mip is infeasible, this is typically detected during fix-and-propagate, or via infeasible root LP. Generate conflict clauses for the original MINLP Add them to the original MINLP. Use them to revise fixing values and/or fixing order Start another fix-and-propagate run If the sub-mip remains infeasible, at least this gives us valid conflicts to prune the search tree in the original problem. 33

88 Computational experiments Test set 149 MIQCPs from GloMIQO test set Comparison to other heuristics Undercover: solution for 76 instances (typically less than 0.1 sec) root heuristics: Baron 65, Couenne 55, SCIP 98 lower success rate on general MINLPs Undercover components 34

89 Take-away messages SCIP can solve nonconvex MINLPs to global optimality like other solvers: Antigone/GloMIQO, BARON, Couenne,... 35

90 Take-away messages SCIP can solve nonconvex MINLPs to global optimality like other solvers: Antigone/GloMIQO, BARON, Couenne,... sometimes problem-specific algorithms can be efficiently generalized to structure-specific algorithms (Undercover) 35

91 Take-away messages SCIP can solve nonconvex MINLPs to global optimality like other solvers: Antigone/GloMIQO, BARON, Couenne,... sometimes problem-specific algorithms can be efficiently generalized to structure-specific algorithms (Undercover) convex MINLPs can be solved much more efficiently convex modelling/reformulation/detection crucial convex solvers can be used heuristically for nonconvex MINLPs 35

92 Take-away messages SCIP can solve nonconvex MINLPs to global optimality like other solvers: Antigone/GloMIQO, BARON, Couenne,... sometimes problem-specific algorithms can be efficiently generalized to structure-specific algorithms (Undercover) convex MINLPs can be solved much more efficiently convex modelling/reformulation/detection crucial convex solvers can be used heuristically for nonconvex MINLPs Thank you very much for your attention! Muito obrigado! 35

93 Solving Mixed-Integer Nonlinear Programs (with SCIP) Ambros M. Gleixner Zuse Institute Berlin MATHEON Berlin Mathematical School 5th Porto Meeting on Mathematics for Industry, April 10 11, 2014, Porto 36