Outline. 1 Introduction. 2 Modeling and Applications. 3 Algorithms Convex. 4 Algorithms Nonconvex. 5 The Future of MINLP? 2 / 126

Transcription

1 Outline 1 Introduction 2 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 2 / 126

2 Larry Leading Off 3 / 126

3 Introduction Outline 1 Introduction Software Tools and Online Resources Basic Concepts 2 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 4 / 126

4 Introduction Mixed-Integer Nonlinear Optimization Mixed Integer Nonlinear Program: (MINLP) z minlp = min x f (x) subject to g i (x) 0, i = 1,..., m x X, x I Z I f : R n R, g : R n R m smooth, sometimes convex functions. X R n bounded, polyhedral set, e.g. X = {x : l A T x u} I {1,..., n} subset of integer variables 5 / 126

5 Introduction Mixed-Integer Nonlinear Optimization Mixed Integer Nonlinear Program: (MINLP) z minlp = min x η subject to g i (x) 0, i = 1,..., m f (x) η x X, x I Z I f : R n R, g : R n R m smooth, sometimes convex functions. X R n bounded, polyhedral set, e.g. X = {x : l A T x u} I {1,..., n} subset of integer variables We may assume that f is a linear function. 5 / 126

6 Introduction Mixed-Integer Nonlinear Optimization Mixed Integer Nonlinear Program: (MINLP) z minlp = min x η subject to g i (x) 0, i = 1,..., m f (x) η x X, x I Z I f : R n R, g : R n R m smooth, sometimes convex functions. X R n bounded, polyhedral set, e.g. X = {x : l A T x u} I {1,..., n} subset of integer variables We may assume that f is a linear function. NP-Super-Hard Combines challenges of handling nonlinearities with combinatorial explosion of integer variables 5 / 126

7 Introduction The MINLP Family The great watershed in optimization isn t between linearity and nonlinearity, but convexity and nonconvexity. - R. Tyrrell Rockafellar If f and g are convex functions, then we have a convex MINLP If f and g are not convex, then we have a nonconvex MINLP 6 / 126

8 Introduction The MINLP Family The great watershed in optimization isn t between linearity and nonlinearity, but convexity and nonconvexity. - R. Tyrrell Rockafellar If f and g are convex functions, then we have a convex MINLP If f and g are not convex, then we have a nonconvex MINLP Optimization 101 Know the convexity properties of your instance! Algorithms for two different classes of problems are different We can solve significantly larger convex MINLP instances that nonconvex MINLP instances Algorithms designed for convex-minlp are often used as heuristics for nonconvex MINLPs 6 / 126

9 Introduction Specializations Functional Form g i (x) = A i x + b i 2 p T i x + q i g i (x) = polynomial Problem Type MISOCP MIPP g i (x) = x T Q i x + p T i + q i MIQP g i (x) = a T i x b i MILP MISOCP: Mixed Integer Second Order Cone Program Are convex-minlp MIPP: Mixed Integer Polynomial Program MIQP: Mixed Integer Quadratic Program May be convex or nonconvex Convex MIQP is a special case of MISOCP If f is convex quadratic and c is an affine mapping, then there are specialized algorithms for convex-miqp MILP: Mixed Integer Linear Program 7 / 126

10 Introduction Aside MISOCP The feasible regions of surprisingly MINLP can be expressed as MISOCP: 8 / 126

11 Introduction Aside MISOCP The feasible regions of surprisingly MINLP can be expressed as MISOCP: Hyperbolic constraints: Product of (non-negative) variables a norm 2. [ ] w T w xy, x 0, y 0 2w x + y x y 8 / 126

12 Introduction Aside MISOCP The feasible regions of surprisingly MINLP can be expressed as MISOCP: Hyperbolic constraints: Product of (non-negative) variables a norm 2. [ ] w T w xy, x 0, y 0 2w x + y x y Convex Quadratic: c i (x) = x T Q i x + p T i x + q i t Q i is positive semidefinite, so it has Cholesky factorization Q i = L T L y = Lx g(x) = y T y + p T i x + q i t {(x, t) R n+1 x T Qx + p T i x + q i t} = Proj (x,t) {(x, y, t, p) R 2n+2 y = Lx, p + q T x + r t, p y 2 } 8 / 126

13 Introduction MINLP Tree MINLP Nonconvex NLP Nonconvex MINLP Convex MINLP Convex NLP Polynomial Opt. MI- Polynomial Opt. MISOCP SOCP Nonconvex QP Nonconvex MIQP Convex MIQP Convex QP MILP LP 9 / 126

14 Introduction How Big Is It? The Leyffer-Linderoth-Luedtke (LLL) Measure of Complexity You have a problem of class X that has Y decision variables? What is the largest value of Y for which one of Sven, Jim, and Jeff would be willing to bet $50 that a state-of-the-art solver could solve the problem? 10 / 126

15 Introduction How Big Is It? The Leyffer-Linderoth-Luedtke (LLL) Measure of Complexity You have a problem of class X that has Y decision variables? What is the largest value of Y for which one of Sven, Jim, and Jeff would be willing to bet $50 that a state-of-the-art solver could solve the problem? Convex Nonconvex Problem Class (X ) # Var (Y ) Problem Class (X ) # Var (Y ) MINLP 500 MINLP 100 NLP NLP 100 MISOCP 1000 MIPP 150 SOCP 10 5 PP 150 MIQP 1000 MIQP 300 QP QP 300 MILP LP / 126

16 Introduction Software Tools and Online Resources Outline 1 Introduction Software Tools and Online Resources Basic Concepts 2 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 11 / 126

17 Introduction Software Tools and Online Resources Solvers for MINLP Convex α-ecp BONMIN DICOPT FilMINT GUROBI KNITRO MILANO MINLPBB MINOPT MINOTAUR SBB Xpress MIQP Nonconvex α-bb ANTIGONE BARON COCONUT COUENNE CPLEX LGO LindoGlobal SCIP MIQP/convex MIQP Open-source and commercial solvers 12 / 126

18 Introduction Software Tools and Online Resources Optimization Modeling Languages Advantages Express optimization languages in natural algebraic form Access to automatic differentiation & presolve techniques 13 / 126

19 Introduction Software Tools and Online Resources Optimization Modeling Languages Advantages Express optimization languages in natural algebraic form Access to automatic differentiation & presolve techniques Modeling Languages for Optimization AIMMS [Bisschop and Entriken, 1993] AMPL [Fourer et al., 1993] GAMS [Brooke et al., 1992] MOSEL [Colombani and Heipcke, 2002] TomLab [Holmström and Edvall, 2004] (On top of Matlab) Pyomo [Hart et al., 2011] 13 / 126

20 Introduction Software Tools and Online Resources Online Resources MINLP Test Problem Libraries wiki.mcs.anl.gov/leyffer/index.php/macminlp minlp.org/... models & descriptions Good source of modeling tricks and practices! Online Solvers Computational Infrastructure for Operations Research 14 / 126

21 Introduction Basic Concepts Outline 1 Introduction Software Tools and Online Resources Basic Concepts 2 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 15 / 126

22 Introduction Basic Concepts Simple Example x 2 (x 1 2) 2 + (x 2 + 1) x 1 x (x 1 1) 2 x x 1, x 2 5 x 2 Z Doubly NP-Hard! Feasible region is not convex for two reasons Integrality of x 2 and nonconvexity of constraint x 1 16 / 126

23 Introduction Basic Concepts Q: What to Do?! Relax! Remove some constraints. Solution of relaxed problem provide a lower bound on optimal solution value 17 / 126

24 Introduction Basic Concepts Natural Relaxation x 2 x 1 18 / 126

25 Introduction Basic Concepts Natural Relaxation x 2 Natural relaxation is to remove constraints x i Z i I But this still leaves a nonconvex NLP relaxation x 1 18 / 126

26 Introduction Basic Concepts Relax More We need a mechanism for relaxing the nonconvex constraints. x 2 x 1 19 / 126

27 Introduction Basic Concepts Relax More We need a mechanism for relaxing the nonconvex constraints. Secant underestimator for concave functions x 2 x 1 19 / 126

28 Introduction Basic Concepts Relax More We need a mechanism for relaxing the nonconvex constraints. Secant underestimator for concave functions Easy to get polyhedral outerapproximation of nonconvex constraints x 2 x 1 19 / 126

29 Introduction Basic Concepts Relax More We need a mechanism for relaxing the nonconvex constraints. Secant underestimator for concave functions Easy to get polyhedral outerapproximation of nonconvex constraints Ideally, we would like to get the convex hull of the feasible points x 2 x 1 19 / 126

30 Introduction Basic Concepts Linear Objective Is Important Here! min(x 1 1/2) 2 + (x 2 1/2) 2 x 2 s.t. x 1 {0, 1}, x 2 {0, 1} (ˆx 1, ˆx 2 ) x 1 20 / 126

31 Introduction Basic Concepts Linear Objective Is Important Here! min(x 1 1/2) 2 + (x 2 1/2) 2 x 2 s.t. x 1 {0, 1}, x 2 {0, 1} (ˆx 1, ˆx 2 ) η (x 1 1/2) 2 + (x 2 1/2) 2 η x 1 Without linear objective, optimal solution may be interior to the convex hull convexifying may do you no good! 20 / 126

32 Introduction Basic Concepts Algorithm Ingredients 1 Relaxation Used to compute a lower bound on the optimum value Obtained by enlarging feasible set; e.g. ignore constraints Should be tractable and tight 21 / 126

33 Introduction Basic Concepts Algorithm Ingredients 1 Relaxation Used to compute a lower bound on the optimum value Obtained by enlarging feasible set; e.g. ignore constraints Should be tractable and tight 2 Constraint Enforcement Exclude solutions from relaxations not feasible to MINLP Refine or tighten of relaxation Branching and Cutting 21 / 126

34 Introduction Basic Concepts Algorithm Ingredients 1 Relaxation Used to compute a lower bound on the optimum value Obtained by enlarging feasible set; e.g. ignore constraints Should be tractable and tight 2 Constraint Enforcement Exclude solutions from relaxations not feasible to MINLP Refine or tighten of relaxation Branching and Cutting 3 Upper Bounds Obtained from any feasible point; e.g. solve NLP for fixed x I. (NLP(x I )). Other heuristic techniques have been developed 21 / 126

35 Outline 1 Introduction 2 Supply Chain Water Network Design Pooling 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 22 / 126

36 Models are Important! Writing good models is extremely important in MILP Probably more important for MINLP 23 / 126

37 Models are Important! Writing good models is extremely important in MILP Probably more important for MINLP More Complexity Tradeoff between physical accuracy and mathematical tractability Convex reformulations? Linearize expressions? 23 / 126

38 Models are Important! Writing good models is extremely important in MILP Probably more important for MINLP More Complexity Tradeoff between physical accuracy and mathematical tractability Convex reformulations? Linearize expressions? MINLP Modeling Preference We want strong formulations Relaxations that can be relaxed to something that closely approximates the convex hull We prefer linear over convex over nonconvex formulations 23 / 126

39 Some MINLP Applications 1 Supply Chain Convex MINLP Discrete: Fixed charges for opening facilities Nonlinear: Nonlinear transportation costs 2 Gas/Water Network Design Nonconvex MINLP Discrete: Pipe connections/sizes Nonlinear: Pressure Loss 3 Pooling/Petrochemical Nonconvex MINLP Discrete: Which process to use? Nonlinear: Product Blending (among others) 24 / 126

40 Some MINLP Applications 1 Supply Chain Convex MINLP Discrete: Fixed charges for opening facilities Nonlinear: Nonlinear transportation costs 2 Gas/Water Network Design Nonconvex MINLP Discrete: Pipe connections/sizes Nonlinear: Pressure Loss 3 Pooling/Petrochemical Nonconvex MINLP Discrete: Which process to use? Nonlinear: Product Blending (among others) Luedtke Theorem 97.5% of all practical MINLPs have integer variables for one of two purposes... 1 Binary variables to make a multiple choice selection 2 Binary indicator variables that turn on/off continuous variables and/or constraints. 24 / 126

41 Supply Chain Outline 1 Introduction 2 Supply Chain Water Network Design Pooling 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 25 / 126

42 Supply Chain Uncapacitated Facility Location Problem introduced by Günlük, Lee, and Weismantel ( 07) and classes of strong cutting planes derived 26 / 126

43 Supply Chain Uncapacitated Facility Location Problem introduced by Günlük, Lee, and Weismantel ( 07) and classes of strong cutting planes derived M: Facility N: Customer x ij : percentage of customer j N demand met by facility i M z i = 1 facility i M is built Fixed cost for opening facility i M Quadratic cost for meeting demand j N from facility i M 26 / 126

44 Supply Chain Quadratic Uncapacitated Facility Location Problem A very simple MIQP subject to z def = min c i z i + i M i M q ij xij 2 j N x ij z i i M, j N x ij = 1 j N i M x ij 0 i M, j N z i {0, 1} i M 27 / 126

45 Supply Chain Base Case of Induction to Luedtke Theorem Note that binary variables are used as indicators: z i = 0 x ij = 0 If z = 1, then we need to model the epigraph of x 2 ij 28 / 126

46 Supply Chain Base Case of Induction to Luedtke Theorem Note that binary variables are used as indicators: z i = 0 x ij = 0 If z = 1, then we need to model the epigraph of x 2 ij Intager Programmerz r Smrt Mixed Integer Linear Programmers carefully study simple problem structures to come up with good formulations for problems Good formulations closely approximate convex hull of feasible solutions Important to understand the structure of a special MINLP with indicator variables 28 / 126

47 Supply Chain A Very Simple Structure R def = { } (x, y, z) R 2 B y x 2, 0 x uz y z = 1 z x 29 / 126

48 Supply Chain A Very Simple Structure R def = { } (x, y, z) R 2 B y x 2, 0 x uz y z = 0 x = 0, y 0 z = 1 x u, y x 2 z z = 1 x 29 / 126

49 Supply Chain A Very Simple Structure R def = { } (x, y, z) R 2 B y x 2, 0 x uz y z = 0 x = 0, y 0 z = 1 x u, y x 2 z z = 1 Deep Insights conv(r) line connecting (0, 0, 0) to y = x 2 in the z = 1 plane x 29 / 126

50 Supply Chain Characterization of Convex Hull Deep Theorem #1 { } R = (x, y, z) R 2 B y x 2, 0 x uz conv(r) = { (x, y, z) R 3 yz x 2, 0 x uz, 0 z 1, y 0 } 30 / 126

51 Supply Chain Characterization of Convex Hull Deep Theorem #1 { } R = (x, y, z) R 2 B y x 2, 0 x uz conv(r) = { (x, y, z) R 3 yz x 2, 0 x uz, 0 z 1, y 0 } x 2 yz, y, z 0 Second Order Cone Programming There are effective, robust algorithms for optimizing over conv(r) 30 / 126

52 Supply Chain In General Giving You Some Perspective For a convex function f : R n R, the perspective function P : R n+1 R of f is { P(x, z) def 0 if z = 0 = zf (x/z) if z > 0 The epigraph of P(x, z) is a cone pointed at the origin whose lower shape is f (x) 31 / 126

53 Supply Chain In General Giving You Some Perspective For a convex function f : R n R, the perspective function P : R n+1 R of f is { P(x, z) def 0 if z = 0 = zf (x/z) if z > 0 The epigraph of P(x, z) is a cone pointed at the origin whose lower shape is f (x) Exploiting Your Perspective If z i is an indicator that the (nonlinear, convex) inequality f (x) 0 must hold, (otherwise x = 0), replace the inequality with its perspective version: z i f (x/z i ) 0 The resulting (convex) inequality is a much tighter relaxation of the feasible region. 31 / 126

54 Supply Chain Perspectivizing Quadratic Uncapacitated Facility Location z def = min c i z i + i M i M q ij xij 2 j N Steps x ij z i i M, j N x ij = 1 j N i M x ij 0 i M, j N z i {0, 1} i M 32 / 126

55 Supply Chain Perspectivizing Quadratic Uncapacitated Facility Location z def = min c i z i + i M i M x ij z i x ij = 1 i M x ij 0 q ij y ij j N i M, j N j N i M, j N z i {0, 1} i M x 2 ij y ij 0 i M, j N Steps 1 Make Objective Linear 32 / 126

56 Supply Chain Perspectivizing Quadratic Uncapacitated Facility Location z def = min c i z i + i M i M x ij z i x ij = 1 i M x ij 0 q ij y ij j N i M, j N j N i M, j N z i {0, 1} i M x 2 ij z i y ij 0 i M, j N Steps 1 Make Objective Linear 2 Apply Perspective 32 / 126

57 Supply Chain Perspectivizing Quadratic Uncapacitated Facility Location z def = min c i z i + i M i M x ij z i x ij = 1 i M x ij 0 q ij y ij j N i M, j N j N i M, j N z i {0, 1} i M x 2 ij z i y ij 0 i M, j N Steps 1 Make Objective Linear 2 Apply Perspective 3 Make Giant Profit 32 / 126

58 Supply Chain Strength of Relaxations z R : Value of NLP relaxation z GLW : Value of NLP relaxation after GLW cuts z P : Value of perspective relaxation z : Optimal solution value 33 / 126

59 Supply Chain Strength of Relaxations z R : Value of NLP relaxation z GLW : Value of NLP relaxation after GLW cuts z P : Value of perspective relaxation z : Optimal solution value M N z R z GLW z P z / 126

60 Supply Chain Strength of Relaxations z R : Value of NLP relaxation z GLW : Value of NLP relaxation after GLW cuts z P : Value of perspective relaxation z : Optimal solution value M N z R z GLW z P z / 126

61 Supply Chain Strength of Relaxations z R : Value of NLP relaxation z GLW : Value of NLP relaxation after GLW cuts z P : Value of perspective relaxation z : Optimal solution value M N z R z GLW z P z Cheers! 33 / 126

62 Supply Chain Impact of SOCP m = 30, n = 100 Convex MINLP, Original: CPU seconds, nodes Convex MINLP, w/glw Ineq.: CPU seconds, nodes 34 / 126

63 Supply Chain Impact of SOCP m = 30, n = 100 Convex MINLP, Original: CPU seconds, nodes Convex MINLP, w/glw Ineq.: CPU seconds, nodes MISOCP, Perspective, 23 CPU seconds, 44 B&B nodes 34 / 126

64 Supply Chain Impact of SOCP m = 30, n = 100 Convex MINLP, Original: CPU seconds, nodes Convex MINLP, w/glw Ineq.: CPU seconds, nodes MISOCP, Perspective, 23 CPU seconds, 44 B&B nodes Larger Instances m n T N Poifect 34 / 126

65 Water Network Design Outline 1 Introduction 2 Supply Chain Water Network Design Pooling 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 35 / 126

66 Water Network Design Design of Water Distribution Networks Goal: Design minimum cost network from discrete pipe diameters to meet water demand 36 / 126

67 Water Network Design Design of Water Distribution Networks Goal: Design minimum cost network from discrete pipe diameters to meet water demand Sets N nodes in network S source nodes A: arcs in the network K: Pipe diameters 36 / 126

68 Water Network Design Design of Water Distribution Networks Variables q ij : flow pipe (i, j) A h i : hydraulic head at node i N d ij : diameter of pipe (i, j) A, where d ij {P 1,..., P r } a ij : area of cross section (i, j) A y ijk : Binary variable = 1 if pipe of size k K chosen for (i, j) A 37 / 126

69 Water Network Design Design of Water Distribution Networks Variables q ij : flow pipe (i, j) A h i : hydraulic head at node i N d ij : diameter of pipe (i, j) A, where d ij {P 1,..., P r } a ij : area of cross section (i, j) A y ijk : Binary variable = 1 if pipe of size k K chosen for (i, j) A Constraints Conservation of flow q ij (i,j) A (j,i) A q ji = D i, i N \ S. Flow bounds: (linear in a, but since a ij = 0.25πdij 2, nonlinear in d) V max a ij q ij V max a ij, (i, j) A. 37 / 126

70 Water Network Design Design of Water Distribution Networks Discrete pipe sizes d ij {P 1,..., P r } Use multiple choice integrality to remove nonlinearity a ij = πd 2 ij /4 This is in general a good modeling practice 38 / 126

71 Water Network Design Design of Water Distribution Networks Discrete pipe sizes d ij {P 1,..., P r } Use multiple choice integrality to remove nonlinearity a ij = πd 2 ij /4 This is in general a good modeling practice 1 Binary variables y ijk {0, 1} for k K 2 Model discrete choice as k K y ijk = 1, and k K 3 Model a ij = πdij 2 /4 (nonlinear) as a ij = k K ( π 4 P k P k y ijk = d ij. (i, j) A ) y ijk (i, j) A 38 / 126

72 Water Network Design Design of Water Distribution Networks Nonsmooth, nonlinear pressure loss as a function of flow q ij along arc (i, j) A and diameter d ij : h i h j = κ ij d c 1 ij sgn(q ij ) q ij c 2 (i, j) A 39 / 126

73 Water Network Design Design of Water Distribution Networks Nonsmooth, nonlinear pressure loss as a function of flow q ij along arc (i, j) A and diameter d ij : h i h j = κ ij d c 1 ij sgn(q ij ) q ij c 2 (i, j) A Another trick: disaggregate flow into flow variables for each pipe size: q ij = k K q ijk ( V max a ij )y ijk q ijk (V max a ij )y ijk 39 / 126

74 Water Network Design Why On Earth? Doing this allows us to write the (nonlinear) headloss as a function of a single variable q ijk Often in applications, the nonlinearity is low-dimensional (low {1, 2}) f (q) = sgn(q) q c 2 q 40 / 126

75 Water Network Design Why On Earth? Doing this allows us to write the (nonlinear) headloss as a function of a single variable q ijk Often in applications, the nonlinearity is low-dimensional (low {1, 2}) Therefore a piecewise linear approximation (or relaxation) may be sufficiently accurate f (q) = sgn(q) q c 2 q 40 / 126

76 Water Network Design Why On Earth? Doing this allows us to write the (nonlinear) headloss as a function of a single variable q ijk Often in applications, the nonlinearity is low-dimensional (low {1, 2}) Therefore a piecewise linear approximation (or relaxation) may be sufficiently accurate Luedtke s Theorem! Note also that y ijk = 0 q ijk = 0 f (q) = sgn(q) q c 2 q 40 / 126

77 Water Network Design Modeling Piecewise-Linear Functions Take Your Pick SOS2 Model [Beale and Tomlin, 1970, Beale and Forrest, 1976] Incremental Model [Markowitz and Manne, 1957] Multiple Choice Model [Jeroslow and Lowe, 1984] Convex Combination Model [Dantzig, 1960, Padberg, 2000] Disaggregated Convex Combination Model [Meyer, 1976] Logarithmic Model [Vielma and Nemhauser, 2011] 41 / 126

78 Water Network Design Multiple Choice Model Want to model ( i) f (q) = m i q + c i, q [B i 1, B i ] f (q) sgn(q) q c 2 B 0 B 1 B 2 B 3 B 4 q q = t = n w i, i=1 n (m i w i + c i b i ) i=1 B i 1 b i w i B i b i n 1 = i=1 b i i 1,..., n 42 / 126

79 Water Network Design Multiple Choice Model Want to model ( i) f (q) = m i q + c i, q [B i 1, B i ] f (q) sgn(q) q c 2 Introduce New Variables w i : = q if q [B i 1, B i ] b i : = 1 if q [B i 1, B i ] t f (q) B 0 B 1 B 2 B 3 B 4 q q = t = n w i, i=1 n (m i w i + c i b i ) i=1 B i 1 b i w i B i b i n 1 = i=1 b i i 1,..., n 42 / 126

80 Water Network Design Multiple Choice Model Want to model ( i) f (q) = m i q + c i, q [B i 1, B i ] f (q) sgn(q) q c 2 Introduce New Variables w i : = q if q [B i 1, B i ] b i : = 1 if q [B i 1, B i ] t f (q) B 0 B 1 B 2 B 3 B 4 q q = t = n w i, i=1 n (m i w i + c i b i ) i=1 B i 1 b i w i B i b i n 1 = i=1 b i i 1,..., n B 0 y q B n y 42 / 126

81 Water Network Design A Simple Trick Instead of modeling the piecewise-linear indicator in the standard way: B 0 y q B n y, 1 = n i=1 b i 43 / 126

82 Water Network Design A Simple Trick Instead of modeling the piecewise-linear indicator in the standard way: B 0 y q B n y, 1 = n i=1 b i Model it as y = n i=1 b i 43 / 126

83 Water Network Design A Simple Trick Instead of modeling the piecewise-linear indicator in the standard way: B 0 y q B n y, 1 = n i=1 b i Model it as y = n i=1 b i Resulting formulation is provably stronger (locally-ideal) All piecewise-linear functions turned on/off by an indicator have a similar modeling trick [Sridhar et al., 2013] It is the exact extension of the perspective reformulation to this nonconvex case 43 / 126

84 Pooling Outline 1 Introduction 2 Supply Chain Water Network Design Pooling 3 Algorithms Convex 4 Algorithms Nonconvex 5 The Future of MINLP? 44 / 126

85 Pooling The Pooling Problem The Pooling Problem A canonical problem in the petrochemical industry Decide how to blend (pool) feeds together to make products meeting quality requirements at minimum cost Feeds I Pools J Products K Generally each product has multiple attributes 45 / 126

86 Pooling Pooling Problem Basic variables x ij : Flow of input i to pool j x ik : Flow of input i to product k x jk : Flow from pool j to product k Key Parameters λ i : Concentration of attribute in feed i l k, υ k : Bounds on attribute concentration in product k C i, C j, C k : Capacities on feeds, pools, and outputs U ij, U ik, U kj : Flow capacities (some may be zero) Simple (Linear) Constraints Linear Constraints: Flow balance, bounds on flows, feed/pool/output capacity 46 / 126

87 Pooling Modeling the Quality Constraints Q-formulation [Ben-Tal et al., 1994] New variables q ij : proportion of flow to pool j coming from feed i: q ij = 1 i I q ij satisfy the relations: q ij = x ij t I x tj x ij = q ij t I x tj 47 / 126

88 Pooling Modeling the Quality Constraints Q-formulation [Ben-Tal et al., 1994] New variables q ij : proportion of flow to pool j coming from feed i: q ij = 1 q ij satisfy the relations: i I q ij = x ij t I x tj x ij = q ij t I x tj Reformulate using flow balance at pool j: t I x tj = k K x jk x ij = q ij k K x jk 47 / 126

89 Pooling Modeling the Quality Constraints Q-formulation [Ben-Tal et al., 1994] New variables q ij : proportion of flow to pool j coming from feed i: q ij = 1 q ij satisfy the relations: i I q ij = x ij t I x tj x ij = q ij t I x tj Reformulate using flow balance at pool j: t I x tj = k K x jk x ij = q ij k K Concentration of attribute at pool i is then: i I λ iq ij x jk 47 / 126

90 Pooling Modeling the Quality Constraints (2) Concentration of attribute at pool i: i I λ iq ij Quality constraint on product k: i I l k λ ix ik + j J i I λ iq ij x jk i I x ik + j J x jk υ k Multiply and gather terms (e.g., for upper bound): (λ i υ k )x ik + (λ i υ k )q ij x jk 0 i I j J i I 48 / 126

91 Pooling Modeling the Quality Constraints (2) Concentration of attribute at pool i: i I λ iq ij Quality constraint on product k: i I l k λ ix ik + j J i I λ iq ij x jk i I x ik + j J x jk υ k Multiply and gather terms (e.g., for upper bound): (λ i υ k )x ik + (λ i υ k )q ij x jk 0 i I j J i I Introduce variables w ijk to represent bilinear terms q ij x jk : x ij = q ij x jk = w ijk k K k K 48 / 126

92 Pooling Modeling the Quality Constraints (3) Bringing it all together q ij = 1, j J i I x ij = w ijk, i I, j J k K (λ i υ k )x ik + (λ i υ k )w ijk 0, k K i I j J i I (l k λ i )x ik + (l k λ i )w ijk 0 k K i I j J i I w ijk = q ij x jk, i I, j J, k K 49 / 126

93 Pooling PQ-Formulation [Tawarmalani and Sahinidis, 2002] 1 For each j J, multiply the constraint i I q ij = 1 with x jk : q ij x jk = x jk i I 2 For each j J, multiply the constraint: k K x jk C j with q ij : q ij x jk C j q ij k K 50 / 126

94 Pooling PQ-Formulation [Tawarmalani and Sahinidis, 2002] 1 For each j J, multiply the constraint i I q ij = 1 with x jk : q ij x jk = x jk i I 2 For each j J, multiply the constraint: k K x jk C j with q ij : q ij x jk C j q ij k K 3 Replace q ij x jk with w ijk : w ijk = x jk, i I w ijk C j q j k K 50 / 126

95 Pooling PQ-Formulation [Tawarmalani and Sahinidis, 2002] 1 For each j J, multiply the constraint i I q ij = 1 with x jk : q ij x jk = x jk i I 2 For each j J, multiply the constraint: k K x jk C j with q ij : q ij x jk C j q ij k K 3 Replace q ij x jk with w ijk : w ijk = x jk, i I w ijk C j q j k K Much better relaxations! Takeaway: Adding redundant constraints can significantly improve relaxations of nonconvex MINLPs. 50 / 126

96 Pooling The Pooling Problem with Binary Variables Arcs connecting the input streams to other nodes may be used only by paying a fixed cost. Network Design Ship assignment 51 / 126

97 Pooling The Pooling Problem with Binary Variables Arcs connecting the input streams to other nodes may be used only by paying a fixed cost. Network Design Ship assignment Add binary variables z ij for these arcs: x ij U ij z ij A real nonconvex, integer problem. 51 / 126

98 Pooling Practical Advice About the Luedtke Theorem We have seen many cases where we have continuous variable 0 x U and associated binary variable z {0, 1}. We see many models that write the expression w = zx. 52 / 126

99 Pooling Practical Advice About the Luedtke Theorem We have seen many cases where we have continuous variable 0 x U and associated binary variable z {0, 1}. We see many models that write the expression w = zx. On The Horror! Never multiply a binary variable by a bounded continuous variable You can linearize this: 0 w Uz U(1 z) x w U(1 z) 52 / 126

100 Pooling Practical Advice About the Luedtke Theorem (2) The same trick applies to binary variables that indicate constraints. Optimal transmission switching. Instead of x ij = α ij z ij (θ j θ i ) (i, j) E Write α ij (θ i θ j ) x ij + M(1 z ij ) 0 α ij (θ i θ j ) x ij M(1 z ij ) 0 (i, j) E (i, j) E Fancier models (disjunctive programming) exist [Grossmann and Lee, 2003]), but you have our permission to use big-m in this case 53 / 126

101 Pooling Other MINLP Applications 1 Portfolio Management Convex MIQP Discrete: Trading Strategy Nonlinear: Utility 2 Reactor Core Reloading Nonconvex MINLP Discrete: Bundle Reordering Nonlinear: Neutron Transport 3 Power Grid Operation Nonconvex MINLP Discrete: Unit Committment Nonlinear: AC Power Flow 4 Building Co-Generation Nonconvex MINLP Discrete: Unit Committment Nonlinear: Ramping Constraints 5 Oil-Spill Response Nonconvex MINLP Discrete: Resource Allocation Nonlinear: Fluid Flow & Chemistry 2. Core-Reloading 4. Building Demand 5. Oil-Spill Response 54 / 126