Mixed-Integer Nonlinear Programming

Transcription

1 Mixed-Integer Nonlinear Programming Claudia D Ambrosio CNRS researcher LIX, École Polytechnique, France pictures taken from slides by Leo Liberti MPRO PMA

2 Motivating Applications

3 Nonlinear Knapsack Problem

4 Nonlinear Knapsack Problem We are given N objects and a container with capacity C. Every object j has a weight w j, a profit p j, and a max available quantity U j (j = 1,, N). Find for each object the quantity that has to be loaded in the container so as the total weight isn t greater than the capacity, and the total profit is maximized.

5 Nonlinear Knapsack Problem Variables: quantities x. Every variable x j must be 0 and U j. The profit is a nonlinear function of the the quantity. The total weight should be less or equal than the capacity C. Maximization of the total profit. MINLP problem.

6 Nonlinear Knapsack Problem max N f j (x j ) j=1 N w j x j C j=1 0 x j U j j = 1,, N. where f j (x j ) is any twice continuously differentiable and univariate function j = 1,, N (e.g., c j (1+b j exp( a j (x j +d j ))) ).

7 Nonlinear Knapsack Problem If a subset J of object can assume only integer value: x j integer j J {1,, N}. Real-world interpretation: Given a budget C we want to decide how to invest our money so that our profit is maximized. We can buy by choosing among N items, each of which is available for a maximum quantity U j and each unit of item j costs w j. Some of the items have to be bought in lots.

8 Nonlinear KP problem: Citations 1. C. D Ambrosio, S. Martello. Heuristic algorithms for the general nonlinear separable knapsack problems, Computers and Operations Research, 38 (2), pp , H. Kellerer, U. Pferschy, D. Pisinger. Knapsack Problems. Springer (2004). 3. S. Martello, P. Toth. Knapsack problems: Algorithms and computer interpretations. Wiley-Interscience (1990).

9 Pooling Problem

10 The Pooling Problem Inputs I Pools L Outputs J Nodes N = I L J Arcs A (i, j) (I L) (L J) (I J) on which materials flow Material attributes: K Arc capacities: u ij, (i, j) A Node capacities: C i, i N Attribute requirements α kj, k K, j J

11 The Pooling Problem: Motivation refinery processes in the petroleum industry

12 The Pooling Problem: Motivation refinery processes in the petroleum industry different specifications: e.g., sulphur/carbon concentrations or physical properties such as density, octane number,...

13 The Pooling Problem: Motivation refinery processes in the petroleum industry different specifications: e.g., sulphur/carbon concentrations or physical properties such as density, octane number,... wastewater treatment, e.g., Karuppiah and Grossmann (2006)

14 The Pooling Problem: Motivation refinery processes in the petroleum industry different specifications: e.g., sulphur/carbon concentrations or physical properties such as density, octane number,... wastewater treatment, e.g., Karuppiah and Grossmann (2006) Formally introduced by Haverly (1978)

15 The Pooling Problem: Motivation refinery processes in the petroleum industry different specifications: e.g., sulphur/carbon concentrations or physical properties such as density, octane number,... wastewater treatment, e.g., Karuppiah and Grossmann (2006) Formally introduced by Haverly (1978) Alfaki and Haugland (2012) formally proved it is strongly NP-hard

16 The Pooling problem: Citations 1. Haverly, Studies of the behaviour of recursion for the pooling problem, ACM SIGMAP Bulletin, Adhya, Tawarmalani, Sahinidis, A Lagrangian approach to the pooling problem, Ind. Eng. Chem., Audet et al., Pooling Problem: Alternate Formulations and Solution Methods, Manag. Sci., Liberti, Pantelides, An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms, JOGO, Misener, Floudas, Advances for the pooling problem: modeling, global optimization, and computational studies, Appl. Comput. Math., D Ambrosio, Linderoth, Luedtke, Valid inequalities for the pooling problem with binary variables, IPCO, Tawarmalani and Sahinidis. Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications, Ch. 9. Kluwer Academic Publishers, 2002.

17 Mathematical Programming

18 Mathematical Programming MP: formal language for expressing optimization problems P Parameters p =problem input p also called an instance of P Decision variables x: encode problem output Objective function min f (p, x) Constraints i m g i (p, x) 0 f, g: explicit mathematical expressions involving symbols p, x If an instance p is given (i.e. an assignment of numbers to the symbols in p is known), write f (x), g i (x) This excludes black-box optimization

19 Main optimization problem classes

20 MP: Notation P: MP formulation with decision variables x = (x 1,..., x n ) Solution: assignment of values to decision variables, i.e. a vector v R n F(P) =set of feasible solutions x R n such that i m (g i (x) 0) G(P) =set of globally optimal solutions x R n s.t. x F(P) and y F(P) (f (x) f (y))

21 Exact reformulations The formulation Q is an exact reformulation of P if an efficiently computable surjective map φ : F(Q) F(P) s.t. φ G(Q) is onto G(P) Informally: any optimum of Q can be mapped easily to an optimum of P, and for any optimum of P there is a corresponding optimum of Q P Q G G phig F F Construct Q so that it is easier to solve than P phi

22 Exact reformulations: example xy when x is binary If bilinear term xy where x {0, 1}, y [0, 1] We can construct an exact reformulation: Replace each term xy by an added variable w Adjoin Fortet s reformulation constraints: w 0 w x + y 1 w x w y Get a MILP reformulation Solve reformulation using CPLEX: more effective than solving MINLP

23 Proof

24 Proof w 0 w x + y 1 w x w y x = 0 w 0 w y 1 w 0 w y w = 0 x = 1 w 0 w y w 1 w y w = y

25 Relaxations The formulation Q is a relaxation of P if min f Q (y) min f P (x) ( ) Relaxations are used to compute worst-case bounds to the optimum value of the original formulation Construct Q so that it is easy to solve Proving ( ) may not be easy in general The usual strategy: Make sure y x and F(Q) F(P) Make sure x F(P) (fq (y) f P (x)) Then it follows that Q is a relaxation of P Example: convex relaxation F(Q) a convex set containing F(P) fq a convex underestimator of f P Then Q is a cnlp and can be solved efficiently

26 Relaxations: example xy when x, y continuous Get bilinear term xy where x [x L, x U ], y [y L, y U ] We can construct a relaxation: Replace each term xy by an added variable w Adjoin following constraints: w x L y + y L x x L y L w x U y + y U x x U y U w x U y + y L x x U y L w x L y + y U x x L y U These are called McCormick s envelopes Get an LP relaxation (solvable in polynomial time)

27 Reformulations: Citations & Software McCormick, Computability of global solutions to factorable nonconvex programs: Part I Convex underestimating problems, Math. Prog Liberti, Reformulations in Mathematical Programming: definitions and systematics, RAIRO-RO 2009 Williams, Model building in mathematical programming, 2002 Liberti, Cafieri, Tarissan, Reformulations in Mathematical Programming: a computational approach, in Abraham et al. (eds.), Foundations of Comput. Intel., 2009 ROSE (

28 Example: the Pooling Problem Simple constraints Variables x ij for flow on arcs Inputs I Pools L Outputs J Flow balance constraints at pools: x il x lj = 0, l L i I l j J l

29 Example: the Pooling Problem Simple constraints Variables x ij for flow on arcs Flow balance constraints at pools: x il x lj = 0, l L i I l j J l Inputs I Pools L Outputs J Capacity constraints: x ij + x il C i, j J i l L i x il C l, i I l x ij + x lj C j, i I j l L j i I l L j J

30 Example: the Pooling Problem Complicating constraints Inputs have associated attribute concentrations λ ki, k K, i I Concentration of attribute in pool is the weighted average of the concentrations of its inputs. This results in bilinear constraints.

31 Example: the Pooling Problem Complicating constraints Inputs have associated attribute concentrations λ ki, k K, i I Concentration of attribute in pool is the weighted average of the concentrations of its inputs. This results in bilinear constraints. P-formulation (Haverly 78): Keep track of concentration p kl of attribute k in pool l

32 Example: the Pooling Problem Complicating constraints Inputs have associated attribute concentrations λ ki, k K, i I Concentration of attribute in pool is the weighted average of the concentrations of its inputs. This results in bilinear constraints. P-formulation (Haverly 78): Keep track of concentration p kl of attribute k in pool l Q-formulation (Ben-Tal et al. 94): Variables q il for proportion of flow into pool l coming from input i

33 Example: the Pooling Problem P-formulation x ij + x il C i, j J i l L i x il C l, i I l x ij + x lj C j, i I j l L j x il x lj = 0, i I l j J l p kl = i I l L j J l L i I l λ ki x il i J l x lj k K, l L i I j λ ki x ij + l L j p kl x lj i I j L j x ij α kj, k K, j J

34 Example: the Pooling Problem P-formulation x ij + x il C i, j J i l L i x il C l, i I l x ij + x lj C j, i I j l L j x il x lj = 0, i I l j J l p kl x lj = λ ki x il i J l i I l λ ki x ij + p kl x lj α kj x ij, i I j l L j i I j L j i I l L j J l L k K, l L k K, j J

35 Example: the Pooling Problem Q-formulation x il = q il q il = 1, i I l j J l x lj, i I, l L i l L

36 Example: the Pooling Problem Q-formulation x il = q il q il = 1, i I l j J l x lj, i I, l L i l L Attribute constraints ( ) λ ki x ij + x lj λ ki q il α kj x ij, i I j l L j i I l i I j L j k K, j J

37 Example: the Pooling Problem Q-formulation x ij + x il C i, j J i l L i x il C l, i I l x ij + x lj C j, i I j l L j x il q il x lj = 0 j J l q il = 1 i I l ( ) λ ki x ij + x lj λ ki q il α kj x ij, i I j l L j i I l i I j L j i I l L j J i I, l L i l L k K, j J

38 Example: the Pooling Problem with binary vars From NLP to MINLP Decide whether to install pipes or not (0/1 decision) Associate a binary variable z ij with each pipe (suppose for now on arcs from input to output) Extra constraints: x ij min(c i, C j )z ij z ij {0, 1} i I, j J i i I, j J i Objective Function Fixed cost for installing pipe min c i x il + x ij p j x ij + x lj + f ij z ij i I l Li j J i j J i Ij l L j i I j J i

39 Global optimization methods

40 Global Optimization methods: Deterministic / Stochastic subregion 1 discarded as h(c) > f(e) subregion 2 f(x) h(x) g(x) b c a d e objective function convex relaxation in whole space a: solution of convex relaxation in whole space b: local solution of objective function in whole space Exact = Deterministic Exact in continuous space: ε-approximate (find solution within pre-determined ε distance from optimum in obj. fun. value) Heuristic = Stochastic Find solution with probability 1 in infinite time For some problems, finite convergence to optimum (ε = 0)

41 GO: Multistart The easiest GO method 1: f = 2: x = (,..., ) 3: while termination do 4: x = (random(),..., random()) 5: x = localsolve(p, x ) 6: if f P (x) < f then 7: f f P (x) 8: x x 9: end if 10: end while Termination condition: e.g. repeat k times

42 Six-hump camelback function f (x, y) = 4x 2 2.1x x 6 + xy 4y 2 + 4y 4 Global optimum (Couenne)

43 Six-hump camelback function f (x, y) = 4x 2 2.1x x 6 + xy 4y 2 + 4y Multistart with IPOPT, k = 5

44 Six-hump camelback function f (x, y) = 4x 2 2.1x x 6 + xy 4y 2 + 4y Multistart with IPOPT, k = 10

45 Six-hump camelback function f (x, y) = 4x 2 2.1x x 6 + xy 4y 2 + 4y Multistart with IPOPT, k = 20

46 Six-hump camelback function f (x, y) = 4x 2 2.1x x 6 + xy 4y 2 + 4y Multistart with IPOPT, k = 50

47 Six-hump camelback function f (x, y) = 4x 2 2.1x x 6 + xy 4y 2 + 4y Multistart with SNOPT, k = 20

48 GO: Citations Schoen, Two-Phase Methods for Global Optimization, in Pardalos et al. (eds.), Handbook of Global Optimization 2, 2002 Liberti, Kucherenko, Comparison of deterministic and stochastic approaches to global optimization, ITOR 2005