Mixed Integer Programming Models for Non-Separable Piecewise Linear Cost Functions

Transcription

1 Mixed Integer Programming Models for Non-Separable Piecewise Linear Cost Functions Juan Pablo Vielma H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology Joint work with Shabbir Ahmed and George Nemhauser. University of Pittsburgh, 2008 Pittsburgh, PA

2 Piecewise Linear Optimization min f 0 (x) s.t. f i (x) 0 i I x X R n is a piecewise linear function (PLF) and is any compact set. Convex = Linear Programming. Non-Convex = NP Hard. Specialized algorithms (Tomlin 1981,..., de Farias et al ) or Mixed Integer Programming Models (12+ papers). 2/26

3 Mixed Integer Models for PLFs Existing studies are for separable functions: Contributions (Vielma et al. 2008a,b): First models with a logarithmic # of binary variables. Theoretical and computational comparison: multivariate (non-separable) and lower semicontinuous functions in a unifying framework. 3/26

4 Outline Applications of Piecewise Linear Functions. Modeling Piecewise Linear Functions. Logarithmic Formulations. Comparison of Formulations. Extension to Lower Semicontinuous Functions. Final Remarks. 4/26

5 Applications of Piecewise Linear Functions Economies of Scale: Concave Single and multi-commodity network flow. Applications in telecommunications, transportation, and logistics. (Balakrishnan and Graves 1989,..., Croxton, et al. 2007). 5 /26

6 Applications of Piecewise Linear Functions Fixed Charges and Discounts x y Fixed Costs in Logistics. 2.Discounts (e.g. Auctions: Sandholm, et al. 2006, CombineNet). 3.Discounts in fixed charges (Lowe 1984). 6/26

7 Applications of Piecewise Linear Functions Non-Linear and PDE Constraints Source Connections Pipes/Valves/ Compressors Pipe A ρ t + ρ 0 p q x = 0, x = λ v v 2D ρ. Demand Points Gas Network Optimization (Martin et al. 2006). 7/26

8 Applications of Piecewise Linear Functions Non-Linear and PDE Constraints Source Connections Pipes/Valves/ Compressors Pipe A ρ t + ρ 0 p q x = 0, x = λ v v 2D ρ. Demand Points Gas Network Optimization (Martin et al. 2006). 7/26

9 Applications of Piecewise Linear Functions Non-Linear and PDE Constraints Source Connections Pipes/Valves/ Compressors Pipe A ρ t + ρ 0 p q x = 0, x = λ v v 2D ρ. Demand Points Gas Network Optimization (Martin et al. 2006). 7/26

10 Applications of Piecewise Linear Functions Numerically Exact Global Optimization Process engineering (Bergamini et al. 2005, 2008, Computers and Chemical Eng.) Wetland restoration (Stralberg et al. 2009). 8/26

11 Applications of Piecewise Linear Functions Numerically Exact Global Optimization Process engineering (Bergamini et al. 2005, 2008, Computers and Chemical Eng.) Wetland restoration (Stralberg et al. 2009). 8/26

12 Modeling Piecewise Linear Functions Piecewise Linear Functions: Definition f(2) = 40 f(1) = 32 f(5) = 15 f(0) = 10 f(4) = Definition 1. Piecewise Linear f : D R n R: { f(x) := m P x+c P x P P P. for finite family of polytopes P such that D = P P P 9/26

13 Modeling Piecewise Linear Functions Piecewise Linear Functions: Definition f(x,y) y x P Definition 1. Piecewise Linear f : D R n R: { f(x) := m P x+c P x P P P. for finite family of polytopes P such that D = P P P 9/26

14 Modeling Piecewise Linear Functions Piecewise Linear Functions: Definition f(x,y) y x P Definition 1. Piecewise Linear f : D R n R: { f(x) := m P x+c P x P P P. for finite family of polytopes P such that D = P P P 9/26

15 Modeling Piecewise Linear Functions Modeling Function = Epigraph f(2) = 40 f(1) = f(5) = 15 f(0) = 10 f(4) = (a) f. (b) epi(f). Example: 10/26

16 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate 3 P 1, /26

17 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate P 1, = /26

18 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate P 1, = /26

19 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate P 1, = x =0λ 0 +2λ 2 +4λ 4 z 1λ 0 +3λ 2 +0λ 4 1=λ 0 + λ 2 + λ 4, λ 0, λ 2, λ 4 0 λ 0 y P1, λ 2 y P1 + y P2, λ 4 y P2 1=y P1 + y P2, y P1,y P2 {0, 1} 11/26

20 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate P 1, = x =0λ 0 +2λ 2 +4λ 4 z 1λ 0 +3λ 2 +0λ 4 1=λ 0 + λ 2 + λ 4, λ 0, λ 2, λ 4 0 λ 0 y P1, λ 2 y P1 + y P2, λ 4 y P2 1=y P1 + y P2, y P1,y P2 {0, 1} 11/26

21 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate P 1, = x =0λ 0 +2λ 2 +4λ 4 z 1λ 0 +3λ 2 +0λ 4 1=λ 0 + λ 2 + λ 4, λ 0, λ 2, λ 4 0 λ 0 y P1, λ 2 y P1 + y P2, λ 4 y P2 1=y P1 + y P2, y P1,y P2 {0, 1} 11/26

22 Modeling Piecewise Linear Functions Convex Combination (CC): Univariate P 1, = x =0λ 0 +2λ 2 +4λ 4 z 1λ 0 +3λ 2 +0λ 4 1=λ 0 + λ 2 + λ 4, λ 0, λ 2, λ 4 0 λ 0 y P1, λ 2 y P1 + y P2, λ 4 y P2 are SOS2 1=y P1 + y P2, y P1,y P2 {0, 1} 11/26

23 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). 12/26

24 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). v V(P) λ vv = x, v V(P) λ v (m P v +c P ) z λ v λ v 0 v V(P) := P P V (P), v V(P) λ v = 1 y P {P P :v V (P)} P P v V(P), y P =1, y P {0,1} P P 12/26

25 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). v V(P) λ vv = x, v V(P) λ v (m P v +c P ) z λ v λ v 0 v V(P) := P P V (P), v V(P) λ v = 1 y P {P P :v V (P)} P P v V(P), y P =1, y P {0,1} P P 12/26

26 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). Original Constraints λ v {P P :v V (P)} v V(P) λ vv = x, v V(P) λ v (m P v +c P ) z λ v 0 v V(P) := P P V (P), v V(P) λ v = 1 y P v V(P), P P y P =1, y P {0,1} P P 12/26

27 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). Extra Constraints λ v {P P :v V (P)} v V(P) λ vv = x, v V(P) λ v (m P v +c P ) z λ v 0 v V(P) := P P V (P), v V(P) λ v = 1 y P v V(P), P P y P =1, y P {0,1} P P 12/26

28 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). Extra Constraints λ v v V(P) λ vv = x, v V(P) λ v (m P v +c P ) z λ v 0 v V(P) := P P V (P), v V(P) λ v = 1 SOS2 y P only v V(P), for univariate {P P :v V (P)} P P y P =1, y P {0,1} P P 12/26

29 Modeling Piecewise Linear Functions Convex Combination (CC): Multivariate Univariate (Dantzig, 1960)... Multivariate (Lee and Wilson (2001). Extra Constraints λ v {P P :v V (P)} v V(P) λ vv = x, v V(P) λ v (m P v +c P ) z λ v 0 v V(P) := P P V (P), v V(P) λ v = 1 y P v V(P), P P y P =1, y P {0,1} P P Nonzero variables are associated to vertices of a single polytope. 12/26

30 Modeling Piecewise Linear Functions Existing Models are Linear on Other models: Multiple Choice (MC), Incremental (Inc), Disaggregated Convex Combination (DCC). Number of binary variables and combinatorial extra constraints f(x,y) are linear in. For multivariate on a y grid. x Logarithmic sized formulations? 13/26

31 Logarithmic Formulations SOS1, SOS2 and CC constraints. SOS1-2 (Beale and Tomlin 1970): SOS1: At most one variable is nonzero. SOS2: Only 2 adjacent variables are nonzero. (0,1,1/2,0,0) (0,1,0,1/2,0), allowed sets. SOS1: SOS2: CC: 14/26

32 Logarithmic Formulations Logarithmic Formulation for SOS1 15/26

33 Logarithmic Formulations Logarithmic Formulation for SOS1 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 15/26

34 Logarithmic Formulations Logarithmic Formulation for SOS1 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 15/26

35 Logarithmic Formulations Logarithmic Formulation for SOS1 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 15/26

36 Logarithmic Formulations Logarithmic Formulation for SOS1 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 15/26

37 Logarithmic Formulations Logarithmic Formulation for SOS1 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 15/26

38 Logarithmic Formulations Logarithmic Formulation for SOS1 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 15/26

39 Logarithmic Formulations Logarithmic Formulation for SOS1 0 0 In general: /26

40 Logarithmic Formulations Logarithmic Formulation for SOS2 16/26

41 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

42 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

43 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

44 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

45 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

46 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

47 Logarithmic Formulations Logarithmic Formulation for SOS2 Injective function: B : {0,...,m 1} {0,1} log 2 m Variables: Idea: w {0,1} log 2 m 16/26

48 Logarithmic Formulations Logarithmic Formulation for SOS /26

49 Logarithmic Formulations Logarithmic Formulation for SOS2 0 0 Where is?! λ /26

50 Logarithmic Formulations Logarithmic Formulation for SOS2 0 0 Where is?! λ /26

51 Logarithmic Formulations Logarithmic Formulation for SOS2 0 0 Where is?! λ /26

52 Logarithmic Formulations Logarithmic Formulation for SOS Where is?! λ 2 In general: B(i) and B(i+1) differ in one component Gray Code. 16/26

53 Logarithmic Formulations Independent Branching: Dichotomies λ 0 λ 1 λ 2 λ 3 λ 4 λ 0 λ 1 λ 2 λ 3 λ 4 λ 0 λ 0 λ 0 λ 0 λ 1 λ 1 λ 1 λ 1 λ 2 λ 2 λ 2 λ 2 λ 3 λ 3 λ 3 λ 3 λ 4 λ 4 λ 4 λ 4 17/26

54 Logarithmic Formulations Independent Branching: Dichotomies λ 0 λ 1 λ 2 λ 3 λ 4 λ 0 λ 1 λ 2 λ 3 λ λ 0 λ 0 λ 0 λ 0 λ 1 λ 1 λ 1 λ 1 λ 2 λ 2 λ 2 λ 2 λ 3 λ 3 λ 3 λ 3 λ 4 λ 4 λ 4 λ 4 17/26

55 Logarithmic Formulations Independent Branching for 2 var CC y Select Triangle by forbidding vertices. 2 stages: Select Square by SOS2 on each variable. Select 1 triangle from each square. 1 T x 18/26

56 Logarithmic Formulations Independent Branching for 2 var CC y Select Triangle by forbidding vertices. 2 stages: Select Square by SOS2 on each variable. Select 1 triangle from each square. 1 T x 18/26

57 Logarithmic Formulations Independent Branching for 2 var CC Select Triangle by forbidding vertices. 2 stages: Select Square by SOS2 on each variable. Select 1 triangle from each square. y T x 4 L = {(r, s) J : r even and s odd} = {square vertices} R = {(r, s) J : r odd and s even} = {diamond vertices} 18/26

58 Comparison of Formulations Strength of LP Relaxations Sharp Models: LP = lower convex envelope. LP relaxation (a) epi(f). (b) conv(epi(f)). All popular models are sharp. Locally Ideal: LP = Integral (All but CC, even Log). Locally ideal implies Sharp. 19/26

59 Comparison of Formulations Strength of LP Relaxations Sharp Models: LP = lower convex envelope. LP relaxation (a) epi(f). (b) conv(epi(f)). All popular models are sharp. Locally Ideal: LP = Integral (All but CC, even Log). Locally ideal implies Sharp. 19/26

60 Comparison of Formulations Computational Results Instances Transportation problems (10x10 & 5x2). Univariate: Concave Separable Objective. Multivariate: 2-commodity. Functions: affine in k segments or k x k grid triangulation (100 instances per k). Solver: CPLEX 11 on 2.4Ghz machine Logarithmic versions of CC = Log, DCC=DLog. 20/26

61 Comparison of Formulations Univariate Case (Separable) Average Time Solve [s] DCC CC MC DLog Log Number of Segments 21/26

62 Comparison of Formulations Univariate Case (Separable) Average Time Solve [s] DCC CC MC DLog Log Number of Segments 21/26

63 Comparison of Formulations Univariate Case (Separable) Average Time Solve [s] DCC CC MC DLog Log Number of Segments 21/26

64 Comparison of Formulations Univariate Case (Separable) Average Time Solve [s] DCC CC MC DLog Log Number of Segments 21/26

65 Comparison of Formulations Multivariate Case (Non-Separable) Average Time Solve [s] DCC CC MC DLog Log 1 4x4 8x8 16x16 Grid Size 22/26

66 Lower Semicontinuous Functions Lower Semicontinuous PLFs { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

67 Lower Semicontinuous Functions Lower Semicontinuous PLFs { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

68 Lower Semicontinuous Functions Lower Semicontinuous PLFs { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

69 Lower Semicontinuous Functions Lower Semicontinuous PLFs x f(x, y) := 3 (x, y) (0,1] 2 2 (x, y) {(x, y) 2 : x =0, y > 0} 2 (x, y) {(x, y) 2 : y =0, x > 0} 0 (x, y) {(0,0)}. y { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

70 Lower Semicontinuous Functions Lower Semicontinuous PLFs x f(x, y) := 3 (x, y) (0,1] 2 2 (x, y) {(x, y) 2 : x =0, y > 0} 2 (x, y) {(x, y) 2 : y =0, x > 0} 0 (x, y) {(0,0)}. y { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

71 Lower Semicontinuous Functions Lower Semicontinuous PLFs x f(x, y) := 3 (x, y) (0,1] 2 2 (x, y) {(x, y) 2 : x =0, y > 0} 2 (x, y) {(x, y) 2 : y =0, x > 0} 0 (x, y) {(0,0)}. y { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

72 Lower Semicontinuous Functions Lower Semicontinuous PLFs x f(x, y) := 3 (x, y) (0,1] 2 2 (x, y) {(x, y) 2 : x =0, y > 0} 2 (x, y) {(x, y) 2 : y =0, x > 0} 0 (x, y) {(0,0)}. y { f(x) := P = {x m P x + c P x P P P n : a i x b i i {1,...,p}, a i x<b i i {p,...,m}} Finite family of copolytopes 23/26

73 Lower Semicontinuous Functions Lower Semicontinuous Models Direct from Disjunctive Programming (Jeroslow and Lowe) Extreme point = DCC. Traditional = Multiple Choice (MC). x Other models can be adapted to special types of y discontinuities (e.g. simple fixed charges). MC, DCC, DLog are locally ideal and sharp. Computations: 2-commodity FC discount function. 24/26

74 Comparison of Formulations Multivariate Lower Semicontinuous DCC MC DLog Average Time Solve [s] x4 8x8 16x16 32x32 Grid Size 25/26

75 Comparison of Formulations Multivariate Lower Semicontinuous DCC MC DLog Average Time Solve [s] x4 8x8 16x16 32x32 Grid Size 25/26

76 Comparison of Formulations Multivariate Lower Semicontinuous DCC MC DLog Average Time Solve [s] x4 8x8 16x16 32x32 Grid Size 25/26

77 Final Remarks Final Remarks Unifying theoretical framework: allows for multivariate non-separable and lower semicontinuous functions. First logarithmic formulations: Theoretically strong and provides significant computational advantage for large. Revive forgotten formulations and functions: MC and fixed charge discount function. x y 26/26