Incremental Formulations for SOS1 Variables

Transcription

1 Incremental Formulations for SOS Variables Juan Pablo Vielma Massachusetts Institute of Technology joint work with Sercan Yıldız arnegie Mellon University INFORMS nnual Meeting, October 22 Phoenix, rizona

2 Outline Introduction Encodings General Incremental Formulation Incremental Formulation and ranching omputational Results Summary 2/3

3 Introduction Logarithmic Formulation for SOS i= i = i= bi i = y 2 R n + y 2 {, } m b i n i= = {, }log 2 n Li and Lu 29, dams and Henry 2, V. and Nemhauser 28. Sommer, TIMS 972. Log = inary Encoding Other choices of lead to standard and b i n i= incremental formulations 3/3

4 Introduction General Logarithmic Formulation P i k i= polytopes kx X v i v = x i= v2ext(p i ) x 2 k[ i= P i, kx i= kx X v2ext(p i ) X b i i v = i v = y i= v2ext(p i ) y 2 {, } dlog 2 (k)e, i v V., hmed and Nemhauser 2; V /3

5 Introduction General Logarithmic Formulation P i k i= polytopes kx X v i v = x i= v2ext(p i ) k[ x 2 P i, i= lso for general polyhedron with common recession cones. kx X i= v2ext(p i ) kx X i= v2ext(p i ) b i y 2 {, } dlog 2 (k)e, i v = i v = y i v V., hmed and Nemhauser 2; V /3

6 Encodings Unary and inary Encodings = y Unary, i = y i = y inary 5/3

7 Encodings Incremental Encoding = y, X 8 i= i =, 2 R 8,y2 {, } 7 y y 2... y 7 Linear transformation of -formulation gives generalization of incremental -formulation of Lee and Wilson /3

8 Incremental Formulation Incremental Delta Formulation v 2 2 v 2 3 P 2 v 2 4 x = v + kx i= r i X j=2 v i j v i i j v kx v i v i r i y i v 2 v 3 i=2 v 3 P 3 P v i j 2 [, ], y i 2 {, } v 2 Yıldız and V. 22 7/3

9 Incremental Formulation Incremental Delta Formulation v 2 2 v 2 3 P 2 v 2 4 x = v + kx i= r i X j=2 v i j v i i j v kx v i v i r i y i v 2 v 3 i=2 v 3 v 2 P v P 3 r i X j=2 i j apple, i j Yıldız and V. 22 7/3

10 Incremental Formulation Incremental Delta Formulation v 2 2 v 2 3 P 2 v 2 4 x = v + kx i= r i X j=2 v i j v i i j v kx v i v i r i y i v 2 v 3 i=2 v 3 P v P 3 r i X i j apple y i v 2 j=2 Yıldız and V. 22 7/3

11 Incremental Formulation Incremental Delta Formulation v 2 2 v 2 3 P 2 v 2 4 x = v + kx i= r i X j=2 v i j v i i j v kx v i v i r i y i v 2 v 3 i=2 v 3 P 3 P v y i+ apple i r i, y i+ apple y i v 2 Yıldız and V. 22 7/3

12 ranching Example: # of & Nodes min s.t. x x 2 { i } n i= /3

13 ranching Example: # of & Nodes min s.t. x x 2 { i } n i= min s.t. i= i= i= t i = i= bi i = y 2R n y 2 {, } m /3 8

14 ranching Example: Unary Encoding min s.t. i= i= i= t i = i= bi i = y 2R n + y 2 {, } m 9/3

15 ranching Example: Unary Encoding t =,t LP = min s.t. i= i= i= t i = i= bi i = y 2R n + y 2 {, } m 9/3

16 ranching Example: Unary Encoding t =,t LP = + min s.t. i= i= i= t i = i= bi i = y 2R n + y 2 {, } m 9/3

17 ranching Example: Unary Encoding min t s.t. t =,t LP = = y y 2 y 3 y 4 i= i= i= i = i= bi i = y 2R n + y 2 {, } m 9/3

18 ranching Example: Unary Encoding min t s.t. t =,t LP = = y = y y 2 y 3 y 4 i= i= i= i = i= bi i = y 2R n + y 2 {, } m 9/3

19 ranching Example: Unary Encoding t =,t LP = = y = y y 2 y 3 y 4 min s.t. i= i= i= t i = i= bi i = y 2R n + y 2 {, } m 9/3

20 ranching Example: Unary Encoding t =,t LP = = y = y y 2 y 3 y 4 min s.t. i= i= i= t i = i= bi i = y 2R n + y 2 {, } m 9/3

21 ranching Example: Unary Encoding t =,t LP = = y y 2 y 3 y 4 min s.t. y = y 4 = i= i= i= t i = i= bi i = y 2R n + y 2 {, } m 9/3

22 = y y 2 y 3 y 4 t =,t LP = t LP =unless: y i = 8i apple n/2 or y i = 8i n/2 /3 ranching Example: Unary Encoding 9 ( ( n/2 n/2

23 ranching Example: Unary Encoding t LP =unless: y i = 8i apple n/2 or y i = 8i n/2 ( ( = n/2 n/2 y y 2 y 3 y 4 Need n/2 branches to solve. t =,t LP = 9/3

24 ranching Example: inary Encoding (... 2 k t =,t LP = = y y 2 y 3 /3

25 ranching Example: inary Encoding (... 2 k = y y 2 y 3 t =,t LP = y = y 2 = /3

26 ranching Example: inary Encoding t LP =unless: y i = 8i (... 2 k = y y 2 y 3 t =,t LP = y = y 2 = /3

27 ranching Example: inary Encoding t LP =unless: y i = 8i... = y y 2 y 3 Need all k = log n 2 branches ( 2 k to solve. t =,t LP = y = y 2 = /3

28 ranching Example: Incremental Encoding t =,t LP = /3

29 /3 ranching Example: Incremental Encoding t =,t LP = = y y 2 y 3 y 4

30 /3 ranching Example: Incremental Encoding t =,t LP = = y y 2 y 3 y 4 y 3 =_ y 3 =

31 t LP =if: y i =_ y i = Only need branch! /3 ranching Example: Incremental Encoding t =,t LP = = y y 2 y 3 y 4 y 3 =_ y 3 =

32 omputational Revisiting Transportation Instances Discontinuous Piecewise Linear Disc. PWL + Semicontinuous 2/3

33 omputational Piecewise Linear L-Inc Log /3

34 omputational Piecewise Linear + Semi ontinuous L-Inc Log /3

35 Summary General encoding formulation General incremental formulation Incremental formulation can be better than logarithmic formulation! Paper ready soon, meanwhile: Survey: V., Mixed Integer Linear Programming Formulation Techniques, Web and Opt-Online. 5/3