Encodings in Mixed Integer Linear Programming
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1 Encodings in Mixed Integer Linear Programming Juan Pablo Vielma Sloan School of usiness, Massachusetts Institute of Technology Universidad de hile, December, 23 Santiago, hile.
2 Mixed Integer er inary Formulations MIP Formulations = Model Finite lternatives x 2 n[ i= P i R d f(x,y) y x P i 2/29 9
3 Textbook Formulation f(d 3 ) f(d 2 ) f(d ) f(d 4 ) d d 2 d 3 d 4 Formulation for f(x)=z 4X 4X d i i = x, f(d i ) i = z i= 4X i= i= i =, i 3X y i =, y i 2 {, } apple y, 2 apple y + y 2 3 apple y 2 + y 3, 4 apple y 3 i= 3/29 9
4 Textbook Formulation f(d 3 ) f(d 2 ) f(d ) f(d 4 ) d d 2 d 3 d 4 Formulation for f(x)=z 4X 4X d i i = x, f(d i ) i = z i= i= 4X i= i= i =, i 3X y i =, y i 2 {, } Weak apple y, 2 apple y + y 2 3 apple y 2 + y 3, 4 apple y 3 3/29 9
5 etter Formulation f(d 3 ) f(d 2 ) f(d ) f(d 4 ) d d 2 d 3 d 4 Formulation for f(x)=z f(d )+ 3X d + (d i+ d i ) i = x, i= 3X (f(d i+ ) f(d i )) i = z i= 3 apple y 2 apple 2 apple y apple y i 2{, } 4/299
6 etter Formulation f(d 3 ) f(d 2 ) f(d ) f(d 4 ) d d 2 d 3 d 4 Formulation for f(x)=z f(d )+ 3X d + (d i+ d i ) i = x, i= 3X (f(d i+ ) i= f(d i )) i = z Integral 3 apple y 2 apple 2 apple y apple y i 2{, } 4/299
7 Solve Times in PLEX Weak Strong Mystery Transportation Problems in V., hmed and Nemhauser. 5/29 9
8 Solve Times in PLEX Weak Strong Mystery 2 s 5 s s s s Transportation Problems in V., hmed and Nemhauser. 5/29 9
9 Solve Times in PLEX Weak Strong Mystery 2 s 5 s s s s Transportation Problems in V., hmed and Nemhauser. 5/29 9
10 Outline MIP v/s constraint branching. Have your cake and eat it too formulation Step : Encoding alternatives. Step 2: ombine with strong standard formulation. Summary, Extensions and More. 6/299
11 MIP v/s constraint branching Formulating Discrete lternatives min s.t. x x 2 { i } n i= /29 9
12 MIP v/s constraint branching Formulating Discrete lternatives min x min x s.t. s.t. Xn x 2 { i } n i= i= X n i= i i = x i = 2 {, } n /29 9
13 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n 8/29 9
14 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n 8/29 9
15 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i i = x i = 2 {, } n + i= IP opt =, LP opt = 8/29 9
16 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by binary ranch-and-ound: 8/29 9
17 MIP v/s constraint branching Formulating Discrete lternatives min x Solve by binary ranch-and-ound: ranch on 2 s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = 8/29 9
18 MIP v/s constraint branching Formulating Discrete lternatives min x + s.t. Xn i= X n i= i i = x 2 =! Feasible with x = ranch on 2 i = 2 {, } n IP opt =, LP opt = Solve by binary ranch-and-ound: 8/29 9
19 MIP v/s constraint branching Formulating Discrete lternatives min x + s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by binary ranch-and-ound: 2 =! Feasible with x = ranch on 2 2 =! est ound = 8/29 9
20 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i i = x i = 2 {, } n + i= IP opt =, LP opt = Solve by binary ranch-and-ound: 2 =! Feasible with x = ranch on 2 2 =! est ound = ranch on 2, 4, 5! est ound = 8/29 9
21 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by binary ranch-and-ound: Worst case: n/2 branches to solve (i.e. 2 n/2 -and- nodes!). 8/29 9
22 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by constraint -and-: /29
23 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by constraint -and-: ranch on + 2 /29
24 MIP v/s constraint branching Formulating Discrete lternatives min x + s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by constraint -and-: ranch on =! Feasible with x = /29
25 MIP v/s constraint branching Formulating Discrete lternatives min x + s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by constraint -and-: + 2 =! Feasible with x = ranch on =! Feasible with x = /29
26 MIP v/s constraint branching Formulating Discrete lternatives min x s.t. Xn i= X n i= i i = x i = 2 {, } n IP opt =, LP opt = Solve by constraint -and-: ranch on =! Feasible with x = + 2 =! Feasible with x = Never more than one branch (2 nodes). /29
27 MIP v/s constraint branching onstraint ranching is the Solution? Ryan and Foster, 98. Discrete lternatives: SOS branching of eale and Tomlin 97. lso SOS2 (. and T, 7) and piecewise linear functions (Tomlin 98). SOS: X t i= i = or m X t i= i = i = 8i >t or i = 8i apple t Problem: Need to re-implement advanced branching selection (e.g. pseudocost ). /29 9
28 MIP v/s constraint branching inary v/s Specialized ranching Weak Integer SOS2 ranching Mystery Integer /299
29 MIP v/s constraint branching inary v/s Specialized ranching PLEX 9: asic SOS2 branching implementation (graph from Nemhauser, Keha and V. 8) Med. Large 5 s 2.5 s 75 s 37.5 s s Weak Integer SOS2 ranching Mystery Integer /299
30 MIP v/s constraint branching inary v/s Specialized ranching PLEX 9: asic SOS2 branching implementation (graph from Nemhauser, Keha and V. 8) Med. Large 5 s 2.5 s 75 s 37.5 s s PLEX : Improved SOS2 branching implementation (graph from Nemhauser, hmed and V. ) , s,5 s, s 5 s s Weak Integer SOS2 ranching Mystery Integer /299
31 MIP v/s constraint branching inary v/s Specialized ranching PLEX 9: asic SOS2 branching implementation (graph from Nemhauser, Keha and V. 8) Med. Large 5 s 2.5 s 75 s 37.5 s s PLEX : Improved SOS2 branching implementation (graph from Nemhauser, hmed and V. ) , s,5 s, s 5 s s Weak Integer SOS2 ranching Mystery Integer /299
32 Formulation Step : Encoding lternatives
33 Step : Encodings Formulation for Discrete lternatives Li and Lu 29, dams and Henry 2, V. and Nemhauser 28. Sommer, TIMS 972. b i n Log = inary Encoding Other choices of lead to standard and b i n i= incremental formulations i= = {, }log 2 n 9 3/29
34 Step : Encodings Formulation for Discrete lternatives Li and Lu 29, dams and Henry 2, V. and Nemhauser 28. Sommer, TIMS 972. b i n Log = inary Encoding Other choices of lead to standard and b i n i= incremental formulations i= = {, }log 2 n 9 3/29
35 Step : Encodings Unary Encoding = y, X 8 i= i =, 2 R 8,y2 {, } 8 m i = y i 4/299
36 Step : Encodings inary Encoding = y, X 8 i= i =, 2 R 8,y2 {, } 3 5/299
37 Step : Encodings Incremental Encoding = y, X 8 i= i =, 2 R 8,y2 {, } 7 + y y 2... y 7 6/299
38 Step : Encodings Example: Unary Encoding min x s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m 7/299
39 Step : Encodings Example: Unary Encoding min x s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m 7/299
40 Step : Encodings Example: Unary Encoding min x + s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m 7/299
41 Step : Encodings Example: Unary Encoding min x = y y 2 y 3 y 4 s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m 7/299
42 Step : Encodings Example: Unary Encoding min x s.t. Xn = y = y y 2 y 3 y 4 i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m 7/299
43 Step : Encodings Example: Unary Encoding min x s.t. Xn = y y 2 y 3 y 4 i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m y = 7/299
44 Step : Encodings Example: Unary Encoding min x s.t. Xn = y y 2 y 3 y 4 i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m y = 7/299
45 Step : Encodings Example: Unary Encoding min x = y y 2 y 3 y 4 s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m y = y 4 = 7/299
46 Step : Encodings Example: Unary Encoding min x = y y 2 y 3 y 4 s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m y = y 4 = 7/299
47 Step : Encodings Example: Unary Encoding min x ( ( = n/2 n/2 y y 2 y 3 y 4 s.t. Xn i= X n X n i= i i = x i = i= bi i = y 2 R n + y 2 {, } m Need n/2 branches 7/299
48 Step : Encodings Example: inary Encoding (... 2 k = y y 2 y 3 8/299
49 Step : Encodings Example: inary Encoding (... 2 k = y y 2 y 3 y = y 2 = 8/299
50 Step : Encodings Example: inary Encoding est ound = unless: y i = 8i (... 2 k = y y 2 y 3 y = y 2 = 8/299
51 Step : Encodings Example: inary Encoding est ound = unless: y i = 8i Need k = log 2 n... = y y 2 y 3 branches (n nodes) ( 2 k y = y 2 = 8/299
52 Step : Encodings Example: Incremental Encoding 9/299
53 9 Step : Encodings Example: Incremental Encoding 9/29 = y y 2 y 3 y 4
54 9 Step : Encodings Example: Incremental Encoding 9/29 = y y 2 y 3 y 4 y 3 =_ y 3 = =
55 y i =_ y i = Only need branch! est ound = if: 9 Step : Encodings Example: Incremental Encoding 9/29 = y y 2 y 3 y 4 y 3 =_ y 3 = =
56 Step : Encodings Induced onstraint ranching Incremental = y inary = y 2/299
57 Step : Encodings Induced onstraint ranching Incremental = y SOS ranching! = 2 = or 3 = 4 = inary = y 2/299
58 Step : Encodings Induced onstraint ranching Incremental SOS ranching = y! = 2 = or 3 = 4 = inary SOS ranching = y! = 2 = or 3 = 4 = 2/299
59 Step : Encodings Induced onstraint ranching Incremental SOS ranching = y! = 2 = or 3 = 4 = inary Odd/Even ranching = y! = 3 = or 2 = 4 = 2/299
60 Formulation Step 2: ombining with Strong Formulation
61 Step 2: Strong Formulation Long Lost Integral Formulation P i n i= polytopes x 2 n[ i= P i, lso for general polyhedra with common recession cones. nx i= X v2ext(p i ) X v2ext(p i ) v i v = x i v = nx y i = i= n y 2 {, }, y i i v Jeroslow and Lowe /299
62 Step 2: Strong Formulation ombining with lternative Encoding P i n i= polytopes x 2 n[ i= P i, lso for general polyhedra with common recession cones. nx i= X v2ext(p i ) X v2ext(p i ) v i v = x i v = nx y i = i= n y 2 {, }, y i i v Jeroslow and Lowe /299
63 Step 2: Strong Formulation ombining with lternative Encoding P i n i= polytopes x 2 n[ i= P i, lso for general polyhedra with common recession cones. nx i= nx i= nx i= X v2ext(p i ) X v2ext(p i ) X v i v = x nx v2ext(p i i= ) m y 2 {, }, b i i v = i v y i = y i v Jeroslow and Lowe /299
64 Step 2: Strong Formulation ombining with lternative Encoding P i n i= polytopes x 2 n[ i= P i, lso for general polyhedra with common recession cones. nx i= nx i= nx i= X v2ext(p i ) X v2ext(p i ) X v i v = x nx v2ext(p i i= ) m y 2 {, }, b i i v = i v y i = y i v V., hmed and Nemhauser 2; V /299
65 Step 2: Strong Formulation Univariate Transportation Problems Discontinuous Piecewise Linear Disc. PWL + Semicontinuous 25/299
66 Step 2: Strong Formulation Piecewise Linear Unary Incremental / Strong Integer inary / Mystery Integer 26/299
67 Step 2: Strong Formulation Piecewise Linear Unary Incremental / Strong Integer inary / Mystery Integer 26/299
68 Step 2: Strong Formulation Piecewise Linear + Semi ontinuous Unary Incremental / Strong Integer inary / Mystery Integer 27/299
69 Summary, Extensions and More. Effective formulations: Encode and Formulate est encoding? Why not try a few. lever combination of encodings can be useful (e.g. V. and Nemhauser 28 for multivariate piecewise linear functions) Smaller formulations for shared vertex case Need encodings with special structure. 28/29
70 More Information 29/299
71 More Information Survey: V., MIP Formulation Techniques : 29/299
72 More Information Survey: V., MIP Formulation Techniques : Next year: automatic formulations for JUMP Julia based modeling language: s simple as MPL + faster than ++ Solver independent call-backs and more! JUMP/Julia tutorial in January 29/29 9
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