Cuts for mixed 0-1 conic programs

Transcription

1 Cuts for mixed 0-1 conic programs G. Iyengar 1 M. T. Cezik 2 1 IEOR Department Columbia University, New York. 2 GERAD Université de Montréal, Montréal TU-Chemnitz Workshop on Integer Programming and Continuous Optimization, Nov. 7-9th 2004 CORC Tech Reports: TR , TR , TR

2 Research Goals What are mixed 0-1 conic programs Are conic IPs useful? How to solve mixed 0-1 conic programs? Gomory cuts and generalizations Cuts from hierarchies of tighter relaxations Computational results Conclusion

3 What are mixed 0-1 conic programs? Mixed 0-1 conic programs (MCP) min c T x, s.t. Ax K b x i {0, 1}, i = 1,..., p. K : partial order w.r.t. a cone K = K 1 K 2... K r Each K j is one of the following Linear cone (LP): K = {y : yi 0, i} Second-order cone (SOCP): K = {y = (y0 ; ȳ) : y 0 ȳ } Semidefinite cone (SDP): K = {Y : Y sym. pos. semidef.}. Each K j = K j is self-dual: K = K is self-dual

4 Why bother with mixed 0-1 conic programs?

5 Why bother with mixed 0-1 conic programs? Compact formulation: TSP, Scheduling min s. t. Tr(CX), ( ) ) 2 cos(2π/n) I + 2 n( 1 cos(2π/n) 11 T (X + X T ) 0 X1 = X T 1 = 1 diag(x) = 0 X ij {0, 1} i, j

6 Why bother with mixed 0-1 conic programs? Compact formulation: TSP, Scheduling min s. t. Tr(CX), ( ) ) 2 cos(2π/n) I + 2 n( 1 cos(2π/n) 11 T (X + X T ) 0 X1 = X T 1 = 1 diag(x) = 0 X ij {0, 1} i, j Tighter formulations of mixed 0-1 LPs: Sherali-Adams lifting Lovasz-Schrijver lifting

7 Why bother with mixed 0-1 conic programs? Compact formulation: TSP, Scheduling min s. t. Tr(CX), ( ) ) 2 cos(2π/n) I + 2 n( 1 cos(2π/n) 11 T (X + X T ) 0 X1 = X T 1 = 1 diag(x) = 0 X ij {0, 1} i, j Tighter formulations of mixed 0-1 LPs: Sherali-Adams lifting Lovasz-Schrijver lifting Naturally arise in many applications: Robust counterparts of uncertain mixed 0-1 LPs Hybrid control

8 Solution technique for mixed 0-1 conic programs

9 Solution technique for mixed 0-1 conic programs Relax into a conic LP and round : e.g. max-cut

10 Solution technique for mixed 0-1 conic programs Relax into a conic LP and round : e.g. max-cut Branch-and-bound: sub-problem is a conic program

11 Solution technique for mixed 0-1 conic programs Relax into a conic LP and round : e.g. max-cut Branch-and-bound: sub-problem is a conic program Branch-and-cut: Cut generation techniques Cut lifting techniques

12 Solution technique for mixed 0-1 conic programs Relax into a conic LP and round : e.g. max-cut Branch-and-bound: sub-problem is a conic program Branch-and-cut: Cut generation techniques Cut lifting techniques

13 Gomory cuts for pure integer conic programs

14 Gomory cuts for pure integer conic programs Pure integer conic program min c T x, s.t. Ax K b x Z n +

15 Gomory cuts for pure integer conic programs Pure integer conic program min c T x, s.t. Ax K b x Z n + Conic duality: y K iff u T y 0 for all u K (= K)

16 Gomory cuts for pure integer conic programs Pure integer conic program min c T x, s.t. Ax K b x Z n + Conic duality: y K iff u T y 0 for all u K (= K) Chvátal-Gomory (C-G) procedure: u K 0: u T Ax = n j=1 (at j u)x j u T b

17 Gomory cuts for pure integer conic programs Pure integer conic program min c T x, s.t. Ax K b x Z n + Conic duality: y K iff u T y 0 for all u K (= K) Chvátal-Gomory (C-G) procedure: u K 0: u T Ax = n j=1 (at j u)x j u T b n x 0: j=1 at j u x j u T b.

18 Gomory cuts for pure integer conic programs Pure integer conic program min c T x, s.t. Ax K b x Z n + Conic duality: y K iff u T y 0 for all u K (= K) Chvátal-Gomory (C-G) procedure: u K 0: u T Ax = n j=1 (at j u)x j u T b n x 0: j=1 at j u x j u T b. x Z n + : n j=1 at j u x j u T b.

19 Gomory cuts for pure integer conic programs Pure integer conic program min c T x, s.t. Ax K b x Z n + Conic duality: y K iff u T y 0 for all u K (= K) Chvátal-Gomory (C-G) procedure: u K 0: u T Ax = n j=1 (at j u)x j u T b n x 0: j=1 at j u x j u T b. x Z n + : n j=1 at j u x j u T b. Identical to the C-G procedure for integer programs.

20 Gomory cuts: relate LP and SDP relaxations

21 Gomory cuts: relate LP and SDP relaxations Mixed 0-1 SDP formulation of TSP ( ) 2 ( ) 2 cos(2π/n) I + 1 cos(2π/n) 11 T (X + X T ) 0 n

22 Gomory cuts: relate LP and SDP relaxations Mixed 0-1 SDP formulation of TSP ( ) 2 ( ) 2 cos(2π/n) I + 1 cos(2π/n) 11 T (X + X T ) 0 n Choose W {1,..., n} and set U = W 1 T W

23 Gomory cuts: relate LP and SDP relaxations Mixed 0-1 SDP formulation of TSP ( ) 2 ( ) 2 cos(2π/n) I + 1 cos(2π/n) 11 T (X + X T ) 0 n Choose W {1,..., n} and set U = W 1 T W Gomory cut: i,j W X ij W 1

24 Gomory cuts: relate LP and SDP relaxations Mixed 0-1 SDP formulation of TSP ( ) 2 ( ) 2 cos(2π/n) I + 1 cos(2π/n) 11 T (X + X T ) 0 n Choose W {1,..., n} and set U = W 1 T W Gomory cut: i,j W X ij W 1 Theorem Sub-tour elimination inequalities for TSP are rank-1 C-G cuts.

25 Gomory cuts: relate LP and SDP relaxations Mixed 0-1 SDP formulation of TSP ( ) 2 ( ) 2 cos(2π/n) I + 1 cos(2π/n) 11 T (X + X T ) 0 n Choose W {1,..., n} and set U = W 1 T W Gomory cut: i,j W X ij W 1 Theorem Sub-tour elimination inequalities for TSP are rank-1 C-G cuts. Theorem Triangle inequalities for max-cut are rank-1 C-G cuts.

26 Subadditive functions and duality f K-non-decreasing: y K z implies f(y) f(z)

27 Subadditive functions and duality f K-non-decreasing: y K z implies f(y) f(z) F: set of subadditive, K-non-decreasing functions

28 Subadditive functions and duality f K-non-decreasing: y K z implies f(y) f(z) F: set of subadditive, K-non-decreasing functions Duality: min c T x, s.t. Ax K b x Z n + max s.t. f(b), f(a i ) c i, i f F

29 Subadditive functions and duality f K-non-decreasing: y K z implies f(y) f(z) F: set of subadditive, K-non-decreasing functions Duality: min c T x, s.t. Ax K b x Z n + Duality useless: F very large! max s.t. f(b), f(a i ) c i, i f F

30 Subadditive functions and duality f K-non-decreasing: y K z implies f(y) f(z) F: set of subadditive, K-non-decreasing functions Duality: min c T x, s.t. Ax K b x Z n + max s.t. f(b), f(a i ) c i, i f F Duality useless: F very large! Lasserre: A much smaller set suffices for integer programs

31 Subadditive functions and duality f K-non-decreasing: y K z implies f(y) f(z) F: set of subadditive, K-non-decreasing functions Duality: min c T x, s.t. Ax K b x Z n + max s.t. f(b), f(a i ) c i, i f F Duality useless: F very large! Lasserre: A much smaller set suffices for integer programs Extends to a special class of integer SDPs SDP constraint: i x ia i A 0 Each Ai non-negative integer matrices Simple extension of Lasserre s ideas

32 Cuts from hierarchies of tighter relaxation MCP: C = conv (x : Ax K b, x i {0, 1}, 1 i p)

33 Cuts from hierarchies of tighter relaxation MCP: C = conv (x : Ax K b, x i {0, 1}, 1 i p) Initial relaxation: C = {x : Ax K b}

34 Cuts from hierarchies of tighter relaxation MCP: C = conv (x : Ax K b, x i {0, 1}, 1 i p) Initial relaxation: C = {x : Ax K b} Basic idea: x (t) argmax{c T x : x C (t) } Construct a tighter relaxation C with x (t) C Compute a valid inequality f(x) 0 for C that cuts x (t) New relaxation: C (t+1) C (t) {x : f(x) 0}

35 Cuts from hierarchies of tighter relaxation MCP: C = conv (x : Ax K b, x i {0, 1}, 1 i p) Initial relaxation: C = {x : Ax K b} Basic idea: x (t) argmax{c T x : x C (t) } Construct a tighter relaxation C with x (t) C Compute a valid inequality f(x) 0 for C that cuts x (t) New relaxation: C (t+1) C (t) {x : f(x) 0} Hierarchies for MLPs extend to MCPs Lovász-Schrijver (LS) and Balas-Ceria-Cornuéjols (BCC) hierarchies Sherali-Adams (SA) and Lasserre hierarchies

36 Cuts from hierarchies of tighter relaxation MCP: C = conv (x : Ax K b, x i {0, 1}, 1 i p) Initial relaxation: C = {x : Ax K b} Basic idea: x (t) argmax{c T x : x C (t) } Construct a tighter relaxation C with x (t) C Compute a valid inequality f(x) 0 for C that cuts x (t) New relaxation: C (t+1) C (t) {x : f(x) 0} Hierarchies for MLPs extend to MCPs Lovász-Schrijver (LS) and Balas-Ceria-Cornuéjols (BCC) hierarchies Sherali-Adams (SA) and Lasserre hierarchies Introduce new vars and connect using conic constraints

37 Convex cuts from LS/BCC hierarchy Choose B {1,..., p} (w.l.o.g. B = {1,..., l})

38 Convex cuts from LS/BCC hierarchy Choose B {1,..., p} (w.l.o.g. B = {1,..., l}) Lifted set M + B (C): (x, Y0, Y 1 ) M B (C) iff yk 0 + y1 k = [1; x] y0 kk = 0 y1 kk = y1 0k n i=1 y0 ik a i K y0k 0 b n i=1 y1 ik a i K y0k 1 b Y 1 B = (Y1 B )T 0 Y 1 = [y 1 1,..., y1 l ] Y0 = [y 0 1,..., y0 l ]

39 Convex cuts from LS/BCC hierarchy Choose B {1,..., p} (w.l.o.g. B = {1,..., l}) Lifted set M + B (C): (x, Y0, Y 1 ) M B (C) iff yk 0 + y1 k = [1; x] y0 kk = 0 y1 kk = y1 0k n i=1 y0 ik a i K y0k 0 b n i=1 y1 ik a i K y0k 1 b Y 1 B = (Y1 B )T 0 Y 1 = [y 1 1,..., y1 l ] Y0 = [y 0 1,..., y0 l ] Conic duality yields necessary and sufficient conditions α T x + β 0 for all x M B (C) Tr(QYB ) + α T x + β 0 for all (x, Y B ) M B (C)

40 Convex cuts from LS/BCC hierarchy Choose B {1,..., p} (w.l.o.g. B = {1,..., l}) Lifted set M + B (C): (x, Y0, Y 1 ) M B (C) iff yk 0 + y1 k = [1; x] y0 kk = 0 y1 kk = y1 0k n i=1 y0 ik a i K y0k 0 b n i=1 y1 ik a i K y0k 1 b Y 1 B = (Y1 B )T 0 Y 1 = [y 1 1,..., y1 l ] Y0 = [y 0 1,..., y0 l ] Conic duality yields necessary and sufficient conditions α T x + β 0 for all x M B (C) Tr(QYB ) + α T x + β 0 for all (x, Y B ) M B (C) Q 0: x T B Qx B + α T x + β 0 valid convex inequality

41 Traveling salesman problem: Random instance % closed % closed instance gap (%) 25 cuts 50 cuts B-and-B B-and-C Random n = n = n = Random n = n = n = Random n = n = n = Random n = n = n = Random n = n = n =

42 Traveling salesman problem: TSPLIB problems % closed % closed instance gap (%) 25 cuts 50 cuts B-and-B B-and-C burma ulysses gr ulysses gr fri bayg bays Small size! No warm start or dual simplex-like method.

43 MLPs: Comparison of linear/sdp cuts Question: Are cuts from SDP relaxations superior? Control the complexity of SDP cut generation At each iteration, project LP into the space of fractional variables Construct an SDP lifting with B 10 Generate disjunctive cut for each index in the set B Lift cut to the space of all variables One linear cut for each fractional variable: no cut lifting L1 normalization: cut generation is an LP L2 normalization: cut generation is an SOCP Did not focus on time. Why not? No dual simplex.

44 Pure 0-1 LPs problem semidefinite linear: L 2 linear : L 1 cuts iter % cuts iter % cuts iter % LSB LSC PE PE PE PE Stein Stein

45 Mixed 0-1 LPs problem semidefinite linear: L 2 linear: L 1 cuts iter % cuts iter % cuts iter % CTN CTN CTN Danoint SDP cuts are superior: Reduce both iters and cuts p small: L 2 normalization close to SDP cuts SDP cuts are not likely to be facet defining Effective because initial relaxation is bad?

46 Conclusion Mixed 0-1 conic programs are interesting!

47 Conclusion Mixed 0-1 conic programs are interesting! All techniques known for mixed 0-1 LPs extend readily

48 Conclusion Mixed 0-1 conic programs are interesting! All techniques known for mixed 0-1 LPs extend readily Quality of conic relaxation is quite good

49 Conclusion Mixed 0-1 conic programs are interesting! All techniques known for mixed 0-1 LPs extend readily Quality of conic relaxation is quite good But... computational techniques are lacking