A Tighter Piecewise McCormick Relaxation for Bilinear Problems

Transcription

1 A Tighter Piecewise McCormick Relaxation for Bilinear Problems Pedro M. Castro Centro de Investigação Operacional Faculdade de Ciências niversidade de isboa

2 Problem definition (NP) Bilinear program min f 0 (x) f q (x) 0 q Q\{0} (P) f q (x) = (i,j) B q a ijq x i x j + h q (x) q Q 0 x x x x R m Relaxation provides a lower bound a ijq - parameters x vector of continuous variables B q - (i, j) index set i j strictly bilinear problems i = j can be allowed (quadratic problems) h q (x)- linear function in x MINP

3 Introduction Bilinear problems occur in a variety of applications Process network problems (Quesada & Grossmann, 1995) Water networks (Bagajewicz, 2000) Pooling and blending (Haverly, 1978) Non-convex, leading to multiple local solutions Gradient based solvers unable to certify optimality Need for global optimization algorithms What do they have in common? inear (P) or mixed-integer linear (MIP) relaxation of (P) B P: Standard McCormick envelopes (1976) MIP: Piecewise McCormick envelopes (Bergamini et al. 2005) MIP: Multiparametric disaggregation (Teles et al. 2013; Kolodziej et al. 2013) Solution of (P) with fast local solver B sing B as starting point Tight relaxation critical to ensure convergence Relative optimality gap (B-B)/B< MINP

4 Piecewise McCormick relaxation w ij = x i x j Domain of divided into N uniform partitions x j x j,1 n = 1 x j,1 n = 2 x j,2 x j,n 1 n = N x j,2... x j,n y j,1 = true y j,2 = true... y j,n = true x j,n x j Optimization will pick single partition Generalized Disjunctive Program (Balas, 1979; Raman & Grossmann, 1994) y jn w ij x i x jn + x i x j x i x jn n w ij x i x jn + x i x j x i x jn w ij x i x jn + x i x j x i x jn w ij x i x jn + x i x j x i x jn x jn x j x jn (PR-MIP) Convex hull x i Standard McCormick N = 1 N = 2 N = 20 x jn x i x i x i = x j + (x j x j ) (n 1)/N x jn = x j + (x j x j ) n/n x j x j x j MINP

5 New tighter piecewise relaxation Basic idea se partition-dependent n bounds also for variable x i x /x lower/upper bound when x j constrained to n Better bounds tighter relaxation lower gap y jn w ij x i x jn + x w ij x i x jn + x w ij x i x jn + x w ij x i x jn + x x jn x x jn x j x x j x x j x x j x x i x x j x jn x jn x jn x jn x jn = x j + (x j x j ) (n 1)/N x jn = x j + (x j x j ) n/n MINP 2014 Convex hull (PRT-MIP) min z R = f 0 x = f q x = x j = n x jn n y jn = 1 i,j B i,j B a ij0 w ij + h 0 x a ijq w ij + h q x 0 q Q\{0} {j (i, j) B} x jn = x j + x j x j n 1 N x jn = x j + x j x j n N x jn y jn x jn x jn y jn 5 w ij w ij w ij w ij x i = x n n n n x n x x jn + x x x jn + x x x jn + x x x jn + x y jn x x x jn x x jn x j x jn y jn x jn y jn x jn x x jn y jn x jn x x jn y jn i, j B, n y jn (i, j) B, n {1,, N} (i, j)

6 How to generate bounds x & x? Optimality bound contraction with McCormick envelopes Solve multiple instances of P problem (BC), two per: Non-partitioned variable x i Partitioned variable x j Partition n Focus on regions that can actually improve current B z obtained from solving (P) with local solver (BC) infeasible? Remove partition n of j from (PRT-MIP) Computation can be time consuming (BC) x i j n f 0 x = = min x i (x i j n = max x i i,j B 0 a ij0 w ij + h 0 (x) z f q (x) = i,j B q a ijq w ij + h q (x) 0 q Q\{0} w ij x i x j + x i x j x i x j w ij x i x j + x i x j x i x j w ij x i x j + x i x j x i x j w ij x i x j + x i x j x i x j x j = x j n x j x j n x x x (i, j) B = x j MINP

7 tighter bounds, Feasible? remove partition New optimization algorithm YES NO update bounds Key features ser selects: N partitions Partitioned variables x j Preliminary bound contraction stage Bounds updated at different levels B from MIP relaxation B from NP (single starting point) Computes an optimality gap tighter bounds, Given: partitions,,,,, SOVE (P) with local solver SOVE (standard bound contract) YES YES NO NO Initialization SOVE (BC) Feasible? YES NO NO upper bound update,,, remove partition update bounds update bounds NO NO YES YES SOVE (PRT-MIP) SOVE (P) with local solver Output Optimal solution Optimality gap Time Elapsed update bounds lower bound upper bound MINP 2014 SOVE YES lower bound June 3, 2014 Tighter (PRT-MIP) Piecewise McCormick Relaxation 7

8 Illustrative example P1 min f 0 x = x 1 + x 1 x 2 x 2 Global optimum z = f 0 x = Partitioned variable: x 1 B = {(2,1)} Non-partitioned variable: x 2 6x 1 + 8x 2 3 3x 1 x x 1, x Standard optimality bound contraction x x (PR-MIP) relaxation z R = (PRT-MIP) relaxation z R = Gap 8.60% 7.38% Partition-dependent bound contraction N = 3 Partition n x 1n x 1n x 21n x 21n P is infeasible MINP

9 How about using more partitions? p to two orders of magnitude reduction in optimality gap Major increase in total computational time Still, new approach performs best! P1 Relaxation Partitions (N) (PR-MIP) (PRT-MIP) Total CPs Optimality gap 1.67% 0.188% % % Optimality gap 0.77% 0.010% % Total CPs P2 Relaxation Partitions (N) min 6x x x 1 x 2 x 1 x x 1, x 2 10 (PR-MIP) (PRT-MIP) Total CPs Optimality gap 8.41% 0.84% % % Optimality gap 0.84% 0.02% % Total CPs MINP

10 Subset of problems selected for illustration Results for larger test problems 16 easiest water-using design problems New (PRT-MIP) approach leads to lower gaps at termination (3600 CPs) Faster in 56% of the cases (finest partition level) Single failure when finding best-known solution for (P) Number of variables (PR-MIP) (PRT-MIP) Non-partitioned I Partitioned J Partitions (N) Solution Total CPs Gap Gap Total CPs Solution Ex Ex Ex Ex Ex % % % % % % % % % % % % % % % % % % % % Ex % % % % Ex % % % % MINP

11 How to determine the number of partitions? Quality of relaxation increases with N Careful not to generate intractable MIPs (Gounaris et al. 2009) Several works identify most appropriate value for a particular problem (Misener et al. 2011; Wittmann-Holhbein and Pistikopoulos, 2013; 2014) No method to estimate N as a function of problem complexity Needed to provide a fair comparison with commercial solvers Proposed formula meets important criteria: Returns an integer value Ensures minimum of two partitions So as to benefit from piecewise relaxation scheme Reduces N with increase in problem size Roughly inversely proportional (r 2 =0.76) N = 1 + α I J α = 1.8E4 MINP

12 Cumulative distribution function Comparison to commercial solvers Performance profiles (Dolan & Moré, 2002) KPI: Optimality gap (PR-MIP) (PRT-MIP) GloMIQO 2.3 BARON GAMS , CPEX , GloMIQO 2.3, BARON , Intel Core i (3.07 GHz), 8 GB RAM, Windows 7 64-bit Termination criteria: gap=0.0001%, Time=3600 CPs (PRT-MIP) better (PR-MIP) GloMIQO outperforms BARON # Failures finding best-known solution (34 problems) Algorithm GloMIQO (PRT-MIP) BARON (PR-MIP) Suboptimal No solution Total MINP

13 (PR-BV-MIP) Why not bivariate partitioning? Domain of x i and x j known a priori for each 2-D partition Slightly better performance than univariate partitioning for P1 Gap for partitions one order of magnitude larger vs. (PRT-MIP) N N n=1 n =1 w ij x i x jn w ij x i x jn w ij x i x jn w ij x i x jn x in x jn y injn + x in x j x in x jn + x in x j x in x jn + x in x j x in x jn + x in x j x in x jn x i x in x j x jn (i, j) Relaxation Partitions (N) nivariate (PR-MIP) Bivariate (PR-BV-MIP) Total CPs Optimality gap 1.70% 0.197% % % N Optimality gap 1.28% 0.188% % % % Total CPs MINP

14 Conclusions Bilinear problems tackled through piecewise relaxation (PR) Novel approach with partition-dependent bounds for all bilinear variables Significant improvement in relaxation quality Need for extensive optimality-based bound contraction (BC) Total computational time is sometimes lower New algorithm outperforms state-of-the-art global solvers Specific class of water-using network design problems PR and BC schemes should be used to greater extent Integrating with spatial B&B subject of future work See CACE paper for further details Acknowledgments: uso-american Foundation (2013 Portugal-.S. Research Networks) Fundação para a Ciência e Tecnologia (Investigador FCT 2013) MINP