Tightening piecewise McCormick relaxations for bilinear problems

Abstract We address nonconvex bilinear problems where the main objective is the computation of a tight lower bound for the objective function to be minimized. This can be obtained through a mixed-integer linear programming formulation relying on the concept of piecewise McCormick relaxation. It works by dividing the domain of one of the variables in each bilinear term into a given number of partitions, while considering global bounds for the other. We now propose using partition-dependent bounds for the latter so as to further improve the quality of the relaxation. While it involves solving hundreds or even thousands of linear bound contracting problems in a pre-processing step, the benefit from having a tighter formulation more than compensates the additional computational time. Results for a set of water network design problems show that the new algorithm can lead to orders of magnitude reduction in the optimality gap compared to commercial solvers.

[1]  Miguel J. Bagajewicz,et al.  Novel bound contraction procedure for global optimization of bilinear MINLP problems with applications to water management problems , 2011, Comput. Chem. Eng..

[2]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[3]  Ignacio E. Grossmann,et al.  A global optimization algorithm for linear fractional and bilinear programs , 1995, J. Glob. Optim..

[4]  Christodoulos A. Floudas,et al.  Global optimization of a combinatorially complex generalized pooling problem , 2006 .

[5]  I. Karimi,et al.  Piecewise MILP under‐ and overestimators for global optimization of bilinear programs , 2008 .

[6]  Ignacio E. Grossmann,et al.  A discretization-based approach for the optimization of the multiperiod blend scheduling problem , 2013, Comput. Chem. Eng..

[7]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[8]  Marianthi G. Ierapetritou,et al.  Mixed-Integer Linear Programming Model for Gasoline Blending and Distribution Scheduling , 2003 .

[9]  Ignacio E. Grossmann,et al.  Logic-based outer approximation for globally optimal synthesis of process networks , 2005, Comput. Chem. Eng..

[10]  Iiro Harjunkoski,et al.  Different transformations for solving non-convex trim-loss problems by MINLP , 1998, Eur. J. Oper. Res..

[11]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[12]  Christodoulos A. Floudas,et al.  Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations , 2012, Mathematical Programming.

[13]  Pedro M. Castro,et al.  Multi-parametric disaggregation technique for global optimization of polynomial programming problems , 2013, J. Glob. Optim..

[14]  I. Karimi,et al.  Piecewise linear relaxation of bilinear programs using bivariate partitioning , 2009 .

[15]  Munawar A. Shaik,et al.  Short-Term Scheduling of Refinery Crude Oil Operations , 2012 .

[16]  Chrysanthos E. Gounaris,et al.  Computational Comparison of Piecewise−Linear Relaxations for Pooling Problems , 2009 .

[17]  Pedro M. Castro,et al.  Univariate parameterization for global optimization of mixed-integer polynomial problems , 2013, Eur. J. Oper. Res..

[18]  Efstratios N. Pistikopoulos,et al.  On the global solution of multi-parametric mixed integer linear programming problems , 2013, J. Glob. Optim..

[19]  A comparative assessment of linearization methods for bilinear models , 2013, Comput. Chem. Eng..

[20]  Mahmoud M. El-Halwagi,et al.  Global optimization of mass and property integration networks with in-plant property interceptors , 2009 .

[21]  Jung-Fa Tsai,et al.  An improved framework for solving NLIPs with signomial terms in the objective or constraints to global optimality , 2013, Comput. Chem. Eng..

[22]  M. Bagajewicz,et al.  A new approach for global optimization of a class of MINLP problems with applications to water management and pooling problems , 2012 .

[23]  Christodoulos A. Floudas,et al.  Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2 , 2015, Optim. Methods Softw..

[24]  Ignacio E. Grossmann,et al.  Exploiting vector space properties to strengthen the relaxation of bilinear programs arising in the global optimization of process networks , 2011, Optim. Lett..

[25]  I. Grossmann,et al.  Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming , 2013 .

[26]  C. Floudas.  Handbook of Test Problems in Local and Global Optimization , 1999 .

[27]  Pedro M. Castro,et al.  Global optimization of water networks design using multiparametric disaggregation , 2012, Comput. Chem. Eng..

[28]  N. Sahinidis,et al.  Computational Experience with Applications of Bilinear Cutting Planes , 2013 .

[29]  Nikolaos V. Sahinidis,et al.  A polyhedral branch-and-cut approach to global optimization , 2005, Math. Program..

[30]  Pedro M. Castro,et al.  Optimality-based bound contraction with multiparametric disaggregation for the global optimization of mixed-integer bilinear problems , 2014, J. Glob. Optim..

[31]  Efstratios N. Pistikopoulos,et al.  Approximate solution of mp-MILP problems using piecewise affine relaxation of bilinear terms , 2014, Comput. Chem. Eng..

[32]  R. Raman,et al.  Modelling and computational techniques for logic based integer programming , 1994 .

[33]  Pedro M. Castro,et al.  Global optimization of bilinear programs with a multiparametric disaggregation technique , 2013, Journal of Global Optimization.

[34]  I. Grossmann,et al.  Optimal Design of Distributed Wastewater Treatment Networks , 1998 .

[35]  J.P.S. Catalão,et al.  Hydro energy systems management in Portugal: Profit-based evaluation of a mixed-integer nonlinear ap , 2011 .

[36]  J. M. Ponce-Ortega,et al.  Global Optimization in Property-Based Interplant Water Integration , 2013 .

[37]  Christodoulos A. Floudas,et al.  GloMIQO: Global mixed-integer quadratic optimizer , 2012, Journal of Global Optimization.

[38]  Christodoulos A. Floudas,et al.  APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes , 2011, Comput. Chem. Eng..

[39]  Pedro M. Castro,et al.  Comparison of global optimization algorithms for the design of water-using networks , 2013, Comput. Chem. Eng..

[40]  Jose M. Pinto,et al.  A PLANNING MODEL FOR REFINERY DIESEL PRODUCTION , 1998 .

[41]  Pedro M. Castro,et al.  MILP-based initialization strategies for the optimal design of water-using networks , 2009 .

[42]  Ignacio E. Grossmann,et al.  Global optimization for the synthesis of integrated water systems in chemical processes , 2006, Comput. Chem. Eng..

[43]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[44]  J. M. Pinto,et al.  Mixed-Integer Linear Programming Model for Refinery Short-Term Scheduling of Crude Oil Unloading with Inventory Management , 1996 .

[45]  Marianthi G. Ierapetritou,et al.  Refinery Short-Term Scheduling Using Continuous Time Formulation: Crude-Oil Operations , 2003 .

[46]  E. Balas.  Disjunctive programming and a hierarchy of relaxations for discrete optimization problems , 1985 .