Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming

Abstract

We propose methods for improving the relaxations obtained by the normalized multiparametric disaggregation technique (NMDT). These relaxations constitute a key component for some methods for solving nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems. It is shown that these relaxations can be more efficiently formulated by significantly reducing the number of auxiliary variables (in particular, binary variables) and constraints. Moreover, a novel algorithm for solving MIQCQP problems is proposed. It can be applied using either its original NMDT or the proposed reformulation. Computational experiments are performed using both benchmark instances from the literature and randomly generated instances. The numerical results suggest that the proposed techniques can improve the quality of the relaxations.

Global optimization of large-scale MIQCQPs via cluster decomposition: Application to short-term planning of an integrated refinery-petrochemical complex

“…Castillo Castillo et al (2017 presented DGO algorithms to solve mixedinteger quadratically constrained quadratic programs (MIQCQPs) arising in large-scale refinery planning that performed similarly to the commercial solvers BARON (Sahinidis, 2004) and ANTIGONE (Misener and Floudas, 2014). The approach relies on mixed-integer linear programming (MILP) relaxations, derived from piecewise McCormick envelopes (Castro, 2015;Gounaris et al, 2009;McCormick, 1976) or the multiparametric disaggregation technique (Andrade et al, 2018;Castro, 2016;Kolodziej et al, 2013;Teles et al, 2013), and on the reduction of the variables domain through optimality-based bound tightening (OBBT) (Castro and Grossmann, 2014;Puranik and Sahinidis, 2017).…”

Computing team-maxmin equilibria in zero-sum multiplayer games

“…It is well-known that the state-of-the-art global optimization solver is BARON [27]. Here we introduce a global optimization technique, the Multiparametric Disaggregation Technique (MDT) [40][41][42] for approximating bilinear terms by using a digit-wise discretization of one variable in the bilinear term and then transforming the non-convex program into a mixed-integer linear program (MILP), which is a strong competitor of BARON and has been applied to solve gametheoretical problems [43,44] for two-player games. The MDT extends the McCormick relaxation [45] and outperforms the piecewise McCormick relaxation 1 [40,41].…”

“…In the domain of global optimization, the MDT [40][41][42] for approximating bilinear terms is a strong competitor of the state-of-the-art global optimization solver BARON.…”

“…Specifically, we develop the Associated Recursive Asynchronous Multiparametric Disaggregation Technique (ARAMDT) to approximate multilinear terms in Eq.(2.6b). The ARAMDT is developed based on the MDT [40,42], which uses a digit-wise discretization of one variable in the bilinear term for approximation and then transforms the bilinear program into a mixed-integer linear program (MILP). The ARAMDT has three additional important features: 1) the Asynchronous MDT (AMDT) (i.e., adding the asynchronous precision method to the MDT) to reduce the number of constraints and new variables for approximating bilinear terms; 2) the Recursive AMDT (RAMDT) (i.e., recursively deploying the AMDT) to approximate multilinear terms which cannot be directly approximated by the AMDT; and 3) the Associated RAMDT (ARAMDT) (i.e., adding associated constraints to the RAMDT) to reduce the feasible solution space of MILP in the RAMDT.…”

“…Other baselines include: 1) IARAMDT-1: at each iteration, we only select bilinear terms with the largest difference between w i (σ T i ) and r i (σ i )w i+1 (σ T i+1 ) similar to the Chapter 5. Computing Team-Maxmin Equilibria in Extensive-Form Games previous approach [44]; 2) IARAMDT+1: we increase the power with 1 instead of Z at lines 10 and 11 of Algorithm 10, which is similar to many approaches [42,44]; 3)…”