Computational advances in polynomial optimization: RAPOSa, a freely available global solver

Abstract

In this paper we introduce , a global optimization solver specifically designed for (continuous) polynomial programming problems with box-constrained variables. Written entirely in , is based on the Reformulation-Linearization (Sherali and Tuncbilek in J Glob Optim 103:225–249, 1992). We present a description of the main characteristics of along with a thorough analysis of the impact on its performance of various enhancements discussed in the literature, such as bound tightening and SDP cuts. We also present a comparative study with three of the main state-of-the-art global optimization solvers: , and .

Assessment of a two-step approach for global optimization of mixed-integer polynomial programs using quadratic reformulation

“…A total of 74 instances of randomly generated POPs are taken from Sherali (2016), andGonzález-Rodríguez et al (2020). These are mostly small problems, comprising between 5 and 16 (continuous) variables and 6-17 (equality and/or inequality) constraints, with a polynomial degree between 3 amd 7 and a variable sparsity ratio between 0.005 and 1-as measured by the ratio between the actual number and total possible number of nonlinear monomials corresponding to the polynomial degree and number of variables.…”

“…Several enhancements of RLT have been proposed in response to this complexity, such as excluding any constraints that may not tighten a relaxation (Sherali et al, 2012b;Dalkiran and Sherali, 2013;Sherali and Dalkiran, 2011) or adding semi-definite cuts to produce tighter relaxations (Sherali and Fraticelli, 2002;Sherali et al, 2012a). Implementations of this approach are available as part of the solvers RLT-POS (Dalkiran and Sherali, 2016) and RAPOSa (González-Rodríguez et al, 2020), although neither solver can handle binary or integer decision variables at present.…”

Efficient linear reformulations for binary polynomial optimization problems

“…Such techniques have been used in continuous polynomial optimization, as in RLT-POS and further developments [9,8], and more recently as the base relaxation in a solver called RAPOSa [19]. In the case of binary variables, Hojny, Pfetsch and Walter [20] use such an approach to tackle BPO.…”