New bounding schemes and algorithmic options for the Branch-and-Sandwich algorithm

Abstract

We consider the global solution of bilevel programs involving nonconvex functions. Deterministic global optimization algorithms for the solution of this challenging class of optimization problems have started to emerge over the last few years. We present new schemes to generate valid bounds on the solution of nonconvex inner and outer problems and examine new strategies for branching and node selection. We integrate these within the Branch-and-Sandwich algorithm (Kleniati and Adjiman in J Glob Opt 60:425-458, 2014), which is based on a branch-and-bound framework and enables the solution of a wide range of problems, including those with nonconvex inequalities and equalities in the inner problem. The impact of the proposed modifications is demonstrated on an illustrative example and 10 nonconvex bilevel test problems from the literature. It is found that the performance of the algorithm is improved for all but one problem (where the CPU time is increased by 2%), with an average reduction in CPU time of 39%. For the two most challenging problems, the CPU time required is decreased by factors of over 3 and 10.

BASBL: Branch-And-Sandwich BiLevel solver. Implementation and computational study with the BASBLib test set

“…In this section, we describe the implementation of the BASBL solver. An understanding of the B&S algorithm, as described in Kleniati and Adjiman (2014a) and Paulavičius and Adjiman (2019b), is assumed. To provide some context to the main concepts, a brief statement of the main steps of the B&S algorithm and references to relevant sources are given in Algorithm 1.…”

“…During its execution, the BASBL solver requires the global solution of nonconvex nonlinear (sub)problems (NLP), as described in Kleniati and Adjiman (2014a); Paulavičius and Adjiman (2019b). Note that even in the case of single-level optimization, algorithms that guarantee convergence to a global optimum in finite time exist only for special cases, e.g., linear or convex problems.…”

“…In the original B&S paper (Kleniati and Adjiman, 2014a) computationally less expensive convex alternatives for the inner lower bounding and inner subproblems were introduced. While this has great potential to reduce significantly the complexity of these BASBL subproblems, algorithmic analysis in Kleniati and Adjiman (2014b); Paulavičius and Adjiman (2019b) revealed that this can increase the total number of BASBL iterations. Here we provide a further experimental investigation of the relaxed scheme and its influence on BASBL performance, focusing on the trade-off between a large number of relatively inexpensive iterations and a smaller number of more computationally expensive iterations.…”

“…This addresses the lack of general deterministic bilevel solvers for problems with nonconvex constraints in the inner and outer problems. General bilevel problems are very challenging and only in the last decade have the first algorithms to tackle them been proposed: the deterministic approach of Mitsos et al (2008), the approximation method of Tsoukalas et al (2008), and the Branch-and-Sandwich (B&S) algorithm introduced in Kleniati and Adjiman (2014a,b,c) and extended in Paulavičius and Adjiman (2019b). The solver presented here, Branch-And-Sandwich BiLevel algorithm (BASBL), is based on this latter work.…”

Experimental Analysis of Algebraic Modelling Languages for Mathematical Optimization

“…Internally, these tools use the combination of a mathematical model with an appropriate solution algorithm (e.g. Cosma et al, 2020;Fernández et al, 2020;Gómez et al, 2019;Lee et al, 2019;Paulavičius and Adjiman, 2020;Stripinis et al, 2019Stripinis et al, , 2021 to solve the problem at hand. Thus, the way mathematical models are formulated is critical to the impact of optimization in real life.…”