A general branch-and-bound framework for continuous global multiobjective optimization

Abstract

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.

The Branch-and-Bound Algorithm in Optimizing Mathematical Programming Models to Achieve Power Grid Observability

“…All optimization problems are written in YALMIP using symbolic syntax, with a strict declaration of the binary nature of the decision variables [26][27][28]. We initially optimize the 0/1 ILP model with a BBA [64][65][66][67][68] implemented by the bmibnb built-in solver in the YALMIP program in collaboration with external optimizer routines to find optimality [69][70][71][72][73][74][75][76][77][78][79].…”

“…6.1. Termination Criteria of BBA for Convergence of the Optimization Run [64][65][66][67] BBA is an in-front algorithm that is utilized to solve the proposed methods due to a good enough stopping criterion to fully accomplish the optimality criterion. As a result, the integer solver successfully returns a globally optimal point, avoiding being trapped in a local minimum or a suboptimal point [74].…”

“…ILP solvers try to deliver a branch-and-bound tree as small as possible without producing many branches [9][10][11]. As a result, a branching regulation with strength is executed, which results in a solution that satisfies the optimality conditions [64][65][66][67].…”

Adaptive piecewise linear relaxations for enclosure computations for nonconvex multiobjective mixed-integer quadratically constrained programs

“…Another approach of numerically solving an (MOP) is-instead of computing a sole approximation-to compute a coverage of the nondominated set N as presented in, e.g., [23][24][25][26]. Besides the introduction of enclosures of the nondominated set, the authors present a branch-and-bound framework for computing such an enclosure of the nondominated set with a prescribed quality.…”

“…In most cases, solving a multiobjective optimization problem (MOP) numerically means that one computes either an enclosure (cf. [23][24][25][26]) or an approximation (cf. [5,10,17,30,33]) of the nondominated set.…”