Polynomial unconstrained binary optimisation – Part 2

The class of problems known as quadratic zeroone (binary) unconstrained optimisation has provided access to a vast array of combinatorial optimisation problems, allowing them to be expressed within the setting of a single unifying model. A gap exists, however, in addressing polynomial problems of degree greater than 2. To bridge this gap, we provide methods for efficiently executing core search processes for the general polynomial unconstrained binary (PUB) optimisation problem. A variety of search algorithms for quadratic optimisation can take advantage of our methods to be transformed directly into algorithms for problems where the objective functions involve arbitrary polynomials. Part 1 of this paper (Glover et al., 2011) provided fundamental results for carrying out the transformations and described coding and decoding procedures relevant for efficiently handling sparse problems, where many coefficients are 0, as typically arise in practical applications. In the present part 2 paper, we provide special algorithms and data structures for taking advantage of the basic results of part 1. We also disclose how our designs can be used to enhance existing quadratic optimisation algorithms

Experiments on local search for bi-objective unconstrained binary quadratic programming

International audienceThis article reports an experimental analysis on stochastic local search for approximating the Pareto set of bi-objective unconstrained binary quadratic programming problems. First, we investigate two scalarizing strategies that iteratively identify a high-quality solution for a sequence of sub-problems. Each sub-problem is based on a static or adaptive definition of weighted-sum aggregation coefficients, and is addressed by means of a state-of-the-art single-objective tabu search procedure. Next, we design a Pareto local search that iteratively improves a set of solutions based on a neighborhood structure and on the Pareto dominance relation. At last, we hybridize both classes of algorithms by combining a scalarizing and a Pareto local search in a sequential way. A comprehensive experimental analysis reveals the high performance of the proposed approaches, which substantially improve upon previous best-known solutions. Moreover, the obtained results show the superiority of the hybrid algorithm over non-hybrid ones in terms of solution quality, while requiring a competitive computational cost. In addition, a number of structural properties of the problem instances allow us to explain the main difficulties that the different classes of local search algorithms have to face

A Study of Memetic Search with Multi-parent Combination for UBQP

We present a multi-parent hybrid genetic–tabu algorithm (denoted by GTA) for the Unconstrained Binary Quadratic Programming (UBQP) problem, by incorporating tabu search into the framework of genetic algorithm. In this paper, we propose a new multi-parent combination operator for generating offspring solutions. A pool updating strategy based on a quality-and-distance criterion is used to manage the population. Experimental comparisons with leading methods for the UBQP problem on 25 large public instances demonstrate the efficacy of our proposed algorithm in terms of both solution quality and computational efficiency

Constrained Entropy Models; Solvability and Sensitivity

The paper presents an analysis of the constrained entropy maximization model from the point of view of geometric programming. While the original entropy maximization model consists of maximizing the entropy of a system subject only to constraints that the solution be a probability measure, the models considered here contain an additional set of linear constraints. These constrained models have been the subject of a wide range of applications in transportation and geographical analysis. Using the duality theory of geometric programming, we develop the dual to the constrained model, which as in the case of the original model is unconstrained except for the positivity restrictions on the dual variables. In addition, this duality theory enables us to study the solvability of the model and the impact of changes in the model parameters on the solution. The sensitivity analysis provides approximations to the optimal solution to problems with perturbed data without requiring the re-solving of the model. This analysis is appropriate for changes in the right hand sides of the constraints, the coefficients in the constraints, and the objective function coefficients. Since the constraint coefficients correspond to the objective function exponents in the primal geometric program, the analysis provides a means of studying such changes is any geometric program. The computational aspects of the procedures are illustrated on a trip distribution problem.entropy, geometric programming