Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

An Improved Excitation Matching Method based on an Ant Colony Optimization for Suboptimal-Free Clustering in Sum-Difference Compromise Synthesis

Dealing with an excitation matching method, this paper presents a global optimization strategy for the optimal clustering in sum-difference compromise linear arrays. Starting from a combinatorial formulation of the problem at hand, the proposed technique is aimed at determining the sub-array configuration expressed as the optimal path inside a directed acyclic graph structure modelling the solution space. Towards this end, an ant colony metaheuristic is used to benefit of its hill-climbing properties in dealing with the non-convexity of the sub-arraying as well as in managing graph searches. A selected set of numerical experiments are reported to assess the efficiency and current limitations of the ant-based strategy also in comparison with previous local combinatorial search methods. (c) 2009 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works

Multi-rendezvous Spacecraft Trajectory Optimization with Beam P-ACO

The design of spacecraft trajectories for missions visiting multiple celestial bodies is here framed as a multi-objective bilevel optimization problem. A comparative study is performed to assess the performance of different Beam Search algorithms at tackling the combinatorial problem of finding the ideal sequence of bodies. Special focus is placed on the development of a new hybridization between Beam Search and the Population-based Ant Colony Optimization algorithm. An experimental evaluation shows all algorithms achieving exceptional performance on a hard benchmark problem. It is found that a properly tuned deterministic Beam Search always outperforms the remaining variants. Beam P-ACO, however, demonstrates lower parameter sensitivity, while offering superior worst-case performance. Being an anytime algorithm, it is then found to be the preferable choice for certain practical applications.Comment: Code available at https://github.com/lfsimoes/beam_paco__gtoc

SamACO: variable sampling ant colony optimization algorithm for continuous optimization

An ant colony optimization (ACO) algorithm offers algorithmic techniques for optimization by simulating the foraging behavior of a group of ants to perform incremental solution constructions and to realize a pheromone laying-and-following mechanism. Although ACO is first designed for solving discrete (combinatorial) optimization problems, the ACO procedure is also applicable to continuous optimization. This paper presents a new way of extending ACO to solving continuous optimization problems by focusing on continuous variable sampling as a key to transforming ACO from discrete optimization to continuous optimization. The proposed SamACO algorithm consists of three major steps, i.e., the generation of candidate variable values for selection, the ants’ solution construction, and the pheromone update process. The distinct characteristics of SamACO are the cooperation of a novel sampling method for discretizing the continuous search space and an efficient incremental solution construction method based on the sampled values. The performance of SamACO is tested using continuous numerical functions with unimodal and multimodal features. Compared with some state-of-the-art algorithms, including traditional ant-based algorithms and representative computational intelligence algorithms for continuous optimization, the performance of SamACO is seen competitive and promising

An ant colony algorithm for the sequential testing problem under precedence constraints.

We consider the problem of minimum cost sequential testing of a series (parallel) system under precedence constraints that can be modeled as a nonlinear integer program. We develop and implement an ant colony algorithm for the problem. We demonstrate the performance of this algorithm for special type of instances for which the optimal solutions can be found in polynomial time. In addition, we compare the performance of the algorithm with a special branch and bound algorithm for general instances. The ant colony algorithm is shown to be particularly effective for larger instances of the problem