Elementary landscape decomposition of the 0-1 unconstrained quadratic optimization

Journal of Heuristics, 19(4), pp.711-728Landscapes’ theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of an especial kind of landscape called elementary landscape. The elementary landscape decomposition of a combinatorial optimization problem is a useful tool for understanding the problem. Such decomposition provides an additional knowledge on the problem that can be exploited to explain the behavior of some existing algorithms when they are applied to the problem or to create new search methods for the problem. In this paper we analyze the 0-1 Unconstrained Quadratic Optimization from the point of view of landscapes’ theory. We prove that the problem can be written as the sum of two elementary components and we give the exact expressions for these components. We use the landscape decomposition to compute autocorrelation measures of the problem, and show some practical applications of the decomposition.Spanish Ministry of Sci- ence and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

Problem Understanding through Landscape Theory

In order to understand the structure of a problem we need to measure some features of the problem. Some examples of measures suggested in the past are autocorrelation and fitness-distance correlation. Landscape theory, developed in the last years in the field of combinatorial optimization, provides mathematical expressions to efficiently compute statistics on optimization problems. In this paper we discuss how can we use optimización combinatoria in the context of problem understanding and present two software tools that can be used to efficiently compute the mentioned measures.Ministerio de Economía y Competitividad (TIN2011-28194

Multi-objectivising Combinatorial Optimisation Problems by means of Elementary Landscape Decompositions

In the last decade, many works in combinatorial optimisation have shown that, due to the advances in multi-objective optimisation, the algorithms from this field could be used for solving single-objective problems as well. In this sense, a number of papers have proposed multi-objectivising single-objective problems in order to use multi-objective algorithms in their optimisation. In this paper, we follow up this idea by presenting a methodology for multi-objectivising combinatorial optimisation prob- lems based on elementary landscape decompositions of their objective function. Under this framework, each of the elementary landscapes obtained from the decomposition is considered as an independent objective function to optimise. In order to illustrate this general methodology, we consider four problems from different domains: the quadratic assignment problem and the linear ordering problem (permutation domain), the 0-1 unconstrained quadratic optimisation problem (binary domain), and the frequency assignment problem (integer domain). We implemented two widely known multi-objective algorithms, NSGA-II and SPEA2, and compared their perfor- mance with that of a single-objective GA. The experiments conducted on a large benchmark of instances of the four problems show that the multi-objective algorithms clearly outperform the single-objective approaches. Furthermore, a discussion on the results suggests that the multi-objective space generated by this decomposition enhances the exploration ability, thus permitting NSGA-II and SPEA2 to obtain better results in the majority of the tested instances.TIN2016-78365R IT-609-1

Exact computation of the expectation curves of the bit-flip mutation using landscapes theory

Chicano, F., & Alba E. (2011). Exact computation of the expectation curves of the bit-flip mutation using landscapes theory. Proceedings of 13th Annual Genetic and Evolutionary Computation Conference, Dublin, Ireland, July 12-16, 2011. pp. 2027–2034.Bit-flip mutation is a common operation when a genetic algorithm is applied to solve a problem with binary representation. We use in this paper some results of landscapes theory and Krawtchouk polynomials to exactly compute the expected value of the fitness of a mutated solution. We prove that this expectation is a polynomial in p, the probability of flipping a single bit. We analyze these polynomials and propose some applications of the obtained theoretical results.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. This research has been partially funded by the Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project) and the Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

Exact Computation of the Fitness-Distance Correlation for Pseudoboolean Functions with One Global Optimum

Chicano, F., & Alba E. (2012). Exact Computation of the Fitness-Distance Correlation for Pseudoboolean Functions with One Global Optimum. (Hao, J-K., & Middendorf M., Ed.).Evolutionary Computation in Combinatorial Optimization - 12th European Conference, EvoCOP 2012, Málaga, Spain, April 11-13, 2012. Proceedings. 111–123.Landscape theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of a special kind of landscapes called elementary landscapes. The decomposition of the objective function of a problem into its elementary components can be exploited to compute summary statistics. We present closed-form expressions for the fitness-distance correlation (FDC) based on the elementary landscape decomposition of the problems defined over binary strings in which the objective function has one global optimum. We present some theoretical results that raise some doubts on using FDC as a measure of problem difficulty.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish Ministry of Science and Innovation and FEDER under contracts TIN2008-06491-C04-01 and TIN2011-28194. Andalusian Government under contract P07-TIC-03044

Quasi-elementary Landscapes and Superpositions of Elementary Landscapes

Whitley, D., & Chicano F. (2012). Quasi-elementary Landscapes and Superpositions of Elementary Landscapes. (Hamadi, Y., & Schoenauer M., Ed.).Learning and Intelligent Optimization - 6th International Conference, LION 6, Paris, France, January 16-20, 2012, Revised Selected Papers. 277–291.There exist local search landscapes where the evaluation function is an eigenfunction of the graph Laplacian that corresponds to the neighborhood structure of the search space. Problems that display this structure are called “Elementary Landscapes” and they have a number of special mathematical properties. The term “Quasi-elementary landscapes” is introduced to describe landscapes that are “almost” elementary; in quasi-elementary landscapes there exists some efficiently computed “correction” that captures those parts of the neighborhood structure that deviate from the normal structure found in elementary landscapes. The “shift” operator, as well as the “3-opt” operator for the Traveling Salesman Problem landscapes induce quasi-elementary landscapes. A local search neighborhood for the Maximal Clique problem is also quasi-elementary. Finally, we show that landscapes which are a superposition of elementary landscapes can be quasi-elementary in structure.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-11-1-0088. Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

Exact computation of the fitness-distance correlation for pseudoboolean functions with one global optimum

Abstract. Landscape theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of a special kind of landscapes called elementary landscapes. The decomposition of the objective function of a problem into its elementary components can be exploited to compute summary statistics. We present closed-form expressions for the fitness-distance correlation (FDC) based on the elementary landscape decomposition of the problems defined over binary strings in which the objective function has one global optimum. We present some theoretical results that raise some doubts on using FDC as a measure of problem difficulty