Competitive coevolutionary algorithm for robust multi-objective optimization: the worst case minimization

Multi-Objective Optimization (MOO) problems might be subject to many modeling or manufacturing uncertainties that affect the performance of the solutions obtained by a multi-objective optimizer. The decision maker must perform an extra step of sensitivity analysis in which each solution should be verified for its robustness, but this post optimization procedure makes the optimization process expensive and inefficient. In order to avoid this situation, many researchers are developing Robust MOO, where uncertainties are incorporated in the optimization process, which seeks optimal robust solutions. We introduce a coevolutionary approach for robust MOO, without incorporating robustness measures neither in the objective function nor in the constraints. Two populations compete in the environment, one representing solutions and minimizing the objectives, another representing uncertainties and maximizing the objectives in a worst case scenario. The proposed coevolutionary method is a coevolutionary version of MOEA/D. The results clearly suggest that these competing co-evolving populations are able to identify robust solutions to multi-objective optimization problems.info:eu-repo/semantics/publishedVersio

On the representation of the search region in multi-objective optimization

Given a finite set $N$ of feasible points of a multi-objective optimization (MOO) problem, the search region corresponds to the part of the objective space containing all the points that are not dominated by any point of $N$, i.e. the part of the objective space which may contain further nondominated points. In this paper, we consider a representation of the search region by a set of tight local upper bounds (in the minimization case) that can be derived from the points of $N$. Local upper bounds play an important role in methods for generating or approximating the nondominated set of an MOO problem, yet few works in the field of MOO address their efficient incremental determination. We relate this issue to the state of the art in computational geometry and provide several equivalent definitions of local upper bounds that are meaningful in MOO. We discuss the complexity of this representation in arbitrary dimension, which yields an improved upper bound on the number of solver calls in epsilon-constraint-like methods to generate the nondominated set of a discrete MOO problem. We analyze and enhance a first incremental approach which operates by eliminating redundancies among local upper bounds. We also study some properties of local upper bounds, especially concerning the issue of redundant local upper bounds, that give rise to a new incremental approach which avoids such redundancies. Finally, the complexities of the incremental approaches are compared from the theoretical and empirical points of view.Comment: 27 pages, to appear in European Journal of Operational Researc

Pareto-Path Multi-Task Multiple Kernel Learning

A traditional and intuitively appealing Multi-Task Multiple Kernel Learning (MT-MKL) method is to optimize the sum (thus, the average) of objective functions with (partially) shared kernel function, which allows information sharing amongst tasks. We point out that the obtained solution corresponds to a single point on the Pareto Front (PF) of a Multi-Objective Optimization (MOO) problem, which considers the concurrent optimization of all task objectives involved in the Multi-Task Learning (MTL) problem. Motivated by this last observation and arguing that the former approach is heuristic, we propose a novel Support Vector Machine (SVM) MT-MKL framework, that considers an implicitly-defined set of conic combinations of task objectives. We show that solving our framework produces solutions along a path on the aforementioned PF and that it subsumes the optimization of the average of objective functions as a special case. Using algorithms we derived, we demonstrate through a series of experimental results that the framework is capable of achieving better classification performance, when compared to other similar MTL approaches.Comment: Accepted by IEEE Transactions on Neural Networks and Learning System

Multi-objective Active Control Policy Design for Commensurate and Incommensurate Fractional Order Chaotic Financial Systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.In this paper, an active control policy design for a fractional order (FO) financial system is attempted, considering multiple conflicting objectives. An active control template as a nonlinear state feedback mechanism is developed and the controller gains are chosen within a multi-objective optimization (MOO) framework to satisfy the conditions of asymptotic stability, derived analytically. The MOO gives a set of solutions on the Pareto optimal front for the multiple conflicting objectives that are considered. It is shown that there is a trade-off between the multiple design objectives and a better performance in one objective can only be obtained at the cost of performance deterioration in the other objectives. The multi-objective controller design has been compared using three different MOO techniques viz. Non Dominated Sorting Genetic Algorithm-II (NSGA-II), epsilon variable Multi-Objective Genetic Algorithm (ev-MOGA), and Multi Objective Evolutionary Algorithm with Decomposition (MOEA/D). The robustness of the same control policy designed with the nominal system settings have been investigated also for gradual decrease in the commensurate and incommensurate fractional orders of the financial system