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Seeding the initial population of a multi-objective evolutionary algorithm using gradient-based information

In the field of single-objective optimization, hybrid variants of gradient-based methods and evolutionary algorithms have been shown to perform better than an evolutionary method by itself. This same idea has been recently used in Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most cases, gradient information is used along the whole process, which involves a high computational cost, mainly related to the computation of the step lengths required. In contrast, in this paper we propose the use of gradient information only at the beginning of the search process. We will show that this sort of scheme maintains results of good quality while considerably decreasing the computational cost. In our work, we adopt a steepest descent method to generate some nondominated points which are then used to seed the initial population of a multi-objective evolutionary algorithm (MOEA), which will spread them along the Pareto front. The MOEA adopted in our case is the NSGA-II, which is representative of the state-of-the-art in the area. To validate our proposal, we adopt box-constrained continuous problems (the ZDT test suite). The gradients required are approximated using quadratic regressions. Our proposed approach performs a total of 2000 objective function evaluations, which is much lower than the number of evaluations normally adopted with the ZDT test suite in the specialized literature. Our results are compared with respect to the ldquopurerdquo NSGA-II (i.e., without using gradient-based information) so that the potential benefit of these initial solutions fed into the population can be properly assessed.

Multiobjective project portfolio selection with fuzzy constraints

Decision makers usually have to face a budget and other type of constraints when they have to decide which projects are going to be undertaken (to satisfy their requirements and guarantee profitable growth). Our purpose is to assist them in the task of selecting project portfolios. We have approached this problem by proposing a general nonlinear binary multi-objective mathematical model, which takes into account all the most important factors mentioned in the literature related with Project Portfolio Selection and Scheduling. Due to the existence of uncertainty in different aspects involved in the aforementioned decision task, we have also incorporated into the model some fuzzy parameters, which allow us to represent information not fully known by the decision maker/s. The resulting problem is both fuzzy and multiobjective. The results are complemented with graphical tools, which show the usefulness of the proposed model to assist the decision maker/s.

The challenge of optimizing expensive black boxes: a scatter search/rough set theory approach

There is renewed interest in the development of effective and efficient methods for optimizing models of which the optimizer has no structural knowledge. This is what in the literature is referred to as optimization of black boxes. In particular, we address the challenge of optimizing expensive black boxes, that is, those that require a significant computational effort to be evaluated. We describe the use of rough set theory within a scatter search framework, with the goal of identifying high-quality solutions with a limited number of objective function evaluations. The rough set strategies that we developed take advantage of the information provided by the best and diverse solutions found during the search, in order to define areas of the solution space that are promising for search intensification. We test our procedure on a set of 92 nonlinear multimodal functions of varied complexity and size and compare the results with a state-of-the-art procedure based on particle swarm optimization.

On the Use of Projected Gradients for Constrained Multiobjective Optimization Problems

Recent works have shown how hybrid variants of gradient-based methods and evolutionary algorithms perform better than a pure evolutionary method both for single-objective and multiobjective optimization. This same idea has been used with Evolutionary Multiobjective Optimization ( EMO ), obtaining also very promising results. In most cases, gradient information is used as part of the mutation operator (and only for unconstrained MOPs), in order to move every generated point to the exact Pareto front. In our approach, we use the Karush-Kuhn-Tucker optimality condition for constrained optimization problems to combine the information provided by the gradient vector of each objective function and the gradient vectors of constraint functions to obtain a feasible movement direction in those points near the border. In our approach, gradients of the objective functions will be approximated using quadratic regressions, trying to avoid local optima. The proposed algorithm is able to converge on several nonlinear constrained multiobjective optimization problems obtained from a benchmark, consuming few objective function evaluations (between 150 and 1000). Our results indicate that our proposed scheme may produce a significant reduction in the computational cost, while producing results of good quality, when it is incorporated into a hybrid MOEA or when it is used to seed an EMO algorithm.

Using a Gradient Based Method to Seed an EMO Algorithm

In the field of single-objective optimization, hybrid variants of gradient based methods and evolutionary algorithms have been shown to performance better than the pure evolutionary method. This same idea has been used with Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most of the cases, gradient information is used as part of the mutation operator, in order to move every generated point to the exact Pareto front. This means that gradient information is used along the whole process, and then consumes computational resources also along the whole process. On the other hand, in our approach we will use gradient information only at the beginning of the process, and will show that quality of the results is not decreased while computational cost is. We will use a steepest descent method to generate some efficient points to be used to seed an EMO method. The main goal will be generating some efficient points in the exact front using the less evaluations as possible, and let the EMO method use these points to spread along the whole Pareto front. In our approach, we will solve box-constrained continuous problems, gradients will be approximated using quadratic regressions and the EMO method will be based on Rough Sets theory Hernandez-Diaz et al. (Parallel Problem Solving from Nature (PPSN IX) 9th International Conference, 2006).