arXiv subjects

Kalyanmoy Deb

Publications and source records attributed to Kalyanmoy Deb.

43 records · Page 3Linked to original sources

arXiv2017

Bilevel Optimization based on Iterative Approximation of Multiple Mappings

A large number of application problems involve two levels of optimization, where one optimization task is nested inside the other. These problems are known as bilevel optimization problems and have been studied by both classical optimization community and evolutionary optimization community. Most of the solution procedures proposed until now are either computationally very expensive or applicable to only small classes of bilevel optimization problems adhering to mathematically simplifying assumptions. In this paper, we propose an evolutionary optimization method that tries to reduce the computational expense by iteratively approximating two important mappings in bilevel optimization; namely, the lower level rational reaction mapping and the lower level optimal value function mapping. The algorithm has been tested on a large number of test problems and comparisons have been performed with other algorithms. The results show the performance gain to be quite significant. To the best knowledge of the authors, a combined theory-based and population-based solution procedure utilizing mappings has not been suggested yet for bilevel problems.

math.OC↗

arXiv2017

Integration of Preferences in Decomposition Multi-Objective Optimization

Most existing studies on evolutionary multi-objective optimization focus on approximating the whole Pareto-optimal front. Nevertheless, rather than the whole front, which demands for too many points (especially in a high-dimensional space), the decision maker might only interest in a partial region, called the region of interest. In this case, solutions outside this region can be noisy to the decision making procedure. Even worse, there is no guarantee that we can find the preferred solutions when tackling problems with complicated properties or a large number of objectives. In this paper, we develop a systematic way to incorporate the decision maker's preference information into the decomposition-based evolutionary multi-objective optimization methods. Generally speaking, our basic idea is a non-uniform mapping scheme by which the originally uniformly distributed reference points on a canonical simplex can be mapped to the new positions close to the aspiration level vector specified by the decision maker. By these means, we are able to steer the search process towards the region of interest either directly or in an interactive manner and also handle a large number of objectives. In the meanwhile, the boundary solutions can be approximated given the decision maker's requirements. Furthermore, the extent of the region of the interest is intuitively understandable and controllable in a closed form. Extensive experiments, both proof-of-principle and on a variety of problems with 3 to 10 objectives, fully demonstrate the effectiveness of our proposed method for approximating the preferred solutions in the region of interest.

cs.NE↗

arXiv2016

Test Problem Construction for Single-Objective Bilevel Optimization

In this paper, we propose a procedure for designing controlled test problems for single-objective bilevel optimization. The construction procedure is flexible and allows its user to control the different complexities that are to be included in the test problems independently of each other. In addition to properties that control the difficulty in convergence, the procedure also allows the user to introduce difficulties caused by interaction of the two levels. As a companion to the test problem construction framework, the paper presents a standard test suite of twelve problems, which includes eight unconstrained and four constrained problems. Most of the problems are scalable in terms of variables and constraints. To provide baseline results, we have solved the proposed test problems using a nested bilevel evolutionary algorithm. The results can be used for comparison, while evaluating the performance of any other bilevel optimization algorithm. The codes related to the paper may be accessed from the website \url{http://bilevel.org}.

cs.MS↗

arXiv2015

Feasibility Preserving Constraint-Handling Strategies for Real Parameter Evolutionary Optimization

Evolutionary Algorithms (EAs) are being routinely applied for a variety of optimization tasks, and real-parameter optimization in the presence of constraints is one such important area. During constrained optimization EAs often create solutions that fall outside the feasible region; hence a viable constraint- handling strategy is needed. This paper focuses on the class of constraint-handling strategies that repair infeasible solutions by bringing them back into the search space and explicitly preserve feasibility of the solutions. Several existing constraint-handling strategies are studied, and two new single parameter constraint-handling methodologies based on parent-centric and inverse parabolic probability (IP) distribution are proposed. The existing and newly proposed constraint-handling methods are first studied with PSO, DE, GAs, and simulation results on four scalable test-problems under different location settings of the optimum are presented. The newly proposed constraint-handling methods exhibit robustness in terms of performance and also succeed on search spaces comprising up-to 500 variables while locating the optimum within an error of 10$^{-10}$. The working principle of the IP based methods is also demonstrated on (i) some generic constrained optimization problems, and (ii) a classic `Weld' problem from structural design and mechanics. The successful performance of the proposed methods clearly exhibits their efficacy as a generic constrained-handling strategy for a wide range of applications.

cs.NE↗

arXiv2013

Efficient Evolutionary Algorithm for Single-Objective Bilevel Optimization

Bilevel optimization problems are a class of challenging optimization problems, which contain two levels of optimization tasks. In these problems, the optimal solutions to the lower level problem become possible feasible candidates to the upper level problem. Such a requirement makes the optimization problem difficult to solve, and has kept the researchers busy towards devising methodologies, which can efficiently handle the problem. Despite the efforts, there hardly exists any effective methodology, which is capable of handling a complex bilevel problem. In this paper, we introduce bilevel evolutionary algorithm based on quadratic approximations (BLEAQ) of optimal lower level variables with respect to the upper level variables. The approach is capable of handling bilevel problems with different kinds of complexities in relatively smaller number of function evaluations. Ideas from classical optimization have been hybridized with evolutionary methods to generate an efficient optimization algorithm for generic bilevel problems. The efficacy of the algorithm has been shown on two sets of test problems. The first set is a recently proposed SMD test set, which contains problems with controllable complexities, and the second set contains standard test problems collected from the literature. The proposed method has been evaluated against two benchmarks, and the performance gain is observed to be significant.

cs.NE↗

arXiv2013

Finding Optimal Strategies in a Multi-Period Multi-Leader-Follower Stackelberg Game Using an Evolutionary Algorithm

Stackelberg games are a classic example of bilevel optimization problems, which are often encountered in game theory and economics. These are complex problems with a hierarchical structure, where one optimization task is nested within the other. Despite a number of studies on handling bilevel optimization problems, these problems still remain a challenging territory, and existing methodologies are able to handle only simple problems with few variables under assumptions of continuity and differentiability. In this paper, we consider a special case of a multi-period multi-leader-follower Stackelberg competition model with non-linear cost and demand functions and discrete production variables. The model has potential applications, for instance in aircraft manufacturing industry, which is an oligopoly where a few giant firms enjoy a tremendous commitment power over the other smaller players. We solve cases with different number of leaders and followers, and show how the entrance or exit of a player affects the profits of the other players. In the presence of various model complexities, we use a computationally intensive nested evolutionary strategy to find an optimal solution for the model. The strategy is evaluated on a test-suite of bilevel problems, and it has been shown that the method is successful in handling difficult bilevel problems.

cs.GT↗

arXiv2013

Multi-objective Stackelberg Game Between a Regulating Authority and a Mining Company: A Case Study in Environmental Economics

Bilevel programming problems are often found in practice. In this paper, we handle one such bilevel application problem from the domain of environmental economics. The problem is a Stakelberg game with multiple objectives at the upper level, and a single objective at the lower level. The leader in this case is the regulating authority, and it tries to maximize its total tax revenue over multiple periods while trying to minimize the environmental damages caused by a mining company. The follower is the mining company whose sole objective is to maximize its total profit over multiple periods under the limitations set by the leader. The solution to the model contains the optimal taxation and extraction decisions to be made by the players in each of the time periods. We construct a simplistic model for the Stackelberg game and provide an analytical solution to the problem. Thereafter, the model is extended to incorporate realism and is solved using a bilevel evolutionary algorithm capable of handling multiple objectives.

cs.GT↗