Bi-level Multi-objective Problem Research Articles

First and Second Order Optimality Conditions using Approximations for Fractional Multiobjective Bilevel Problems under Fractional Constraints

First and Second Order Optimality Conditions using Approximations for Fractional Multiobjective Bilevel Problems under Fractional Constraints

Necessary optimality conditions for a bilevel multiobjective problem in terms of approximations

In this work, we are concerned with a bilevel multiobjective optimization problem. To get the Karush–Kuhn–Tucker-type necessary optimality condition, with the help of the generalized Motzkin's Theorem, we introduced an appropriate generalized constraint qualification. Our findings are based on a mean-value theorem in terms of approximations for continuous functions. Examples that illustrate our results are also given.

Integrated optimization of railway freight operation planning and pricing based on carbon emission reduction policies

Cargo transportation is one of the major sources of carbon emissions. To reduce carbon emissions from inland freight, one way is to encourage rail transportation instead of truck transportation. In this paper, a new approach combining railway operation management, revenue management, and policy formulation is proposed to increase the market share of railway freight and reduce carbon emissions. Through the integrated optimization of operation planning and pricing, the competitiveness of railway freight services is improved. The carbon tax policy acting on the demand-side can affect consigners’ choice behavior. The low-carbon subsidy policy acting on the supply-side can encourage rail transport. Then a bi-level multi-objective model is proposed to achieve the trade-off among the three stakeholders (the government, rail service operators, and consigners). The complexity of the bi-level multi-objective problem motivates the development of a hybrid optimization algorithm that combines the Non-dominated sorting genetic algorithm III (NSGA-III) and the Descent algorithm based on sensitivity analysis (SAB). Through a real case based on a rail service network with nine nodes in China, the rationality of the proposed methodology is verified. The results show that the approach proposed can achieve a 39.27% modal shift (MS) from road to rail, as well as a 37.09% carbon emission reduction.

Solving a class of multiobjective bilevel problems by DC programming

In this paper, we consider a class of multiobjective bilevel programming problems in which the first level objective function is assumed to be a vector valued DC function and the second level problem is a linear multiobjective program. The problem is transformed into a standard single optimisation problem by using a preference function. We give a characterisation to the induced region and reformulate the problem as a problem of optimising a function over the efficient set. Next, a well-known representation of the efficient set is used which will allow to transform the problem, using an exact penalisation, into a DC program. Finally, we apply the DC algorithm to solve the resulting DC program.

Solving a class of multiobjective bilevel problems by DC programming

Solving a class of multiobjective bilevel problems by DC programming

Optimality conditions for nonsmooth multiobjective bilevel optimization problems

This article is devoted to the study of a nonsmooth multiobjective bilevel optimization problem, which involves the vector-valued objective functions in both levels of the considered program. We first formulate a relaxation multiobjective formulation for the multiobjective bilevel problem and examine the relationships of solutions between them. We then establish Fritz John (FJ) and Karush–Kuhn–Tucker (KKT) necessary conditions for the nonsmooth multiobjective bilevel optimization problem via its relaxation. This is done by studying a related multiobjective optimization problem with operator constraints.

Approximated set-valued mapping approach for handling multiobjective bilevel problems

A significant amount of research has been done on bilevel optimization problems both in the realm of classical and evolutionary optimization. However, the multiobjective extensions of bilevel programming have received relatively little attention from researchers in both the domains. The existing algorithms are mostly brute-force nested strategies, and therefore computationally demanding. In this paper, we develop insights into multiobjective bilevel optimization through theoretical progress made in the direction of parametric multiobjective programming. We introduce an approximated set-valued mapping procedure that would be helpful in the development of efficient evolutionary approaches for solving these problems. The utility of the procedure has been emphasized by incorporating it in a hierarchical evolutionary framework and assessing the improvements. Test problems with varying levels of complexity have been used in the experiments.

Necessary Optimality Conditions for Multiobjective Bilevel Programs

The multiobjective bilevel program is a sequence of two optimization problems, with the upper-level problem being multiobjective and the constraint region of the upper level problem being determined implicitly by the solution set to the lower-level problem. In the case where the Karush-Kuhn-Tucker (KKT) condition is necessary and sufficient for global optimality of all lower-level problems near the optimal solution, we present various optimality conditions by replacing the lower-level problem with its KKT conditions. For the general multiobjective bilevel problem, we derive necessary optimality conditions by considering a combined problem, with both the value function and the KKT condition of the lower-level problem involved in the constraints. Most results of this paper are new, even for the case of a single-objective bilevel program, the case of a mathematical program with complementarity constraints, and the case of a multiobjective optimization problem.

Fuzzy Optimality Conditions for Fractional Multiobjective Bilevel Problems Under Fractional Constraints

In this article, we are concerned with fractional multiobjective optimization problems. In order to derive optimality conditions, we consider a new single level problem [12], which is locally equivalent to the bilevel fractional multiobjective problem (P) at the optimal solution. Our approach consists of using another approach initiated by Mordukhovich [7, 8], which does not involve any convex approximations and convex separation arguments, called the extremal principle [5, 6, 9], for the study of necessary optimality conditions in fractional vector optimization.

Towards Understanding Evolutionary Bilevel Multi-Objective Optimization Algorithm

A number of studies can be found in the context of bilevel single objective optimization problems, but not many exist, which tackle the bilevel multi-objective problems. Deb and Sinha (October, 2008) proposed a bilevel multi-objective optimization algorithm based on evolutionary multi-objective optimization (EMO) principles and discussed the issues involved in solving such a problem. In this paper we suggest an improved version of the previous algorithm which leads to high number of savings in function evaluations. Simulation results have been presented for two test problems and a comparison with the previous version has been done. The paper also discusses the complexity of bilevel problems and challenges involved in handling such problems. The existence of these problems in many practical problem solving tasks like optimal control, process optimization, game-playing strategy development, transportation problems, and others make it an important area which still needs to be considered by researchers. A two level optimization task involved in solving such problems makes the problem difficult and poses a number of challenges in getting close to the pareto front which has been addressed in the paper.