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Penalty Function Method for Bilevel Multiobjective Programming

Zhao Wei,

自动化学报 , 1998,
Abstract: In this paper, a class of nonlinear bilevel multiobjective programming problems is studied. Under the assumptions that the objective functions are strictly convex and the constraint set of decision variables is convex, by transforming the bilevel multiobjective programming problem into a series of one level multiobjective programming problems, the penalty function method for bilevel multiobjective programming is established, and the convergence of the method is proved. This method complements the theory of bilevel multiobjective programming and provides a powerful means to solve the practical bilevel multiobjective decision making problems
Journal of Science and Arts , 2011,
Abstract: This paper is concerning on the multiobjective programming problem where the function involved are nondifferentiable. We present and prove the sufficient conditions for a feasible point to be weakly efficient. Our research starts from the invexity proposed by H. Slimani and M.S. Rajdef and extend their concept for the case when the functions are nondifferentiable. To solve the problem in this new framework we show how local cone approximation concept can be used. Thus we provide a new approach for the nondifferentiable multiobjective programming problems that can be easily applied in the practical problems.
Optimality conditions and duality in multiobjective programming with invexity  [PDF]
Ferrara Massimiliano,Viorica-Stefanescu Maria
Yugoslav Journal of Operations Research , 2008, DOI: 10.2298/yjor0802153f
Abstract: (', ρ)-invexity has recently been introduced with the intent of generalizing invex functions in mathematical programming. Using such conditions we obtain new sufficiency results for optimality in multiobjective programming and extend some classical duality properties.
An Improved Particle Swarm Optimization for Solving Bilevel Multiobjective Programming Problem
Tao Zhang,Tiesong Hu,Yue Zheng,Xuning Guo
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/626717
Abstract: An improved particle swarm optimization (PSO) algorithm is proposed for solving bilevel multiobjective programming problem (BLMPP). For such problems, the proposed algorithm directly simulates the decision process of bilevel programming, which is different from most traditional algorithms designed for specific versions or based on specific assumptions. The BLMPP is transformed to solve multiobjective optimization problems in the upper level and the lower level interactively by an improved PSO. And a set of approximate Pareto optimal solutions for BLMPP is obtained using the elite strategy. This interactive procedure is repeated until the accurate Pareto optimal solutions of the original problem are found. Finally, some numerical examples are given to illustrate the feasibility of the proposed algorithm.
Fuzzy Goal Programming Procedure to Bilevel Multiobjective Linear Fractional Programming Problems  [PDF]
Mahmoud A. Abo-Sinna,Ibrahim A. Baky
International Journal of Mathematics and Mathematical Sciences , 2010, DOI: 10.1155/2010/148975
Abstract: This paper presents a fuzzy goal programming (FGP) procedure for solving bilevel multiobjective linear fractional programming (BL-MOLFP) problems. It makes an extension work of Moitra and Pal (2002) and Pal et al. (2003). In the proposed procedure, the membership functions for the defined fuzzy goals of the decision makers (DMs) objective functions at both levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by first-level decision maker are developed first in the model formulation of the problem. Then a fuzzy goal programming model to minimize the group regret of degree of satisfactions of both the decision makers is developed to achieve the highest degree (unity) of each of the defined membership function goals to the extent possible by minimizing their deviational variables and thereby obtaining the most satisfactory solution for both decision makers. The method of variable change on the under- and over-deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology. Illustrative numerical example is given to demonstrate the procedure. 1. Introduction Bi-level mathematical programming (BLMP) is identified as mathematical programming that solves decentralized planning problems with two decision makers (DMs) in a two-level or hierarchical organization [1]. The basic connect of the BLMP technique is that a first-level decision maker (FLDM) (the leader) sets his goals and/or decisions and then asks each subordinate level of the organization for their optima which are calculated in isolation; the second-level DM (SLDM) (the follower) decisions are then submitted and modified by the FLDM with consideration of the overall benefit for the organization; the process continued until a satisfactory solution is reached. In other words, although the FLDM independently optimizes its own benefits, the decision may be affected by the reaction of the SLDM. As a consequence, decision deadlock arises frequently and the problem of distribution of proper decision power is encountered in most of the practical decision situations. Most of the developments on BLMP problems focus on bi-level linear programming [2–5], and many others for bilevel nonlinear programming and bi-level multiobjective programming [2, 6–11]. A bibliography of references on bi-level programming in both linear and non-linear cases, which is updated biannually, can be found in [12]. The use of the fuzzy set theory [13] for
Solving Bilevel Linear Multiobjective Programming Problems  [PDF]
Calice Olivier Pieume, Patrice Marcotte, Laure Pauline Fotso, Patrick Siarry
American Journal of Operations Research (AJOR) , 2011, DOI: 10.4236/ajor.2011.14024
Abstract: This study addresses bilevel linear multi-objective problem issues i.e the special case of bilevel linear programming problems where each decision maker has several objective functions conflicting with each other. We introduce an artificial multi-objective linear programming problem of which resolution can permit to generate the whole feasible set of the upper level decisions. Based on this result and depending if the leader can evaluate or not his preferences for his different objective functions, two approaches for obtaining Pareto- optimal solutions are presented.
Optimality Conditions for Bilevel Programming Problem in Asplund Spaces  [PDF]
Suvendu Pattanaik
Mathematics , 2011,
Abstract: Here, necessary optimal condition for Optimistic Bilevel programming problem is obtained in Asplund spaces. Also we have got necessary optimal conditions in finite dimensional spaces, by assuming differentiability on the given functions.
Geometric Characteristics and Optimality Conditions of Quadratic Bilevel Programming Models

Li Zhihui,Teng Chunxian,Li Lei,Li Haobai,

系统工程理论与实践 , 1999,
Abstract: Bilevel programming models can efficiently describe the management decision system with multilevel. In this paper, we introduce a class of quadratic bilevel programming models, which have extensive representativeness, and discuss their geometric characteristics. In the end, we give optimality conditions of the quadratic bilevel programming models.
Optimality and Duality for Multiobjective Fractional Programming Involving Nonsmooth Generalized -Univex Functions  [PDF]
Jen-Chwan Liu,Chun-Yu Liu
International Journal of Mathematics and Mathematical Sciences , 2013, DOI: 10.1155/2013/284046
Abstract: We establish properly efficient necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth generalized -univex functions. Utilizing the necessary optimality conditions, we formulate the parametric dual model and establish some duality results in the framework of generalized -univex functions. 1. Introduction In this paper, we consider the following nondifferentiable nonconvex multiobjective fractional programming problem: where (a1) , , and , are Lipschitz on , and is an open subset of ; (a2) , , .Minimize means obtaining efficient solution in the following sense. A point is said to be an efficient solution for (MFP) if there is no such that with at least one strict inequality. A point is said to be a??properly efficient solution for (MFP) which was introduced by Geoffrion [1] if and only if (a) is an efficient solution for (MFP), (b) there exists a scalar such that for each , we have ?for some such that , whenever and . An efficient point for (MFP) that is not properly efficient is said to be improperly efficient. Thus, for to be improperly efficient for (MFP) means that to every scalar , there is a point and an such that and for all such that . Many papers have been devoted to the multiobjective fractional programming problem in recent decades; see for example [1–13]. In [8], Preda introduced generalized -convexity, an extension of -convexity and generalized -convexity defined by Vial [14, 15]. Bhatia and Jain [2] defined generalized -convexity for nonsmooth functions, an extension of generalized -convexity defined by Preda [8], and they derived some duality theorems for nonsmooth multiobjective programs. In [5, 6], Liu also established the Kuhn-Tucker type necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth pseudoinvex functions in [5] or -convex functions in [6] and considered the parameter dual problem in the framework of generalized convex functions. Recently, Zalmai [13, 16, 17] introduced generalized -univex -set functions and he also established sufficient efficiency conditions in multiobjective fractional subset programming [13] and sufficient optimality conditions in minimax fractional subset programming [16, 17] under various generalized -univexity assumptions. In [10], Preda et al. obtained duality results for a dual model of Zalmai [13] replacing the assumptions of sublinearity or convexity by that of quasiconvexity in the third argument. In this paper, we are inspired to consider the optimality and duality of properly efficient for
Optimality and Duality for an Efficient Solution of Multiobjective Nonlinear Fractional Programming Problem Involving Semilocally Convex Functions  [cached]
P. Bhatnagar
International Journal of Applied Operational Research , 2012,
Abstract: In this paper, the problem under consideration is multiobjective non-linear fractional programming problem involving semilocally convex and related functions. We have discussed the interrelation between the solution sets involving properly efficient solutions of multiobjective fractional programming and corresponding scalar fractional programming problem. Necessary and sufficient optimality conditions are obtained for efficient and properly efficient solutions. Some duality results are established for multiobjective Schaible type dual. Keywords: Nonlinear Programming, Multi-Objective Fractional Programming, Semilocally Convex Functions, Pseudoconvex Functions.
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