共查询到20条相似文献,搜索用时 833 毫秒
1.
In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented. 相似文献
2.
Patrick Mehlitz 《Optimization》2016,65(6):1203-1227
This article is dedicated to the study of bilevel optimal control problems equipped with a fully convex lower level of special structure. In order to construct necessary optimality conditions, we consider a general bilevel programming problem in Banach spaces possessing operator constraints, which is a generalization of the original bilevel optimal control problem. We derive necessary optimality conditions for the latter problem using the lower level optimal value function, ideas from DC-programming and partial penalization. Afterwards, we apply our results to the original optimal control problem to obtain necessary optimality conditions of Pontryagin-type. Along the way, we derive a handy formula, which might be used to compute the subdifferential of the optimal value function which corresponds to the lower level parametric optimal control problem. 相似文献
3.
Yibing Lv Tiesong Hu Zhongping Wan 《Journal of Computational and Applied Mathematics》2008,220(1-2):175-180
In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem. 相似文献
4.
《Operations Research Letters》2023,51(1):84-91
In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given. 相似文献
5.
The bilevel programming problem: reformulations, constraint qualifications and optimality conditions
We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich. 相似文献
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7.
I. G. Ismailov 《Computational Mathematics and Modeling》1999,10(1):44-54
We study optimization problems in the presence of connection in the form of operator equations defined in Banach spaces. We
prove necessary conditions for optimality of first and second order, for example generalizing the Pontryagin maximal principle
for these problems. It is not our purpose to state the most general necessary optimality conditions or to compile a list of
all necessary conditions that characterize optimal control in any particular minimization problem. In the present article
we describe schemes for obtaining necessary conditions for optimality on solutions of general operator equations defined in
Banach spaces, and the scheme discussed here does not require that there be no global functional constraints on the controlling
parameters. As an example, in a particular Banach space we prove an optimality condition using the Pontryagin-McShane variation.
Bibliography: 20 titles.
Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 55–67. 相似文献
8.
《Optimization》2012,61(4-5):605-616
In this article, we first examine some modeling scenarios for a multistage bilevel programming problem and develop the solution techniques based on certain reformulations of the original problem. The optimality conditions obtained for a class of multistage problems are given in terms of the second order subdifferentials of Mordukhovich. 相似文献
9.
In this article, we consider bilevel optimization problems with discrete lower level and continuous upper level problems. Taking into account both approaches (optimistic and pessimistic) which have been developed in the literature to deal with this type of problem, we derive some conditions for the existence of solutions. In the case where the lower level is a parametric linear problem, the bilevel problem is transformed into a continuous one. After that, we are able to discuss local optimality conditions using tools of variational analysis for each of the different approaches. Finally, we consider a simple application of our results namely the bilevel programming problem with the minimum spanning tree problem in the lower level. 相似文献
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《Optimization》2012,61(8):1029-1047
In this article, we investigate bilevel programming problems with discrete lower level and continuous upper level problems. We will analyse the structure of these problems and discuss both the optimistic and the pessimistic solution approach. Since neither the optimistic nor the pessimistic solution functions are in general lower semicontinuous, we introduce weak solution function. By using these functions we are able to discuss optimality conditions for local and global optimality. 相似文献
13.
Jean Bosco Etoa Etoa 《Applied mathematics and computation》2011,217(15):6680-6690
In this paper, we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem. We use the Karush-Kuhn-Tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints; the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty. These complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using Fisher-Burmeister functional. Then, using a quadratic sequential programming method, we solve a series of smooth, regular problems that progressively approximate the nonsmooth problem. Some preliminary computational results are reported, showing that our approach is efficient. 相似文献
14.
Parametric global optimisation for bilevel programming 总被引:2,自引:2,他引:0
Nuno P. Faísca Vivek Dua Berç Rustem Pedro M. Saraiva Efstratios N. Pistikopoulos 《Journal of Global Optimization》2007,38(4):609-623
We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based
on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower’s)
problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables
of the outer (leader’s) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem
is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved
to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without
right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global
optimisation strategy. 相似文献
15.
以下层问题的K-T最优性条件代替下层问题,将线性二层规划转化为相应的单层规划问题,通过分析单层规划可行解集合的结构特征,设计了一种求解线性二层规划全局最优解的割平面算法.数值结果表明所设计的割平面算法是可行、有效的. 相似文献
16.
下层问题以上层决策变量作为参数,而上层是以下层问题的最优值作为响应
的一类最优化问题——二层规划问题。我们给出了由一系列此类二层规划去逼近原二层规划的逼近法,得到了这种逼近的一些有趣的结果. 相似文献
17.
We consider a hierarchical system where a leader incorporates into its strategy the reaction of the follower to its decision.
The follower's reaction is quite generally represented as the solution set to a monotone variational inequality. For the solution
of this nonconvex mathematical program a penalty approach is proposed, based on the formulation of the lower level variational
inequality as a mathematical program. Under natural regularity conditions, we prove the exactness of a certain penalty function,
and give strong necessary optimality conditions for a class of generalized bilevel programs. 相似文献
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The penalty function method, presented many years ago, is an important numerical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty function approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach. 相似文献
20.
Zhiqing Meng Chuangyin Dang Rui Shen Ming Jiang 《Journal of Optimization Theory and Applications》2012,153(2):377-387
Penalty methods are very efficient in finding an optimal solution to constrained optimization problems. In this paper, we
present an objective penalty function with two penalty parameters for inequality constrained bilevel programming under the
convexity assumption to the lower level problem. Under some conditions, an optimal solution to a bilevel programming defined
by the objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based
on the objective penalty function, an algorithm is developed to obtain an optimal solution to the original bilevel programming,
with its convergence proved under some conditions. 相似文献