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Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. This model has been applied to decentralized planning problems involving a decision process with a hierarchical structure. In this paper, we consider the bilevel linear fractional/linear programming problem, in which the objective function of the first-level is linear fractional, the objective function of the second level is linear, and the common constraint region is a polyhedron. For this problem, taking into account the relationship between the optimization problem of the second level and its dual, a global optimization approach is proposed that uses an exact penalty function based on the duality gap of the second-level problem.

A class of proximal gradient-type algorithm for bilevel nonlinear nondifferentiable programming problems with smooth substructure is developed in this paper. The original problem is approximately reformulated by explicit slow control technique to a parameterized family function which makes full use of the information of smoothness. At each iteration, we only need to calculate one proximal point analytically or with low computational cost. We prove that the accumulation iterations generated by the algorithms are solutions of the original problem. Moreover, some results of complexity of the algorithms are presented in convergence analysis. Numerical experiments are implemented to verify the efficiency of the proximal gradient algorithms for solving this kind of bilevel programming problems.

Bilevel programming deals with hierarchical optimization problems in which the leader at the upper level attempts to optimize his or her objectives, but subject to a set of constraints and the follower's reactions. Typical bilevel programming considers one leader one follower situation and supposes each of them has only one objective. In real world situations, multiple followers may be involved and they may be with different relationships such as sharing decision variables or not, sharing objectives or not. Therefore, the leader's decision will be affected not only by those followers' reactions but also by their relationships. In addition, any of the leader and/or these followers may have multiple conflict objectives that should be optimized simultaneously. Furthermore, the parameters of a bilevel programming model may be described by uncertain values. This paper addresses all these three issues as a whole by particularly focusing on the situation of sharing decision variables among followers. It first proposes a set of fuzzy multi-objective multi-follower bilevel programming (FMMBP) models to describe the complex issue. It then presents an approximation branch-and-bound algorithm to solve the FMMBP problems. Finally, two examples illustrate the proposed models and algorithm.

This paper deals with optimal design of time-censored step-stress partially accelerated life test sampling plan (PALTSP) using Burr type-XII life distribution. The Burr type XII distribution has been found appropriate for modeling failures that occur with less frequency and also when there is high occurrence of early failures. This distribution has been found appropriate for accelerated life testing experiments. The optimum sampling plan obtained using bilevel programming approach consists in finding optimum sample size and optimum stress change point by minimizing expected total cost per lot comprising warranty costs with respect to acceptance or rejection of the lot, sampling cost and testing cost such that the producer's and consumer's interested are safeguarded. The methods developed has been illustrated using an example and sensitivity analyses carried out.

This research develops a decision support for strategic pricing and aggregate production distribution planning for a small-scale supplier intending to penetrate into a potential market engendered by a single buyer. A novel mixed integer bilevel programming model is proposed to formulate the problem in which the supplier is considered as a leader and the buyer is a follower. The proposed model subsumes the assessment of demand share against the price quotation, enabling the supplier to prepare an aggregate production distribution plan accordingly. An integer coded genetic algorithm is used to solve the model and its implementation is exhibited through a test scenario.

The bilevel programming problem is a leader–follower game in which two players try to maximize their own objective functions over a common constraint region. This paper discusses an integer nonlinear bilevel programming problem with box constraints by exploiting the quasimonotone character of the indefinite quadratic fractional function, considered as leader's objective. By making use of the duality theory, given nonlinear bilevel programming problem is transformed into single level programming problem. Various cuts have been discussed in this paper which successively rank and scan all integer feasible points of the single level programming problem in the decreasing value of objective function. An iterative algorithm is proposed, which by making use of these cuts repeatedly, solves the problem.

This paper presents an extended Branch-and-Bound algorithm for solving fuzzy linear bilevel programming problems. In a fuzzy bilevel programming model, the leader attempts to optimize his/her fuzzy objective with a consideration of overall satisfaction, and the follower tries to find an optimized strategy, under himself fuzzy objective, according to each of possible decisions made by the leader. This paper first proposes a new solution concept for fuzzy linear bilevel programming. It then presents a fuzzy number based extended Branch-and-bound algorithm for solving fuzzy linear bilevel programming problems.

In this paper, we consider tariff-traffic equilibria in a two-level hierarchical system where the leader wants to maximize its revenues while the follower strives to minimize its activity costs. The problem is formulated as a bilevel bilinear program (BBP). We provide an algorithm based on a recent non-convex optimization approach. The (BBP) is reformulated as a d.c. program (DCP), i.e. minimizing a difference of convex functions (d.c. functions) on a polyhedral set and a descent method is developed to solve (DCP). Computational results for some numerical examples are reported.

The paper is devoted to applications of advanced tools of modern variational analysis and generalized differentiation to problems of optimistic bilevel programming. In this way, new necessary optimality conditions are derived for two major classes of bilevel programs: those with partially convex and with fully convex lower-level problems. We provide detailed discussions of the results obtained and their relationships with known results in this area.

Capacity control is a classical revenue management problem. Researchers and practitioners have proposed many mathematical models including dynamic programs for solving the capacity control problem. In this paper, we present an equivalent reformulation for the dynamic programming model for the capacity control problem in the single-leg flight setting.