Class Of Bilevel Programming Problems Research Articles

Proximal Gradient-Type Algorithms for a Class of Bilevel Programming Problems

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.

A Multi-Parametric optimization approach for bilevel mixed-integer linear and quadratic programming problems

Optimization problems involving two decision makers at two different decision levels are referred to as bi-level programming problems. In this work, we present novel algorithms for the exact and global solution of two classes of bi-level programming problems, namely (i) bi-level mixed-integer linear programming problems (B-MILP) and (ii) bi-level mixed-integer convex quadratic programming problems (B-MIQP) containing both integer and bounded continuous variables at both optimization levels. Based on multi-parametric programming theory, the main idea is to recast the lower level problem as a multi-parametric programming problem, in which the optimization variables of the upper level problem are considered as bounded parameters for the lower level. The resulting exact multi-parametric mixed-integer linear or quadratic solutions are then substituted into the upper level problem, which can be solved as a set of single-level, independent, deterministic mixed-integer optimization problems. Extensions to problems including right-hand-side uncertainty on both lower and upper levels are also discussed. Finally, computational implementation and studies are presented through test problems.

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.

Evolutionary technique based goal programming approach to chance constrained interval valued bilevel programming problems

The real world multiobjective decision environment involves great complexity and uncertainity. Many decision making problems often need to be modelled as a class of bilevel programming problems with inexact coefficients and chance constraints. To deal with these problems, a genetic algorithm (GA) based goal programming (GP) procedure for solving interval valued bilevel programming (BLP) problems in a large hierarchical decision making and planning organization is proposed. In the model formulation of the problem the chance constraints are converted to their deterministic equivalent using the notion of mean and variance. Further, the individual best and least solutions of the objectives of the decision makers (DMs) located at different hierarchical levels are determined by using GA method. The target intervals for achievement of each of the objectives as well as the target interval of the decision vector controlled by the upper-level DM are defined. Then, using interval arithmetic technique the interval valued objectives and control vectors are transformed into the conventional form of goal by introducing under- and over-deviational variables to each of them. In the solution process, both the aspects of minsum and minmax GP formulations are adopted to minimize the lower bounds of the regret intervals for goal achievement within the specified interval from the optimistic point of view. The potential use of the approach is illustrated by a numerical example

一类带区间系数的分式双层规划问题的遗传算法

针对上层为区间系数分式规划、下层为线性规划的一类双层规划问题，提出了一种基于四个适应度评估函数的遗传算法。首先，利用上层系数区间的上下端点将原问题转化成四个系数确定的分式双层规划问题；其次，利用四个确定问题的特征和线性规划的最优性条件设计了一个基于四个目标函数评估的遗传算法，通过该算法获得原问题的最好最优解和最差最优解。最后，数值仿真结果表明，该算法是可行有效的。 For a class of bilevel programming problems, in which the upper-level problem is an interval coef-ficients fractional program, whereas the lower-level problem is linear, a genetic algorithm based on four fitness functions is presented. Firstly, four certain programs can be gotten by taking up-per-lower bounds of the coefficient intervals of the upper level objective. In addition, using the characteristics of the four problems and the optimality conditions of linear programming, a genetic algorithm which takes four objective functions as evaluation is designed, and the best and the worst optimal solutions can be obtained by using the proposed algorithm. Finally, the simulation results show that the proposed algorithm is feasible and efficient.

A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection

In this paper, we consider a class of bilevel programming problems where the upper objective function is convex quadratic while the lower objective function and the constraints are linear. The problem is first rewritten as minimizing a convex quadratic function subject to linear constraints and one concave constraint. Then, we use an exact penalty technique to reformulate the problem as a DC program. Afterward, DCA, an efficient algorithm in nonconvex programming, is developed to solve the resulting problem. &nbsp &nbsp For globally solving the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). DCA is applied to compute upper bounds, while lower bounds are calculated via a DC relaxation of the DC constraint. The proposed algorithms, DCA and BB-DCA, are compared with the Branch-and-Bound algorithm without DCA (BB) on random data. The numerical results show that the proposed algorithms are efficient. &nbsp &nbsp Finally, we consider an application of portfolio selection problems with the historical data related to the prices in the market of France, Luxembourg, United States and England during the period 2000-2007. The experimental results confirm the efficiency and the rapidity of DCA.

Numerical solution of a class of bilevel programming problems

A quadratic-linear bilevel programming problem is considered. Its optimistic statement is reduced to a series of nonconvex unilevel problems. An approximate algorithm for global search in reduced problems is proposed. Numerical solutions of randomly generated test problems are given and analyzed.

Solving bilevel programming problems with multicriteria optimization techniques

We introduce in this paper a new relation between bilevel programming and multicriteria optimization. We show that solving a certain class of bilevel programming problem can be equivalent to solve two independents multicriteria optimization problems. The optimal solutions of the bilevel problem are then the Pareto optimal points corresponding to the nondominated points belonging to the intersection of the two efficient sets. Comment on the practical implementation of the obtained relation is discussed. A generalisation of the relation between bilevel programming optimization and multicriteria optimization, first presented by Fulop [1] is also discussed in the paper.

Parametric global optimisation for bilevel programming

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.

A Tabu Search Based Approach for Solving a Class of Bilevel Programming Problems in Chemical Engineering

In this paper an approach based on the tabu search paradigm to tackle the bilevel programming problems is presented. The algorithm has been tested for a number of benchmark problems and the results obtained show superiority of the approach over the conventional methods in solving such problems.

Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints

In this paper we study optimization problems with variational inequality constraints in finite dimensional spaces. Kuhn-Tucker type necessary optimality conditions involving coderivatives are given under certain constraint qualifications including one that ensures nonexistence of non-trivial abnormal multipliers. The result is applied to bilevel programming problems to obtain Kuhn-Tucker type necessary optimality conditions. The Kuhn-Tucker type necessary optimality conditions are shown to be satisfied without any constraint qualification by the class of bilevel programming problems where the lower level is a parametric linear quadratic problem.

On a new class of bilevel programming problems and its use for reformulating mixed integer problems

We extend some known results about the Bilevel Linear Problem (BLP), a hierarchical two-stage optimization problem, showing how it can be used to reformulate any Mixed Integer (Linear) Problem; then, we introduce some new concepts, which might be useful to fasten almost all the known algorithms devised for BLP. As this kind of reformulation appears to be somewhat artificial, we define a natural generalization of BLP, the Bilevel Linear/Quadratic Problem (BL/QP), and show that most of the exact and/or approximate algorithms originally devised for the BLP, such as GSA or K-th Best, can be extended to this new class of Bilevel Programming Problems. For BL/QP, more ‘natural’ reformulations of MIPs are available, leading to the use of known (nonexact) algorithms for BLP as (heuristic) approaches to MIPs: we report some contrasting results obtained in the Network Design Problem case, showing that, although the direct application of our (Dual) GSA algorithm is not of any practical use, we obtain as a by-product a good theoretical characterization of the optimal solutions set of the NDP, along with a powerful scheme for constructing fast algorithms for the Minimum Cost Flow Problem with piecewise convex linear cost functions.