Local search in a quadratic-linear bilevel programming problem

A quadratic-linear bilevel programming problem is considered. Its optimistic statement is reduced to a nonconvex mathematical programming problem with a quadratic-bilinear structure. An approximate algorithm of a local search in the problem obtained is proposed, proved, and tested.     

Optimality conditions for pessimistic bilevel problems using convexificator

Positivity - The purpose of this paper is to study the pessimistic version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for...     

Test problem construction for linear bilevel programming problems

A method of constructing test problems for linear bilevel programming problems is presented. The method selects a vertex of the feasible region, far away from the solution of the relaxed linear programming problem, as the global solution of the bilevel problem. A predetermined number of constraints are systematically selected to be assigned to the lower problem. The proposed method requires only local vertex search and solutions to linear programs.     

An equivalent one level optimization problem to a semivectorial bilevel problem

In this paper, we are concerned with a bilevel optimization problem \(P_{0}\), where the lower level problem is a vector optimization problem. First, we give an equivalent one level optimization problem for which the nonsmooth Mangasarian–Fromowitz constraint qualification can hold at feasible solution. Using a special scalarization function, one deduces necessary optimality condition for the initial problem.     

On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem

This paper is mainly concerned with the classical KKT reformulation and the primal KKT reformulation (also known as an optimization problem with generalized equation constraint (OPEC)) of the optimistic bilevel optimization problem. A generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem. M- and S-type stationarity conditions tailored for the problem are derived as well. Considering the close link between the aforementioned reformulations, similarities and relationships between the corresponding CQs and optimality conditions are highlighted. In this paper, a concept of partial calmness known for the optimal value reformulation is also introduced for the primal KKT reformulation and used to recover the M-stationarity conditions.     

线性半向量二层规划问题的全局优化方法

研究了线性半向量二层规划问题的全局优化方法. 利用下层问题的对偶间隙构造了线性半向量二层规划问题的罚问题, 通过分析原问题的最优解与罚问题可行域顶点之间的关系, 将线性半向量二层规划问题转化为有限个线性规划问题, 从而得到线性半向量二层规划问题的全局最优解. 数值结果表明所设计的全局优化方法对线性半向量二层规划问题是可行的.     

Multiobjective bilevel optimization

In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.     

Two-level value function approach to non-smooth optimistic and pessimistic bilevel programs

ABSTRACT

The authors' paper in Dempe et al. [Necessary optimality conditions in pessimistic bilevel programming. Optimization. 2014;63:505–533], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analysed in Dempe et al. [Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22 (2012), 1309–1343], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve functions that are non-differentiable at some points of their domain, the main goal of the current paper is to extend the two-level value function approach by deriving new necessary optimality conditions for both optimistic and pessimistic versions in bilevel programming with non-smooth data.     

A bilevel optimization model for the newsvendor problem with the focus theory of choice

4OR - The newsvendor problem is formulated with the focus theory of choice in which instead of using the expected utility in the existing approaches, the optimal order quantity is determined as per...     

Necessary optimality conditions for a semivectorial bilevel optimization problem using the kth-objective weighted-constraint approach

Positivity - In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. https://doi.org/10.1007/s11117-019-00685-1 ) is erroneous. Using...     

New optimality conditions and a scalarization approach for a nonconvex semi-vectorial bilevel optimization problem

In this paper, we are concerned with the optimistic formulation of a semivectorial bilevel optimization problem. Introducing a new scalarization technique for multiobjective programs, we transform our problem into a scalar-objective optimization problem by means of the optimal value reformulation and establish its theoretical properties. Detailed necessary conditions, to characterize local optimal solutions of the problem, were then provided, while using the weak basic CQ together with the generalized differentiation calculus of Mordukhovich. Our approach is applicable to nonconvex problems and is different from the classical scalarization techniques previously used in the literature and the conditions obtained are new.

     

Double penalty method for bilevel optimization problems

A penalty function method approach for solving a constrained bilevel optimization problem is proposed. In the algorithm, both the upper level and the lower level problems are approximated by minimization problems of augmented objective functions. A convergence theorem is presented. The method is applicable to the non-singleton lower-level reaction set case. Constraint qualifications which imply the assumptions of the general convergence theorem are given.A part of this paper was presented in a talk at the 11th Symposium on Mathematical Programming with Data Perturbations, Washington, DC, May 1989.     

On generalized semi-infinite optimization and bilevel optimization

The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems.     

Necessary optimality conditions of a D.C. set-valued bilevel optimization problem

In this article, we consider a bilevel vector optimization problem where objective and constraints are set valued maps. Our approach consists of using a support function [1–3,14,15,32] together with the convex separation principle for the study of necessary optimality conditions for D.C. bilevel set-valued optimization problems. We give optimality conditions in terms of the strong subdifferential of a cone-convex set-valued mapping introduced by Baier and Jahn 6 Baier, J and Jahn, J. 1999. On subdifferentials of set-valued maps. J. Optim. Theory Appl., 100: 233–240. [Crossref], [Web of Science ®] , [Google Scholar] and the weak subdifferential of a cone-convex set-valued mapping of Sawaragi and Tanino 28 Sawaragi, Y and Tanino, T. 1980. Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl., 31: 473–499.  [Google Scholar]. The bilevel set-valued problem is transformed into a one level set-valued optimization problem using a transformation originated by Ye and Zhu 34 Ye, JJ and Zhu, DL. 1995. Optimality conditions for bilevel programming problems. Optimization, 33: 9–27. [Taylor & Francis Online] , [Google Scholar]. An example illustrating the usefulness of our result is also given.     

An overview of bilevel optimization

This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints). B. Colson now at SAMTECH s.a., Liège, Belgium. This article is an updated version of a paper that appeared in 4OR 3, 87–107, 2005.     

Towards global bilevel dynamic optimization

The global solution of bilevel dynamic optimization problems is discussed. An overview of a deterministic algorithm for bilevel programs with nonconvex functions participating is given, followed by a summary of deterministic algorithms for the global solution of optimization problems with nonlinear ordinary differential equations embedded. Improved formulations for scenario-integrated optimization are proposed as bilevel dynamic optimization problems. Solution procedures for some of the problems are given, while for others open challenges are discussed. Illustrative examples are given.     

A new equivalent single-level problem for bilevel problems

In this article, we give necessary optimality conditions for a bilevel optimization problem (P). An intermediate single-level problem (Q), which is equivalent to the bilevel optimization problem (P), has been introduced.     

A smoothing heuristic for a bilevel pricing problem

In this paper, we provide a heuristic procedure, that performs well from a global optimality point of view, for an important and difficult class of bilevel programs. The algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques. Although the algorithm cannot guarantee global optimality, very good solutions can be obtained through the use of a suitable set of parameters. The algorithm has been tested on large-scale instances of a network pricing problem, an application that fits our modeling framework. Preliminary results show that on hard instances, our approach constitutes an alternative to solvers based on mixed 0–1 programming formulations.     

Discrete linear bilevel programming problem

In this paper, we analyze some properties of the discrete linear bilevel program for different discretizations of the set of variables. We study the geometry of the feasible set and discuss the existence of an optimal solution. We also establish equivalences between different classes of discrete linear bilevel programs and particular linear multilevel programming problems. These equivalences are based on concave penalty functions and can be used to design penalty function methods for the solution of discrete linear bilevel programs.Support of this work has been provided by the INIC (Portugal) under Contract 89/EXA/5, by INVOTAN, FLAD, and CCLA (Portugal), and by FCAR (Québec), NSERC, and DND-ARP (Canada).     

Necessary optimality conditions for bilevel set optimization problems

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.