Generating equidistant representations in biobjective programming

In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective optimization problems. This paper presents two methods for generating discrete representations with equidistant points for biobjective problems with solution sets determined by convex, polyhedral cones. The Constraint Controlled-Spacing method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. The Bilevel Controlled-Spacing method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing, and is extended to multiobjective problems. Both methods are proven to produce (weakly) nondominated points and are demonstrated on a variety of test problems.     

Solving Ill-posed Bilevel Programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.     

二层多目标最优化问题解集的连通性

研究了二层多目标最优化模型(BLMOP)解集的连通性问题,其中(BLMOP)的上层集值目标函数由下层问题的有效点确定.把(BLMOP)看作成单层的集值函数优化问题,借助集值函数优化问题各种有效解集的连通性的结论,得到了(BLMOP)相应的有效解集连通性的结论.     

Optimality Conditions for Bilevel Vector Optimization Problems with a Variable Ordering Structure

The article is concerned with the optimistic formulation of a multiobjective bilevel optimization problem with locally Lipschitz continuous inclusion constraints. Using a variable ordering structure defined by a Bishop–Phelps cone, we investigate necessary optimality conditions for locally weakly nondominated solutions. Reducing the problem into a one-level nonlinear and nonsmooth program, we use the extremal principle by Mordukhovich to get fuzzy optimality conditions. More explicit conditions with the initial data are obtained using both the Ekeland’s variational principle and the support function. Fortunately, the Lipschitz property of a set-valued mapping is conserved for its support function. An appropriate regularity condition is given to help us discern the Lagrange-Kuhn-Tucker multipliers.     

Is bilevel programming a special case of a mathematical program with complementarity constraints?

Bilevel programming problems are often reformulated using the Karush–Kuhn–Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is “No” even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater’s constraint qualification. We also show by examples that this correspondence can fail if the Slater’s constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.     

Bilevel Decision with Generalized Semi-infinite Optimization for Fuzzy Mappings as Lower Level Problems

In this paper, we introduce the bilevel decision problems with parametric generalized semi-infinite optimization for fuzzy mappings as the lower-level problem, and its corresponding MPEC problems. For these problems, we establish two models which are different in the feasible region setting of lower-level problems. Some new existence results are obtained in rather weak conditions.     

Semivectorial Bilevel Optimization Problem: Penalty Approach

We consider a bilevel optimization problem where the upper level is a scalar optimization problem and the lower level is a vector optimization problem. For the lower level, we deal with weakly efficient solutions. We approach our problem using a suitable penalty function which vanishes over the weakly efficient solutions of the lower-level vector optimization problem and which is nonnegative over its feasible set. Then, we use an exterior penalty method.Communicated by H. P. Benson(Formerly Serban Bolintinéanu) Professor, University of New Caledonia, ERIM, Nouméa, New Caledonia. This author thanks the University of Naples Federico II for its support and the Department of Mathematics and Statistics for its hospitality.     

New Optimality Conditions for the Semivectorial Bilevel Optimization Problem

The paper is concerned with the optimistic formulation of a bilevel optimization problem with multiobjective lower-level problem. Considering the scalarization approach for the multiobjective program, we transform our problem into a scalar-objective optimization problem with inequality constraints by means of the well-known optimal value reformulation. Completely detailed first-order necessary optimality conditions are then derived in the smooth and nonsmooth settings while using the generalized differentiation calculus of Mordukhovich. Our approach is different from the one previously used in the literature and the conditions obtained are new. Furthermore, they reduce to those of a usual bilevel program, if the lower-level objective function becomes single-valued.     

基于粒子群算法的非线性二层规划问题的求解算法

粒子群算法（Particle Swarm Optimization，PSO）是一种新兴的优化技术，其思想来源于人工生命和演化计算理论。PSO通过粒子追随自己找到的最好解和整个群的最好解来完成优化。该算法简单易实现，可调参数少，已得到了广泛研究和应用。本文根据该算法能够有效的求出非凸数学规划全局最优解的特点，对非线性二层规划的上下层问题求解，并根据二层规划的特点，给出了求解非线性二层规划问题全局最优解的有效算法。数值计算结果表明该算法有效。     

Optimality conditions for vector optimization problems with variable ordering structures

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319–377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop–Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.     

求解弱线性双层规划问题的一种全局优化方法

双层规划在经济、交通、生态、工程等领域有着广泛而重要的应用.目前对双层规划的研究主要是基于强双层规划和弱双层规划.然而,针对弱双层规划的求解方法却鲜有研究.研究求解弱线性双层规划问题的一种全局优化方法,首先给出弱线性双层规划问题与其松弛问题在最优解上的关系,然后利用线性规划的对偶理论和罚函数方法,讨论该松弛问题和它的罚问题之间的关系.进一步设计了一种求解弱线性双层规划问题的全局优化方法,该方法的优势在于它仅仅需要求解若干个线性规划问题就可以获得原问题的全局最优解.最后,用一个简单算例说明了所提出的方法是可行的.     

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.     

On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification

This paper focuses on bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. We relax the constrained domain of the lower-level problem. Then, an approximate solution of the original bilevel program can be obtained by solving this perturbed bilevel program. As the lower-level problem of the perturbed bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a mathematical program with complementarity constraints which can be solved by standard algorithms. The lower convergence properties of the constraint set mapping and the solution set mapping of the lower-level problem of the perturbed bilevel program are expanded. We show that the solutions of a sequence of the perturbed bilevel programs are convergent to that of the original bilevel program under some appropriate conditions. And this convergence result is applied to simple trilevel programs.     

Approximate proximal methods in vector optimization

This paper studies the vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Y with a nonempty interior. The proximal method in vector optimization is extended to develop an approximate proximal method for this problem by virtue of the approximate proximal point method for finding a root of a maximal monotone operator. In this approximate proximal method, the subproblems consist of finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version, in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established. In addition, we also discuss an extension to Bregman-function-based proximal algorithms for finding weakly efficient points for mappings.     

Convergence Analysis of a Class of Penalty Methods for Vector Optimization Problems with Cone Constraints

In this paper, we consider convergence properties of a class of penalization methods for a general vector optimization problem with cone constraints in infinite dimensional spaces. Under certain assumptions, we show that any efficient point of the cone constrained vector optimization problem can be approached by a sequence of efficient points of the penalty problems. We also show, on the other hand, that any limit point of a sequence of approximate efficient solutions to the penalty problems is a weekly efficient solution of the original cone constrained vector optimization problem. Finally, when the constrained space is of finite dimension, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original cone constrained vector optimization problem if Mangasarian–Fromovitz constraint qualification holds at the limit point.This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

Duality theory in multiobjective programming

In this paper, a multiobjective programming problem is considered as that of finding the set of all nondominated solutions with respect to the given domination cone. Two point-to-set maps, the primal map and the dual map, and the vector-valued Lagrangian function are defined, corresponding to the case of a scalar optimization problem. The Lagrange multiplier theorem, the saddle-point theorem, and the duality theorem are derived by using the properties of these maps under adequate convexity assumptions and regularity conditions.     

Nonlinear robust optimization via sequential convex bilevel programming

In this paper, we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min–max problems which arise in the context of semi-infinite programming. Here, our main motivation are nonlinear inequality constrained robust optimization problems. In the first part of the paper, we propose a conservative approximation strategy for such nonlinear and non-convex robust optimization problems: under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given, we show how to formulate a lower-level concave min–max problem which approximates the robust counterpart in a conservative way. This approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties. In the second part of the paper, we review existing theory on optimality conditions for nonlinear lower-level concave min–max problems which arise in the context of semi-infinite programming. Regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem, we end up with a structured mathematical program with complementarity constraints (MPCC). The special hierarchical structure of this MPCC can be exploited in a novel sequential convex bilevel programming algorithm. We discuss the surprisingly strong global and locally quadratic convergence properties of this method, which can in this form neither be obtained with existing SQP methods nor with interior point relaxation techniques for general MPCCs. Finally, we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example.     

On a class of bilevel linear mixed-integer programs in adversarial settings

We consider a class of bilevel linear mixed-integer programs (BMIPs), where the follower’s optimization problem is a linear program. A typical assumption in the literature for BMIPs is that the follower responds to the leader optimally, i.e., the lower-level problem is solved to optimality for a given leader’s decision. However, this assumption may be violated in adversarial settings, where the follower may be willing to give up a portion of his/her optimal objective function value, and thus select a suboptimal solution, in order to inflict more damage to the leader. To handle such adversarial settings we consider a modeling approach referred to as \(\alpha \)-pessimistic BMIPs. The proposed method naturally encompasses as its special classes pessimistic BMIPs and max–min (or min–max) problems. Furthermore, we extend this new modeling approach by considering strong-weak bilevel programs, where the leader is not certain if the follower is collaborative or adversarial, and thus attempts to make a decision by taking into account both cases via a convex combination of the corresponding objective function values. We study basic properties of the proposed models and provide numerical examples with a class of the defender–attacker problems to illustrate the derived results. We also consider some related computational complexity issues, in particular, with respect to optimistic and pessimistic bilevel linear programs.     

Optimality conditions for mixed discrete bilevel optimization problems

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.     

Computation of the optimal tolls on the traffic network

The present paper is devoted to the computation of optimal tolls on a traffic network that is described as fuzzy bilevel optimization problem. As a fuzzy bilevel optimization problem we consider bilinear optimization problem with crisp upper level and fuzzy lower level. An effective algorithm for computation optimal tolls for the upper level decision-maker is developed under assumption that the lower level decision-maker chooses the optimal solution as well. The algorithm is based on the membership function approach. This algorithm provides us with a global optimal solution of the fuzzy bilevel optimization problem.