Necessary and sufficient optimality conditions for fractional multi-objective problems

Using a special scalarization, we give necessary optimality conditions for fractional multiobjective optimization problems. Under a generalized invexity, sufficient optimality conditions are also given. All over the article, the data are assumed to be continuous but not necessarily Lipschitz.     

Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.     

弧连通锥-凸数学规划的最优性条件

在弧连通锥-凸假设下讨论Hausdorff局部凸空间中的一类数学规划的最优性条件问题.首先,利用择一定理得到了锥约束标量优化问题的一个必要最优性条件.其次,利用凸集分离定理证明了无约束向量优化问题关于弱极小元的标量化定理和一个一致的充分必要条件.所得结果深化和丰富了最优化理论及其应用的内容.     

New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization

We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are studied. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), under projects MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010), and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for valuable comments and suggestions.     

Efficient and weakly efficient solutions for the vector topical optimization

In this paper, we consider some scalarization functions, which consist of the generalized min-type function, the so-called plus-Minkowski function and their convex combinations. We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space. Then, we establish some versions of Farkas type results for the infinite inequality system involving vector topical functions. As applications, we obtain the necessary and sufficient conditions of efficient solutions and weakly efficient solutions for a vector topical optimization problem, respectively.     

Global optimality conditions and optimization methods for quadratic integer programming problems

In this paper, we first establish some sufficient and some necessary global optimality conditions for quadratic integer programming problems. Then we present a new local optimization method for quadratic integer programming problems according to its necessary global optimality conditions. A new global optimization method is proposed by combining its sufficient global optimality conditions, local optimization method and an auxiliary function. The numerical examples are also presented to show that the proposed optimization methods for quadratic integer programming problems are very efficient and stable.     

Optimality conditions for minimizing the difference of nonconvex vector-valued mappings

In this paper, we consider vector optimization problems involving the difference of nonconvex vector-valued mappings. By a nonconvex scalarization function, we establish necessary optimality conditions in terms of the Mordukhovich subdifferential, strong subdifferential and Ioffe subdifferential without any convexity assumption. As an application, we discuss the optimality condition on a nonconvex multiobjective fractional programming problem.     

Global Optimality Conditions for Some Classes of Optimization Problems

We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity.     

Scalarization of constraints system in some vector optimization problems and applications

In this work we employ a new method to penalize a constrained non solid vector optimization problem by means of a scalarization functional applied to the constraints system. Then, we formulate optimality conditions which mainly use several types of regularity for single and set-valued maps. In order to motivate our demarche, we discuss in detail the assumptions used in the main results and we show how it can be verified.     

When the Karush–Kuhn–Tucker Theorem Fails: Constraint Qualifications and Higher-Order Optimality Conditions for Degenerate Optimization Problems

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush–Kuhn–Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush–Kuhn–Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush–Kuhn–Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality.     

Generalized penalization and maximization of vectorial nonsmooth functions

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study.     

New Results on Second-Order Optimality Conditions in Vector Optimization Problems

In this paper, we study second-order optimality conditions for multiobjective optimization problems. By means of different second-order tangent sets, various new second-order necessary optimality conditions are obtained in both scalar and vector optimization. As special cases, we obtain several results found in the literature (see reference list). We present also second-order sufficient optimality conditions so that there is only a very small gap with the necessary optimality conditions. The authors thank Professor P.L. Yu and the referees for valuable comments and helpful suggestions.     

带变动序结构的向量优化问题的E-最优元

推广固定锥序下的改进集概念到变动序关系.引入了带变动序结构的向量优化问题的E-最优元.应用Tammer-Weidner意义下的非线性标量化函数,给出了向量优化问题E-最优元的标量化刻画,建立了带变动序结构的向量优化问题的E-最优元的必要和充分最优性条件.     

Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.     

Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma

In this paper, we present Lagrange multiplier necessary conditions for global optimality that apply to non-convex optimization problems beyond quadratic optimization problems subject to a single quadratic constraint. In particular, we show that our optimality conditions apply to problems where the objective function is the difference of quadratic and convex functions over a quadratic constraint, and to certain class of fractional programming problems. Our necessary conditions become necessary and sufficient conditions for global optimality for quadratic minimization subject to quadratic constraint. As an application, we also obtain global optimality conditions for a class of trust-region problems. Our approach makes use of outer-estimators, and the powerful S-lemma which has played key role in control theory and semidefinite optimization. We discuss numerical examples to illustrate the significance of our optimality conditions. The authors are grateful to the referees for their useful comments which have contributed to the final preparation of the paper.     

The Ekeland variational principle for set-valued maps involving coderivatives

In this paper we use the Fréchet, Clarke, and Mordukhovich coderivatives to obtain variants of the Ekeland variational principle for a set-valued map F and establish optimality conditions for set-valued optimization problems. Our technique is based on scalarization with the help of a marginal function associated with F and estimates of subdifferentials of this function in terms of coderivatives of F.     

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.

     

Boundary control of non-well-posed elliptic equations with nonlinear Neumann boundary conditions

In this paper, we consider a boundary control problem governed by a class of non-well-posed elliptic equations with nonlinear Neumann boundary conditions. First, the existence of optimal pairs is proved. Then by considering a well-posed penalization problem and taking limit in the optimality system for penalization problem, we obtain the necessary optimality conditions for optimal pairs of initial control problem.     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

Vectorial penalization for generalized functional constrained problems

In this paper we use a double penalization procedure in order to reduce a set-valued optimization problem with functional constraints to an unconstrained one. The penalization results are given in several cases: for weak and strong solutions, in global and local settings, and considering two kinds of epigraphical mappings of the set-valued map that defines the constraints. Then necessary and sufficient conditions are obtained separately in terms of Bouligand derivatives of the objective and constraint mappings.