Optimality conditions in multiobjective differentiable programming

Necessary conditions not requiring convexity are based on the convergence of a vector at a point and on Motzkin's theorem of the alternative. A constraint qualification is also involved in the establishment of necessary conditions. Three theorems on sufficiency require various levels of convexity on the component of the functions involved, and the equality constraints are not necessarily linear. Scalarization of the objective function is used only in the last sufficiency theorem.The author is thankful to the unknown referce whose comments improved the quality of the paper.     

Convergence rates of a global optimization algorithm

Second order conditions for the (pseudo-) convexity of a function restricted to an affine subspace are obtained by extending those already known for functions on ⁿ . These results are then used to analyse the (pseudo-) convexity of potential functions of the type introduced by Karmarkar.This research was completed while the first author was on sabbatical leave at the Département d'Informatiques et de Recherche Opérationelle, Université de Montréal, and supported by NSERC (grant Q015807). This research was also supported by NSERC (grants A8312 and A5408) and la Coopération franco-québécoise (project 20-02-13).     

Duality and existence theory for nondifferentiable programming

Necessary and sufficient conditions of optimality are given for a nonlinear nondifferentiable program, where the constraints are defined via closed convex cones and their polars. These results are then used to obtain an existence theorem for the corresponding stationary point problem, under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the programming problem. Furthermore, a dual problem is defined, and a strong duality theorem is obtained under the assumption that the constraint sets of the primal and dual problems are nonempty.     

Approximation of convex set-valued functions

Approximation of set-valued functions is introduced and discussed under a convexity assumption. In particular, a theorem on positive linear operators is given.     

B-vex functions

A class of functions, called b-vex functions, is introduced by relaxing the definition of convexity of a function. Both the differentiable and nondifferentiable cases are presented. Members of this class satisfy most of the basic properties of convex functions. This class forms a subset of the sets of both semistrictly quasiconvex as well as quasiconvex functions, but are not necessarily included in the class of preinvex functions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319. The authors are also grateful to an anonymous referee for the constructive criticism of the first version of the paper, to Dr. Collen Knickerbocker of St. Lawrence University, and to Mrs. Meena K. Bector for their help in sharpening Examples 2.1 and 2.2, respectively.     

回收函数与函数的无界性

主要利用回收锥和回收函数来研究函数的下无界性。首先, 针对凸函数在非可微条件下,利用中值定理和回收锥刻画了凸函数次微分的性质, 并在此基础上给出了基于次可微条件下回收向量的充要条件。其次,将凸性推广到E-凸, 在一定条件下,利用回收函数研究了E-凸函数的下无界性。最后,通过举例说明这些结果不能推广到拟凸条件.     

Generalized Monotonicity of Subdifferentials and Generalized Convexity

Characterizations of convexity and quasiconvexity of lower semicontinuous functions on a Banach space X are presented in terms of the contingent and Fréchet subdifferentials. They rely on a general mean-value theorem for such subdifferentials, which is valid in a class of spaces which contains the class of Asplund spaces.     

Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.      

Optimality and duality for the multiobjective fractional programming with the generalized (F,ρ) convexity

A class of multiobjective fractional programmings (MFP) are first formulated, where the involved functions are local Lipschitz and Clarke subdifferentiable. In order to deduce our main results, we give the definitions of the generalized (F,ρ) convex class about the Clarke subgradient. Under the above generalized convexity assumption, the alternative theorem is obtained, and some sufficient and necessary conditions for optimality are also given related to the properly efficient solution for the problems. Finally, we formulate the two dual problems (MD) and (MD1) corresponding to (MFP), and discuss the week, strong and reverse duality.     

Optimality Criteria and Duality in Multiobjective Programming Involving Nonsmooth Invex Functions

In this paper a generalization of invexity is considered in a general form, by means of the concept of K-directional derivative. Then in the case of nonlinear multiobjective programming problems where the functions involved are nondifferentiable, we established sufficient optimality conditions without any convexity assumption of the K-directional derivative. Then we obtained some duality results.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.     

On Duality in Nonconvex Vector Optimization in Banach Spaces Using Augmented Lagrangians

This paper shows how the use of penalty functions in terms of projections on the constraint cones, which are orthogonal in the sense of Birkhoff, permits to establish augmented Lagrangians and to define a dual problem of a given nonconvex vector optimization problem. Then the weak duality always holds. Using the quadratic growth condition together with the inf-stability or a kind of Rockafellar's stability called stability of degree two, we derive strong duality results between the properly efficient solutions of the two problems. A strict converse duality result is proved under an additional convexity assumption, which is shown to be essential.     

ρ-Invex Functions and ( F , ρ)Convex Functions: Properties and Equivalences

-invexity, -pseudo invexity and -quasi invexity (and their extentions to nondifferentiable Lipschitz functions) have been used to weaken the assumption of convexity in solving duality problems or to state sufficient optimality conditions in nonlinear programming. An attempt to generalize further these concepts has been done with the introduction of ( F , )convexity for differentiable and nondifferentiable Lipschitz functions. Theorems and results regarding both the duality problems and the sufficience of Kuhn-Tucker conditions have been reproduced for these new classes of functions. The aim of this article is to show that for both differentiable and nondifferentiable Lipschitz functions ( F , )convexity is not a generalization of -invexity, but these families of functions coincide.     

Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

Quasiconvex,pseudoconvex, and strictly pseudoconvex quadratic functions

The purpose of this paper is twofold. Firstly, criteria for quasiconvex and pseudoconvex quadratic functions in nonnegative variables of Cottle, Ferland, and Martos are derived by specializing criteria proved by the author. We do not make use of the concept of positive subdefinite matrices. Instead, we are specializing criteria that were derived for quadratic functions on arbitrary convex sets to the special case of quadratic functions in nonnegative variables. The second purpose of this paper is to present several new criteria involving also strictly pseudoconvex quadratic functions.The author wishes to thank Professor R. W. Cottle for helpful discussions.     

On optimality conditions in nonsmooth inequality constrained minimization

First order necessary optimality conditions for a minimum of an inequality constrained minimization problem are given in terms of approximate quasidifferentials, without the usual differentiability, convexity or locally Lipschitz assumptions. The main result is obtained with the help of a semi-infinite Gordan type alternative theorem. Sufficient conditions for a minimum are also given with the usual convexity assumption replaced by an invex condition.     

A Note on the Existence of Nonsmooth Nonconvex Optimization Problems

Sufficient conditions for the existence of a solution to an abstract optimization problem in Banach spaces are given, which do not rely on convexity, regularity properties or a straightforward coerciveness assumption. Applications to sparsity-constrained optimization and to problems from mechanics are provided.     

Generalized asymptotic functions in nonconvex multiobjective optimization problems

In this paper, we use generalized asymptotic functions and second-order asymptotic cones to develop a general existence result for the nonemptiness of the proper efficient solution set and a sufficient condition for the domination property in nonconvex multiobjective optimization problems. A new necessary condition for a point to be efficient or weakly efficient solution is given without any convexity assumption. We also provide a finer outer estimate for the asymptotic cone of the weakly efficient solution set in the quasiconvex case. Finally, we apply our results to the linear fractional multiobjective optimization problem.     

Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems

The aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics. In the first part we characterize the value function for the Mayer problem and the supremum cost problem using viscosity tools and the notion of ε-viability (near viability). These value functions are given with respect to discontinuous cost functionals. In the second part we obtain results describing the ε-viability (near viability) of singularly perturbed control systems.     

二元凸函数的判别条件

给出了二元凸函数的定义,导出了二元凸函数的判别条件,该判别条件由二元函数的二阶导数给出.用二元凸函数的判别条件和半正定的(半负定)矩阵的性质,得到了二元二次多项式凸性的简单判别形式.