Second-order characterizations of pseudoconvex quadratic functions

The paper deals with strictly pseudoconvex quadratic functions on open convex sets. It presents second-order conditions in terms of an extended Hessian and in terms of bordered determinants. All of the conditions are both necessary and sufficient.The author wishes to thank Professor M. Avriel, visiting Stanford University, for helpful discussions. He is also grateful to Professor R. W. Cottle, Stanford University, for his suggestions.     

Second order characterizations of pseudoconvex functions

Second order characterizations for (strictly) pseudoconvex functions are derived in terms of extended Hessians and bordered determinants. Additional results are presented for quadratic functions.     

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.     

The complex Monge–Ampère equation with infinite boundary value

In this article we consider the complex Monge–Ampère equation with infinite boundary value in bounded pseudoconvex domains. We prove the existence of strictly plurisubharmonic solution to the problem in convex domains under suitable growth conditions. We also obtain, for general pseudoconvex domains, some nonexistence results which show that these growth conditions are nearly optimal.     

Local patching for convex sets in Cn

We consider open sets Ω₁ and Ω₂ in ? _n so that $\bar \Omega _1 \cap \bar \Omega _2 = \{ 0\} $ . We investigate the existence of a basis of pseudoconvex neighborhoods for $\bar \Omega _1 \cup \bar \Omega _2 $ and of local (pseudoconvex or polynomially convex) “patching.” Positive results are given for strictly convex sets, and counterexamples are given for strictly pseudoconvex sets (even convex ones).     

On Some Special Pseudoconvex Spaces

We present two ways of constructing pseudoconvex spaces. The first theorem states that an interval space is pseudoconvex if the intervals are range sets of continuous functions and a cone erected to a convex set is convex. Secondly, if H is pseudoconvex and in the Cartesian product H × Rthe intervals are defined by going first along the H-segment, then along the R-segment (the other coordinates fixed) then H × Ris pseudoconvex.     

COHOMOLOGY VANISHING IN HILBERT SPACES

§1. Introduction Oka [40] proved that any additive Cousin problem is solvable in a domain of holomorphyin Cn. Oka [41]-Cartan [2, 3]-Serre [46] generalized this as the theorem B for analyticcoherent sheaves over Stein spaces. J. P. Serre [47] proved…     

Uniqueness of solutions of aH ∞ optimization problem and complex geometric convexity

We study uniqueness of an H^∞ optimization problem which is central to the worst case frequency domain system design. It was known that if the so-called sublevel sets are strictly convex inC ^N, then the uniqueness holds. On the other hand, there are examples of non-uniqueness if the sublevel sets are just strictly pseudoconvex. In this paper we prove that uniqueness holds for a type of convexity which is strictly in-between geometric and pseudoconvexity.     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

Optimality Conditions for Nonsmooth Equilibrium Problems via Hadamard Directional Derivative

In this paper, we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. We obtain necessary conditions of Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) types for a nonsmooth (MPEC) problem in terms of the lower Hadamard directional derivative. In particular sufficient conditions for MPECs are given where the involved functions have pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets is a class of generalized convex functions that include quasiconvex functions.     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

Invex-convexlike functions and duality

We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

集合函数多目标规划的一阶最优性条件

在文（１）－（４）的基础上，本文通过引入集团函数的伪凸，严格伪凸，拟凸，严格拟凸等新概念，给出了集合函数多目标规划问题有效解的一阶充分条件，弱有效解的阶必要条件，弱有交解的一阶必要条件以及强有效解的一阶充分条件。     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Generalized convex functions and generalized differentials

We study some classes of generalized convex functions, using a generalized differential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdifferential or a pseudo-differential in the sense of Jeyakumar and Luc. Such a general framework allows us to avoid technical assumptions related to specific constructions. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced. We devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange–Kuhn–Tucker multipliers for mathematical programming problems.     

非光滑多目标广义本性凸规划的最优性条件与对偶

In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective programming and its saddle point theorem about cone efficient solution. We set up Mond-Weir type duality and Craven type duality for nonsmooth multiobjective programming with generalized essentially convex functions, and prove them.     

Note on generalized convex functions

In this note, an important class of generalized convex functions, called invex functions, is defined under a general framework, and some properties of the functions in this class are derived. It is also shown that a function is (generalized) pseudoconvex if and only if it is quasiconvex and invex.     

Sufficient optimality criteria in nonlinear programming for generalized equality-inequality constraints

Sufficient optimality criteria of the Kuhn-Tucker and Fritz John type in nonlinear programming are established in the presence of equality-inequality constraints. The constraint functions are assumed to be quasiconvex, and the objective function is taken to be pseudoconvex (or convex).