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Characterization of Radially Lower Semicontinuous pseudoconvex functions

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2020年第2期184卷 368-383页

作者： Ivanov, Vsevolod I. Tech Univ Varna Dept Math Varna Bulgaria

In this paper, we prove that a radially lower semicontinuous function of one variable, defined on some interval, is pseudoconvex, if and only if its domain of definition can be split into three parts such that the function is strictly monotone decreasing without stationary points over the first subinterval, it is constant over the second one, and it is strictly monotone increasing without stationary points over the third subinterval. Each one or two of these parts may be empty or degenerate into a single point. The proof of this property is easy, when the function is differentiable. We consider functions, which are pseudoconvex with respect to the lower Dini directional derivative. This result follows from some known claims, but our theorem is a shot, directed to the target. We apply this characterization to obtain a complete characterization of strictly pseudoconvex functions. We also derive the respective results, when the function is radially lower semicontinuous in a real linear space. Several applications of the characterization are provided. A result due to Diewert, Avriel and Zang is extended to radially continuous functions.

关键词： pseudoconvex function Nonsmooth analysis Nonsmooth optimization Lower Dini directional derivative

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functionS WHICH ARE QUASICONVEX UNDER LINEAR PERTURBATIONS

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SIAM JOURNAL ON OPTIMIZATION 2012年第3期22卷 1089-1108页

作者： Barron, E. N. Goebel, R. Jensen, R. R. Loyola Univ Dept Math & Stat Chicago IL 60660 USA

A quasiconvex function is a function which has convex sublevel sets. This paper studies robustly quasiconvex functions, that is, quasiconvex functions which remain quasiconvex under small linear perturbations. Relations to pseudoconvexity and other generalized convexity concepts and necessary and sufficient first-order conditions for robust quasiconvexity of smooth functions are presented. Convex-analytic properties and convexification of robustly quasiconvex functions are studied. Supporting robustly quasiconvex functions by simpler functions is discussed, with duality theory as motivation. Through the use of a second-order condition for robust quasiconvexity of nonsmooth functions, nontrivial examples of such functions are given.

关键词： quasiconvex function convex set linear perturbation pseudoconvex function generalization of convexity duality

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Boundary control of PDEs via curvature flows:: The view from the boundary, II

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APPLIED MATHEMATICS AND OPTIMIZATION 2002年第2-3期46卷 167-178页

作者： Betelú, S Gulliver, R Littman, W Univ Minnesota Sch Math Minneapolis MN 55455 USA Univ Minnesota Inst Math & Applicat Minneapolis MN 55455 USA

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.

关键词： boundary control wave equation Riemannian manifold curvature flow pseudoconvex function

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On the Nonemptiness and Boundedness of Solution Sets of Weakly Homogeneous Optimization Problems

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第3期30卷 1105-1116页

作者： Vu Trung Hieu Sorbonne Univ LIP6 Paris France

In this paper, we introduce a new class of optimization problems whose objective functions are weakly homogeneous relative to the constraint sets. By using the normalization argument in asymptotic analysis, we prove two criteria for the nonemptiness and boundedness of the solution set of a weakly homogeneous optimization problem. Moreover, we discuss the existence and stability of the solution sets of linearly parametric optimization problems. Several examples are given to illustrate the derived results.

关键词： Weakly homogeneous optimization problem Asymptotic cone pseudoconvex function Nonemptiness Boundedness Upper semicontinuity

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Neural network for nonsmooth pseudoconvex optimization with general constraints

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NEUROCOMPUTING 2014年 131卷 336-347页

作者： Li, Qingfa Liu, Yaqiu Zhu, Liangkuan Northeast Forestry Univ Sch Informat & Comp Engn Harbin 150040 Peoples R China

In this paper, a one-layer recurrent projection neural network is proposed for solving pseudoconvex optimization problems with general convex constraints. The proposed network in this paper deals with the constraints into two parts, which brings the network simpler structure and better properties. By the Tikhonov-like regularization method, the proposed network need not estimate the exact penalty parameter in advance. Moreover, comparing with some existing neural networks, the proposed network can solve more general constrained pseudoconvex optimization problems. When the solution of the proposed network is bounded, it converges to the optimal solution set of considered optimization problem, which may be nonsmooth and nonconvex. Meantime, some sufficient conditions are presented to guarantee the boundedness of the solution of the proposed network. Numerical examples with simulation results are given to illustrate the effectiveness and good characteristics of the proposed network for solving constrained pseudoconvex optimization. (C) 2013 Published by Elsevier B.V.

关键词： Neural network Nonsmooth optimization pseudoconvex function Differential inclusion Regularization method

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Pseudo-differential and characterization of generalized convex functions on Riemannian manifolds

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OPTIMIZATION 2024年第1期73卷 113-133页

作者： Ghahraei, Elham Univ Isfahan Dept Math Esfahan Iran Inst Res Fundamental Sci IPM Sch Math Tehran Iran

A notion of pseudo-differential on Riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented and then by definition of pseudo-Hessian, a version of Taylor's expansion theorem is obtained. Moreover, characterizations of generalized convexity of continuous functions are obtained in terms of the generalized monotonicity of pseudo-differentials.

关键词： Pseudo-differential quasiconvex function pseudoconvex function Riemannian manifolds

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Generalized convex functions and generalized differentials

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OPTIMIZATION 2013年第7期62卷 943-959页

作者： Nguyen Thi Hong Linh Penot, Jean-Paul Vietnam Natl Univ Int Univ Linh Trung Ward Ho Chi Minh City Vietnam Univ Paris 06 Lab JL Lions F-75005 Paris France

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.

关键词： colinvex generalized differential mathematical programming optimality conditions protoconvex function pseudoconvex function quasiconvex function 26B25 46G05 49K27 90C26 90C32

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Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2013年第1期158卷 65-84页

作者： Ivanov, Vsevolod I. Tech Univ Varna Dept Math Varna 9010 Bulgaria

We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known.

关键词： Global optimization pseudoconvex function Variational inequality Pseudomonotone-star map

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GENERALIZATION OF SOME DUALITY THEOREMS IN NONLINEAR-PROGRAMMING

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MATHEMATICAL PROGRAMMING 1977年第3期12卷 293-317页

作者： MAHAJAN, DG VARTAK, MN INDIAN INST TECHNOL BOMBAY 76 INDIA

pseudoconvexity of a function on one set with respect to some other set is defined and duality theorems are proved for nonlinear programming problems by assuming a certain kind of convexity property for a particular linear combination of functions involved in the problem rather than assuming the convexity property for the individual functions as is usually done. This approach generalizes some of the well-known duality theorems and gives some additional strict converse duality theorems which are not comparable with the earlier duality results of this type. Further it is shown that the duality theory for nonlinear fractional programming problems follows as a particular case of the results established here.

关键词： Duality Lagrangian Nonlinear Programming Fractional Programming pseudoconvex function Sufficient Conditions for Optimality

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Extended cutting plane method for a class of nonsmooth nonconvex MINLP problems

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OPTIMIZATION 2015年第3期64卷 641-661页

作者： Eronen, Ville-Pekka Makela, Marko M. Westerlund, Tapio Univ Turku Dept Math & Stat Turku Finland Abo Akad Univ Proc Design & Syst Engn Lab Turku Finland

In this article, a generalization of the ECP algorithm to cover a class of nondifferentiable Mixed-Integer NonLinear Programming problems is studied. In the generalization constraint functions are required to be -pseudoconvex instead of pseudoconvex functions. This enables the functions to be nonsmooth. The objective function is first assumed to be linear but also -pseudoconvex case is considered. Furthermore, the gradients used in the ECP algorithm are replaced by the subgradients of Clarke subdifferential. With some additional assumptions, the resulting algorithm shall be proven to converge to a global minimum.

关键词： generalized convexity mixed-integer programming pseudoconvex function alpha ECP nonsmooth optimization nonsmooth MINLP subgradient extended cutting plane algorithm 26A27 90C56 90C11 90C26

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