Department of Computer Science & Engineering

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IIT Jodhpur

School of Artificial Intelligence and Data Science (AIDE)

In this article we intend to study the necessary and sufficient conditions for global maximization of locally Lipschitz functions. We develop a necessary and sufficient condition for a class of Lipschitz pseudoconvex function in terms of the Clarke subdifferential. We also develop sufficient conditions for optimality in terms of very general non-smooth objects called convexifactors. © 2005 Taylor & Francis Group Ltd.

Optimization

Optimality conditions for maximizing a locally Lipschitz function

In this article, we study increasing and positively homogeneous functions defined on convex cones of locally convex spaces. This work is the first part in a series of studies to have a general view of the emerging area of Monotonic Analysis. We develop a general notion of so-called elementary functions, so that the generalized increasing and positively homogeneous functions can be represented as upper-envelopes of families of such functions. We also study many other associated properties like the description of support sets and normal and co-normal sets in a very general setting.

World University root node

Monotonic analysis over cones: I

Some inequalities of Hermite-Hadamard type for increasing convex-along-rays functions are given. Examples for particular domains including triangles, squares, and the part of the unit disk in the first quadrant are also presented. © 2004, Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München. All rights reserved.

Fulltext

Analysis (Germany)

Hermite-hadamard-type inequalities for increasing convex-along-rays functions

The aim of this article is to study necessary optimality conditions for a vector minimization program involving locally Lipschitz functions under certain general regularity conditions. We study problems involving only inequality and both inequality and equality constraints.

Numerical Functional Analysis and Optimization

Regularity conditions and optimality in vector optimization

In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoconvex function is introduced and its properties are studied in terms of convexifactors. We also present some optimality conditions for vector minimization in terms of convexifactors.

Convexifactors, generalized convexity and vector optimization

In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).

Monotonic analysis over cones: II

The main aim of this paper is to obtain optimality conditions for a constrained set-valued optimization problem. The concept of Clarke epiderivative is introduced and is used to derive necessary optimality conditions. In order to establish sufficient optimality criteria we introduce a new class of set-valued maps which extends the class of convex set-valued maps and is different from the class of invex set-valued maps.

Journal of the Australian Mathematical Society

Optimality criteria in set-valued optimization

In this article we study a Hadamard type inequality for nonnegative evenly quasiconvex functions. The approach of our study is based on the notion of abstract convexity. We also provide an explicit calculation to evaluate the asymptotically sharp constant associated with the inequality over a unit square in the two-dimensional plane. © 2002 Elsevier Science (USA). All rights reserved.

Journal of Mathematical Analysis and Applications

Hadamard type inequality for quasiconvex functions in higher dimensions

The recently introduced notion of a convexifactor is further studied, and quasiconvex and pseudoconvex functions are characterized in terms of convexifactors. As an application to a chain rule, a necessary optimality condition is deduced for an inequality constrained mathematical programming problem.

Journal of Optimization Theory and Applications

Convexifactors, generalized convexity, and optimality conditions

The concepts for vector optimization problems by perturbing the ordering cone in the objective space were introduced. Types of ε-weak solutions associated with a vector minimum problem were also studied. The necessary and sufficient optimality conditions were studied for the existence of an approximate vector minimum for the problem (VP). The notions of approximate saddle points were described and certain approximate saddle point theorems were presented.