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IIT Jodhpur

In this article we develop certain necessary optimality condition for bilevel programming problems with convex lower-level problem. The results are abstract in nature and depend on an important construction in nonsmooth analysis called the coderivative of a set-valued map. © 2006 Springer Science + Business Media, LLC.

School of Artificial Intelligence and Data Science (AIDE)

World University root node

Springer Optimization and Its Applications

Bilevel programming with convex lower level problems

We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions for weakly efficient solutions of non-convex location problems. © Springer-Verlag 2006.

Mathematical Methods of Operations Research

Lagrangian conditions for vector optimization in Banach spaces

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.

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.

On approximate minima in vector optimization

In this paper, a very recently introduced class of convexlike functions known as the generalized pre-invex function is studied. It is demonstrated that among convexlike functions the generalized pre-invex functions are important since they can be profitably used in sufficiency conditions in vector optimization. Various properties of the class of generalized pre-invex function are also studied.

Asia-Pacific Journal of Operational Research

On generalized pre-invex functions

In this article we present a simple method to deduce necessary conditions for weak minimisation of a convex vector program in a Banach space. Our main tool here will be the generalised Jabcobian of Ralph.

Bulletin of the Australian Mathematical Society

On convex vector optimisation

We introduce the notion of semi-invex function (non-smooth) and the associated subdifferential. We study their properties and establish the conditions for optimality in constrained and unconstrained minimisation problems.

Semi-invex functions and their subdifferentials

The equivalence between saddle points and optima and duality theorem (without constraint qualification) are established for a large class of non-smooth and non-convex programs.