Convexity plays a very important role in various branches of mathematical, natural and social sciences. In an effort to extend existing results depending on convexity there has been a steady interest over the years towards its generalizations. Many excellent books, monographs and numerous articles have been written pursuing generalizations of the basic concepts, their characterizations. and studying properties under different conditions. The purpose of this contribution is to gather information on certain generalizations of convexity and their applications to duality theory and optimality conditions

A survey of recent[1985-1995]advances in generalized convexity with applications to duality theory and optimality conditions

In this paper, we formulate and prove necessary and sufficient optimality conditions in multiobjective control problems which involve multiple integrals. Under $$ \displaystyle (\rho , b) $$
 -quasiinvexity assumptions, sufficient efficiency conditions for a feasible solution are derived.

Efficiency conditions in vector control problems governed by multiple integrals

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (?, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.

Duality theorems for a new class of multitime multiobjective variational problems

Consider the multi-time multi-objective variational problem (MFP) of minimizing a vector of quotients of path independent curvilinear functionals subject to PDE and/or PDI constraints. The goal of our work is to develop an optimization theory for the second order jet bundle. While the background in Sect. 1 is introductory, the theory in Sects. 2 and 3 is new as a whole, containing our results.

Minimization of vectors of curvilinear functionals on the second order jet bundle

The necessary conditions for (normal) efficient solutions to a class of multi-objective fractional variational problems (MFP) with nonlinear equality and inequality constraints are established using a parametric approach to relate efficient solutions of a fractional problem and a non-fractional problem. Based on these normal efficiency criteria a Mond-Weir type dual is formulated and appropriate duality theorems are proved assuming - quasi-invexity of the functions involved.

Efficiency and duality for multiobjective fractional variational problems with (ρ,b) - quasiinvexity

This paper uses Clarke’s generalised directional derivative to describe several types of invexity, pseudoinvexity and quasiinvexity at a point of a nonlinear function. Direct implications of the relations existing between the various types of invexity and generalised invexity are presented, as well as a block diagram of these implications. In particular, similar results in the class of quasiconvex functions are obtained.

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I.M. Stancu-Minasian

Papers

Invexity at a point: generalisations and classification

Efficiency and duality for multiobjective fractional variational problems with (ρ,b) - quasiinvexity