Mathematics : Optimization Theory

The specific interest lies in Convex and Nonsmooth Optimization, Vector Optimization, Bilevel Programming, variational inequalities. The main thrust area of research has been convex optimization and nonsmooth optimization. The other specific interests like vector optimization, bilevel programming, variational inequalities has been motivated from either convex or nonsmooth optimization.

In convex optimization interest has been largely the extensions of the ideas of convex optimization to areas like vector optimization and also studying the convergence of inexact proximal point methods for variational inequalities involving maximal monotone maps. Recently considerable effort has been devoted to develop the theory of gap functions for vector variational inequalities and also use the gap functions to develop error bounds. Gap functions and associated error bounds were also developed for variational inequalities involving set-valued map. Attempts are now made to develop a gap function theory for vector equilibrium problems.

Another major area of research is bilevel programming and efforts are mainly concentrated on developing necessary optimality conditions for bilevel problems. It is important to note that that bilevel problems are intrinsically non-convex and non-smooth problems even if the problem date is convex. Developing optimality conditions for bilevel programming prblem remains a challenge since most standard optimality conditions are not satisfied there. Further a very simplified version of a bilevel problem with only one variable is also being investigated, where the convexity of the problem is maintained if the problem data is convex. In fact even such a simple convex bilevel problem has its own difficulties which again linked to the fact the Slater constraint qualification fails for it.

The development of monotonic analysis which deals with increasing functions with additional properties, is also part of the interests. These functions have properties almost paralleling that of convex functions. These special functions has been used by many researchers to develop some algorithms for global optimization.

Future research interests involve the study of global optimization of optimization problems involving polynomials. This area of optimization draws in lots of tools from algebraic geometry. Another area of interest would be Stochastic optimization.