Multiobjective Nonsmooth Programming.

It is well known that much of the theory of optimality in constrained optimization has evolved under traditional smoothness (differentiability) assumptions, discussed in previous chapters. As nonsmooth phenomena in optimization occur naturally and frequently, the attempts to weaken these smoothness requirements have received a great deal of attention during the last two decades (Ben-Tal and Zowe (1982), Clarke (1983), Kanniappan (1983), Jeyakumar (1987, 1991), Rockaffelar (1988), Burke (1987), Egudo and Hanson (1993), Bhatia and Jain (1994), Mishra and Mukherjee (1996). Necessary optimality conditions for nonsmooth locally Lipschitz problems have been given in terms of the Clarke generalized subdifferentials (Jeyakumar (1987), Egudo and Hanson (1993), Mishra and Mukherjee (1996)). The Clarke subdifferential method has been proved to be a powerful tool in many nonsmooth optimization problems, see for example Giorgi and others (2004). This Chapter is organized as follows: In Section 3, we establish sufficient optimality conditions to nonsmooth context using conditional proper efficiency. Using the concept of quasi-differentials due to Borwein (1979), Fritz John and Kuhn-Tucker type sufficient optimality conditions for a feasible point to be efficient or conditionally properly efficient for a subdifferentiable multiobjective fractional problem are obtained without recource to an equivalent V-invex program or parametric transformation. In Section 4, Mond-Weir type duality results are established for the nonsmooth multiobjective programming problem, under generalized V-invexity conditions, using conditional proper efficiency. Further, various duality results are established under similar assumptions for subdifferentiable multiobjective fractional programming problems. In Section 5, a vector valued ratio type Lagrangian is considered and vector valued saddle point results are presented under V-invexity conditions and its generalizations. [ABSTRACT FROM AUTHOR]