OPTIMALITY CONDITIONS AND DUALITY IN NONDIFFERENTIABLE ROBUST OPTIMIZATION PROBLEMS

Abstract

Abstract. We consider a nondi erentiable robust optimization problem,which has a maximum function of continuously di erentiable functionsand support functions as its objective function, continuously di erentiablefunctions as its constraint functions. We prove optimality conditions forthe nondi erentiable robust optimization problem. We formulate a Wolfetype dual problem for the nondi erentiable robust optimization problemand prove duality theorems. 1. IntroductionA standard form of nonlinear programming problem with inequality con-straints(P) inf x2R n ff(x) : g i (x) 0; i= 1; ;mg;where f: R n !R and g i : R n !R are continuously di erentiable functions.The problem in the face of data uncertainty in the constraints can be capturedby the following nonlinear programming problem:(UP) inf x2R n ff(x;u) : g i (x;v i ) 0; i= 1; ;mg;where u, v i are uncertain parameters and u2U, v i 2V i ;i= 1; ;mforsome convex compact sets U ˆR p ;V i ˆR q ;i= 1; ;m;respectively andf : R n nR p !R, g i : R R