Focus programming: a bi-level programming approach to static stochastic optimization problems

Static stochastic optimization problems are formulated with the focus theory of choice where the optimal solution is determined as per which solution's focus (the most salient realization of a random vector) is the most preferred. The new formulation that we call the focus programming is a bi-level programming approach in which the lower-level program is used to choose the focus of each feasible solution and the upper-level program is to determine the optimal solution. Since in focus programming models upper-level and lower-level programs are maximin or minimax problems, they are nonsmooth and sometimes even nonconvex so that the existing optimization methods cannot solve such bi-level programming problems. We propose several single-level reformulation methods for such problems.