Optimization with some uncontrollable variables: a min-equilibrium approach

Motivated by instability analysis of unstable (excited state) solutions in computational physics/chemistry, in this paper, the minimax method for solving an optimal control problem with partially uncontrollable variables is embedded into a more general min-equilibrium problem. Results in saddle critical point analysis and computation are modified to provide more information on the minimized objective values and their corresponding riskiness for one to choose in decision making. A numerical algorithm to compute such minimized objective values and their corresponding riskiness is devised. Some convergence results of the algorithm are also established.