A new solution to optimization-satisfaction problems by a penalty method

This paper is concerned with an optimization-satisfaction problem to determine an optimal solution such that a certain objective function is minimized, subject to satisfaction conditions against uncertainties of any disturbances or opponents' decisions. Such satisfaction conditions require that plural performance criteria are always less than specified values against any disturbances or opponents' decisions. Therefore, this problem is formulated as a minimization problem with the constraints which include max operations with respect to the disturbances or the opponents' decision variables. A new computational method is proposed in which a series of approximate problems transformed by applying a penalty function method to the max operations within the satisfaction conditions are solved by usual nonlinear programming. It is proved that a sequence of approximated solutions converges to a true optimal solution. The proposed algorithm may be useful for systems design under unknown parameters, process control under uncertainties, general approximation theory, and strategic weapons allocation problems.