Non-Linear Programming with Randomization

This paper considers non-linear programming models in which randomized solutions are allowed. That is, we consider the minimization of the expected value of a non-linear function subject to constraints on the expectation of nonlinear functions, where the expectations are taken with respect to randomized decision strategies. Saddle-point and related optimality conditions similar to those given in [Karlin, S. 1959. Mathematical Methods and Theory in Games, Programming, and Economics, Vol. 1, Chapter 7, Addison-Wesley.] and [Kuhn, H. W., A. W. Tucker. 1951. Nonlinear programming. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, 481–492.] for convex functions are obtained with only very mild restrictions on the shape of the optimizing and constraint functions. We also consider when the optimal solution need not be randomized and how the notions of Lagrange multipliers and game theory are related to our results. We offer an economic interpretation of our solution. A few simple examples are given to illustrate the implications of randomized strategies. It is also pointed out how the results of this paper are related to the decomposition methods widely used to solve mathematical programming problems.