Saddle Point Condition Research Articles

Differential games with open-loop saddle point conditions

Zero-sum differential games are considered in which one or both of the players are restricted to use open-loop control. It is shown that the first-order necessary conditions for such problems are identical to the first-order necessary conditions for the usual form of a differential game, where both players use closed-loop control laws. An investigation of the conjugate point condition for a special class of games shows that this condition is not the same but depends on the type of solution sought. For games where one or both of the players use open-loop control, there are two conjugate point conditions that must be satisfied. This differs from games in which both players use closed-loop control, where there is only one conjugate point necessary condition.

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.

Bounds for convex variational programming problems arising in power system scheduling and control

The convex variational programming problem has been found of practical importance to power system scheduling and control. Convex variational programming problems are closely related to the convex programming problems treated by Kuhn and Tucker [8]. Among their positive attributes are the absence of difficulties with weak relative minima and with saddle-point or conjugate-point conditions. As a result it is feasible to develop dual-variational problems and to provide estimates for bracketing the extreme value of the functional of the variational problem when gradient or successive approximations methods are employed for determination of near optimal control schedules. This paper develops necessary and sufficient conditions for the optimal control schedule and a means for numerically determining bounds to the extreme value of the functional. Applications are given to the problem of hydrothermal power system coordination.