Infinite-dimensional Linear Programming Problem Research Articles

Abstract A method of reduction of a finite statistical game is described. The method allows to reduce the solution of this game to the solution of a statistical game with a “smaller” space of observations in that case if the optimal strategy is partly known for the statistician. In particular, this.situation arises if a given optimal strategy of the nature is checked. Considering some infinite-dimensional linear programming problems associated to the original game, we prove theorems about the connection between the solutions of the original and the reduced game. A complete analysis of 2 × 2-statistical games is given using the described reduction. AMS I980 Subject Classifications: Primary: 49 D 05Secondary: 90 E 05Keywords: Game theorystatistic gamereduction of statistical games

A class of infinite dimensional linear programming problems

We study the infinite dimensional linear programming problem. The previous work done on this subject defined the dual problem in a small space and derived duality results for such pairs of problems. But because of that and of the strong requirements on the functions involved, those theorems do not actually hold in many applications. With our formulation, we define the dual problem in a larger space and obtain new duality results under, generally, mild assumptions. Furthermore, the solutions turn out to be extreme points of the unbounded, but w ∗-locally compact, feasibility set. For this purpose, we did not try a constructive proof of our duality results, but instead we examine the problem from a more abstract point of view and derive results using general ideas from the theory of convex analysis in normed spaces [ R. T. Rockafellar, “Conjugate Duality and Optimization,” SIAM, Philadelphia, Penn., 1973, and R. Holmes, “Geometric Functional Analysis,” Springer-Verlag, New York, 1975]. Our work extends previous results in this area, which appeared in [ N. Levinson, J. Math. Anal. Appl. 16 (1965) 73–83, and W. Tundall, SIAM J. Appl. Math. 13 (1965) , 644–666].

Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.

Duality of infinite-dimensional linear programming problems and some related matters

Duality of infinite-dimensional linear programming problems and some related matters