Nagata Furukawa

Optimality conditions in nondifferentiable programming and their applications to best approximations

This paper studies constrained optimization problems in Banach spaces without usual differentiability and convexity assumptions on the functionals involved in the data. The aim is to give optimality conditions for the problems from which one can derive the characterization of best approximations. The objective and the inequality constraint functionals are assumed to have one-sided directional derivatives. First-order necessary conditions are given in terms of subdifferentials of the directional derivatives. The notion of “max-pseudoconvexity” weaker than pseudoconvexity is introduced for sufficiency. The optimality conditions are applied to linear and nonlinear Tchebycheff approximation problems to derive the characterization of best approximations.

Stopped decision processes on complete separable metric spaces

In this paper we shall treat the combined problems of optimal control and optimal stopping in the discrete time stochastic systems. These combined problems are instituted in view of the practical use. When concerned with the dynamic programming problems or more generally with the multistage stochastic decision problems, even if in the case of the infinite horizon, we are often in the situation that it is forced or profitable to stop the choice of control actions some day. Such the combined ones come under the optimal control problems, if we define anew a decision process having the union of a “fictitious” absorbing state and the original state space as a new state space (cf. [7]). But following a control policy in terms of the new-defined decision process may not become to stop with probability one in terms of the original decision process. Therefore, if the interest is restricted to the control policies that stop with probability one, the approach by the new-defined decision process as above may be inappropriate. Roughly speaking, a stopped decision process is a decision process which stops with probability one, and a stopped policy is a policy associated with a stopping time which is finite with probability one (for precise definitions, see Section 2). In this paper we study the existence theorems of an optimal stopped policy associated with some optimality criterions in multistage stochastic decision problems, and there the methods of [3], [4], [5], and [9] are available. In Section 2, we give the notations and definitions to be used throughout this paper, and in Section 3 we prepare the fundamental Lemmas to be used in Section 4. Section 4 is devoted to the existence of an optimal stopping time associated with a policy. The study of such the optimal stopping time is not our main object, but the preparatory consideration for the existence of the optimal stopped policy in the subsequent sections. In Section 5, we give the existence theorems of a (p, c)-optimal stopped policy and of a (p, E, S)-optimal

Recurrence set-relations in stochastic multiobjective dynamic decision processes

Finite horizon stochastic dynamic decision processes with Rp valued additive returns are considered. The optimization criterion is a partial-order preference relation induced from a convex cone in Rp . The state space is a countable set, and the action space is a compact metric spaces. The optimal value function, which is of a set-valued mapping, is defined. Under certain assumptions on the continuity of the reward vector and the transition probability, a system of a recurrence set-relations concerning the optimal value functions is given.