Hidefumi Kawasaki

Second order necessary optimality conditions for minimizing a sup-type function

The second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions. By taking account of this fact, we derive new second order necessary optimality conditions for minimization of a sup-type function. The conditions involve an extra term besides the second derivative of the Lagrange function. Furthermore, we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup-type function.

A Duality Theorem in Multiobjective Nonlinear Programming

We give a duality theorem in multiobjective nonlinear programming problems by means of conjugate set-valued functions which were introduced in Kawasaki, H. 1981. Conjugate relations and weak subdifferentials of relations. Math. Oper. Res.6 593--607.. The duality theorem is reflexive in the sense analogous to Ekeland, I., R. Temam. 1976. Convex Analysis and Variational Problems. North-Holland, Amsterdam..

The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t ∈ T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, ∂f/∂x and∂2f/∂x2 are continuous on ℝn× T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.

Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualifications

We are concerned with a nonlinear programming problem with equality and inequality constraints. We shall give second-order necessary conditions of the Kuhn-Tucker type and prove that the conditions hold under new constraint qualifications. The constraint qualifications are weaker than those given by Ben-Tal (Ref. 1).

Second-order necessary and sufficient optimality conditions for minimizing a sup-type function

In this paper, we give second-order necessary and sufficient optimality conditions for a minimization problem of a sup-type functionS(x)=sup{f(x,t);tε T}, whereT is a compact set in a metric space and f is a function defined on ℝn ×T. Our conditions are stated in terms of the first and second derivatives of f(x, t) with respect tox, and involve an extra term besides the second derivative of the ordinary Lagrange function. The extra term is essential when {f(x,t)}t forms an envelope. We study the relationship between our results, Wetterling [14], and Hettich and Jongen [6].

Conjugate Relations and Weak Subdifferentials of Relations

The duality theory for convex programs is based on a concept “conjugate” of functions, and the set of solutions is characterized by “subdifferential” of a function. The purpose of this paper is to extend the concepts “conjugate” and “subdifferential” to those of set-valued functions relations and to investigate their properties.

A Conjugate Points Theory for a Nonlinear Programming Problem

The conjugate point is an important global concept in the calculus of variations and optimal control. In these extremal problems, the variable is not a vector in Rn but a function. So a simple and natural question arises. Is it possible to establish a conjugate points theory for a nonlinear programming problem, Min f(x) on x\in R^n

Conjugate Points for Variational Problems with Equality and Inequality State Constraints

? This paper positively answers this question. We introduce the Jacobi equation and conjugate points for the nonlinear programming problem, and we describe necessary and sufficient optimality conditions in terms of conjugate points.

AN APPLICATION OF A DISCRETE FIXED POINT THEOREM TO A GAME IN EXPANSIVE FORM

In this paper, variational problems with equality and inequality state constraints are considered. The theory of conjugate points for these problems is developed, and necessary conditions for weak local optimality are derived in terms of this concept and the Legendre condition. For the case of inequality constraints, the envelope-like effect is taken into consideration in the accessory problem.

Discrete fixed point theorems and their application to Nash equilibrium

In this paper, we first present a discrete fixed point theorem for contraction mappings from the product set of integer intervals into itself, which is an extension of Roberts discrete fixed point theorem. Next, we derive an existence theorem of a pure-strategy Nash equilibrium for a noncooperative n-person game from our fixed point theorem. Finally, we show that Kuhns theorem for a game in expansive form can be explained by our existence theorem.