On the reduction to a complete class in multiple decision problems

Summary This paper is concerned with the reduction to a complete class of decision rule in case where actions, observations and states are finite in number. Discussions are made, first, from the viewpoint of the distribution of random observation and, secondly, from the viewpoint of loss function. With respect to loss function, “regret-relief ratio” criterion and “incremental loss-gain ratio” criterion are introduced and these usefulness are shown in 2-state of nature case. The Institute of Statistical Mathematics     

On the application of TRIMAP to problems with multiple decision makers

In this paper the potentialities of TRIMAP to provide decision support in multiobjective problems with multiple decision makers are exploited. TRIMAP is an interactive three-objective linear programming package which enables a progressive and selective learning of the nondominated solution set. The aim is to aid the opposing parties in exploring their own preferences and to explore the dynamic nature of the negotiation process.     

On the empirical Bayes approach to multiple decision problems with sequential components

The empirical Bayes approach to multiple decision problems with a sequential decision problem as the component is studied. An empirical Bayesm-truncated sequential decision procedure is exhibited for general multiple decision problems. With a sequential component, an empirical Bayes sequential decision procedure selects both a stopping rule function and a terminal decision rule function for use in the component. Asymptotic results are presented for the convergence of the Bayes risk of the empirical Bayes sequential decision procedure.     

An application of Lemke's method to a class of Markov decision problems

This paper presents an application of Lemke's method to a class of Markov decision problems, appearing in the optimal stopping problems, and other well-known optimization problems. We consider a special case of the Markov decision problems with finitely many states, where the agent can choose one of the alternatives; getting a fixed reward immediately or paying the penalty for one term. We show that the problem can be reduced to a linear complementarity problem that can be solved by Lemke's method with the number of iterations less than the number of states. The reduced linear complementarity problem does not necessarily satisfy the copositive-plus condition. Nevertheless we show that the Lemke's method succeeds in solving the problem by proving that the problem satisfies a necessary and sufficient condition for the extended Lemke's method to compute a solution in the piecewise linear complementarity problem.     

On the problem of weights in multiple criteria decision making (the noncompensatory approach)

This paper shows how the notion of ‘relative importance of attributes’ can be defined within the framework of the noncompensatory approach to multiple criteria decision making. The problem of weights then appears as a problem of functional representation of relations. We state some theoretical results concerning this problem and outline a practical decision-aid (Tactic) based on the ideas introduced in the paper.     

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

On the number of solutions to a class of complementarity problems

In this paper we consider the problem of establishing the number of solutions to the complementarity problem. For the case when the Jacobian of the mapping has all principal minors negative, and satisfies a condition at infinity, we prove that the problem has either 0, 1, 2 or 3 solutions. We also show that when the Jacobian has all principal minors positive, and satisfies a condition at infinity, the problem has a unique solution.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. This material is based upon work supported by the National Science Foundation under Grant No. MCS77-03472 and Grant No. MCS78-09525. This work appeared as an MRC Technical Report No. 1964, University of Wisconsin, Madison, WI, June 1979.     

On a class of extremal problems

In the paper we introduce a class of trigonometrical polynomial extremal problems depending on a continuous parameter 0≤r≤1. It turns out that the two border cases r=0 and r=1 are known problems investigated earlier by Kamae, Mendes-France, Ruzsa and the present author. We also introduce another set of extremal problems for measures with similar parametrization, and prove a duality relationship between the two type of extremal quantities. The proof relies on a minimax theorem proved earlier by the author. The known duality results are proved as corollaries. 1980 MS Classification. Primary 42A05; Secondary 46B25, 46N05.     

On a class of extremal problems in statistics

Let m denote the infimum of the Integral of a function q w r t all probability measures with given marginals. The determination of m is of interest for a series of stochastic problems. In the present paper we prove a duality theorem for the determination of m and give some examples for its application. We consider especially the problem of extremal variance of sums of random variables and prove a theorem for the existence of random variables with given marginal distributions, such that their sum has variance zero.     

On the number of solutions to a class of linear complementarity problems

In this note, we consider the linear complementarity problemw = Mz + q, w 0, z 0, w ^T z = 0, when all principal minors ofM are negative. We show that for such a problem for anyq, there are either 0, 1, 2, or 3 solutions. Also, a set of sufficiency conditions for uniqueness is stated.The work of both authors is partially supported by a grant from the National Science Foundation, MCS 77-03472.     

On a class of equilibrium problems in the real axis

In a series of seminal papers, Thomas J. Stieltjes (1856-1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lamé equations), the so-called Heine-Stieltjes polynomials.In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x₁,…,x_n∈R are in presence of a finite family of “attractors” (i.e., negative charges) z₁,…,z_m∈C?R, is considered and its connection with certain class of Lamé-type equations is shown. In addition, we study the situation when both n→∞ and m→∞, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.     

On a class of restricted phase problems in the plane

Certain geometrical necessary and sufficient conditions are developed for the optimal control of a class of second order systems with restrictions on the phase coordinates. The phase constraint set in the plane is assumed to be convex, and under additional hypotheses, it is shown that an optimal trajectory is the one that encloses the least possible area with the optimal trajectory in the unrestricted case.