Existence Theorem for the Discontinuous Generalized Quasivariational Inequality Problem

We consider the following generalized quasivariational inequality problem: given a real Banach space E with topological dual E* and given two multifunctions G:X2 ^X and F:X2 ^E *, find such that

On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E ^*, a multifunction and a multifunction Γ:X→2 ^X , find a point such that , . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous. The authors express their sincere gratitude to the referees for helpful suggestions and comments. This research was partially supported by a grant from the National Science Council of Taiwan, ROC. B.T. Kien was on leave from National University of Civil Engineering, Hanoi, Vietnam.     

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X E and C F, two multifunctions Γ : X → 2 ^X and Ф : X → 2 ^C , and a single-valued map ψ : , find a pair such that , Ф and for all . We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Ф. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213–228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E ^* and ).     

Optimizing a linear function over an efficient set

The problem (P) of optimizing a linear functiond ^T x over the efficient set for a multiple-objective linear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function directiond and the set of domination directionsD, ifd ^T 0 for all nonzero D, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point , if there is no adjacent efficient edge yielding an increase ind ^T x, then a cutting plane is used to obtain a multiple-objective linear program ( ) with a reduced feasible set and an efficient set . To find a better efficient point, we solve the problem (I_i) of maximizingc _i ^T x over the reduced feasible set in ( ) sequentially fori. If there is a that is an optimal solution of (I_i) for somei and , then we can choosex ⁱ as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.The authors would like to thank an anonymous referee, H. P. Benson, and P. L. Yu for numerous helpful comments on this paper.     

Lagrangian Globalization Methods for Nonlinear Complementarity Problems

This paper extends the Lagrangian globalization (LG) method to the nonsmooth equation arising from a nonlinear complementarity problem (NCP) and presents a descent algorithm for the LG phase. The aim of this paper is not to present a new method for solving the NCP, but to find such that when the NCP has a solution and is a stationary point but not a solution.     

Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

Let H be a real Hilbert space and let be a function that we wish to minimize. For any potential and any control function which tends to zero as t+, we study the asymptotic behavior of the trajectories of the following dissipative system:

Repleteness and regular lattice measures

Let X be an abstract set and L a lattice of subsets of X. To each lattice regular measure µ, we associate two induced measures and on suitable lattices of the Wallman space IR(L) and another measure µ on the spaceIR(L). We will investigate the reflection of smoothness properties of p onto , and µ; and try to set some new criterion for repleteness and measure repleteness.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

ＱＵＡＤＲＡＴＩＣＥＳＴＩＭＡＴＯＲＳＯＦＱＵＡＤＲＡＴＩＣＦＵＮＣＴＩＯＮＳＷＩＴＨＰＡＲＡＭＥＴＥＲＳＩＮＮＯＲＭＡＬＬＩＮＥＡＲＭＯＤＥＬＳ￥ＷＵＱＩＧＵＡＮＧ（吴启光）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｅＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅ...     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.     

Some extremal results on circles containing points

We define (n) to be the largest number such that for every setP ofn points in the plane, there exist two pointsx, y P, where every circle containingx andy contains (n) points ofP. We establish lower and upper bounds for (n) and show that [n/27]+2(n)[n/4]+1. We define for the special case where then points are restricted to be the vertices of a convex polygon. We show that .     

On the existence of value in the Varaiya-Lin sense in differential games of pursuit and evasion

This paper is strictly related to Ref. 1. A pursuit-evasion game described in part by the system and is considered. The state variablesx andy are restricted, in the sense that (x(t),t) N ₁ and (y(t),t) N ₂. The existence of a value in the sense of Varaiya and Lin is proved under the assumption that the sets of all admissible trajectories for the two players are compact and the lower value is not greater than the upper value.     

Spherical elastic-plastic waves

Zusammenfassung Es wird die Fortpflanzung elastisch-plastischer Spannungswellen in einem unendlichen Medium betrachtet, welches einer idealen Spannungs-Verformungs-Kurve folgt, Trescas Fliesskriterium unterworfen ist und einen sphärischen Hohlraum enthält, wobei an der Fläche des Hohlraumes ein Stoss angenommen wird. Ein rechnerisches Verfahren, basiert auf endliche Differenzen, wird entwickelt and ein Beispiel gegeben.

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Notation radial stress - tangential stress - K yield stress - _rr non-dimensional radial stress ( /K) - non-dimensional tangential stress ( /K) - , Lame's constants - K _b Bulk constant (=(3+2)/3) - v Poisson's constant - Material density - C Elastic wave velocity (=((+2)/)^1/2) - C _p Plastic wave velocity (=(K _b /)^1/2) - distance from center of cavity - r ₀ cavity radius - v non-dimensional radial co-ordinate (= /r ₀) - time - t non-dimensional time (=C /r ₀) - radial displacement - u non-dimensional radial displacement (=/r ₀) - particle velocity - v non-dimensional particle velocity (= /C) - pressure - P(t) non-dimensional pressure (= /K)     

Nonconvergence results for the application of least-squares estimation to Ill-posed problems

One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in , where corresponds to a parametric representation providing sufficiently good approximation to the true solutionx*. Call the minimizerx=A( ). Take = _N for a sequence { _N } of subspaces becoming dense, and so determine an approximating sequences {x _N A ( _N )}. It is shown, withf linear and one-to-one, that one need not havex _Nx* iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).     

M-estimators of structural parameters in pseudolinear models

Real valued M-estimators in a statistical model 1 with observations are replaced by -valued M-estimators in a new model with observations where are regressors, is a structural parameter and a structural function of the new model. Sufficient conditions for the consistency of are derived, motivated by the sufficiency conditions for the simpler parent estimator The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If F has a natural exponential density e^{x–b( x )} then our pseudolinear model with u = (g o )^–1 reduces to the well known generalized linear model, provided () = db()/d and g is the so-called link function of the generalized linear model. General results are illustrated for special pairs and leading to some classical M-estimators of mathematical statistics, as well as to a new class of generalized -quantile estimators.     

Quasilinearization and Rate of Convergence for Higher-Order Nonlinear Periodic Boundary-Value Problems

We study the convergence of a sequence of approximate solutions for thefollowing higher-order nonlinear periodic boundary–value problem:

A stability theorem for a class of linear evolution systems

Dynamical systems of the form ,u(0)=u ₀, where,B is a densely defined linear operator mapping its domainD (B ) into — the infinitesimal generator of a semigroup of operatorsT (t, B) of classC ₀ — are investigated, such that for each solutionu to , whereP is the spectral eigenprojection onto the null space ofB.It is shown that under some general hypotheses concerning spectral properties ofB the above stability condition is equivalent with the following situation: There exist (i) a normal generating coneK such thatT(t;B)KK fort0 and (ii) a strictly positive element in the dual coneK such that , whereB denotes the dual ofB. Condition (ii) implies the so called total concentration time preservation, i. e. .     

Generalized monotone bifunctions and equilibrium problems

Using quasimonotone and pseudomonotone bifunctions, we derive existence results for the following equilibrium problem: given a closed and convex subsetK of a real topological vector space, find such that for allyK. In addtion, we study the solution set and the uniquencess of a solution. The paper generalizes results obtained recently for variational inequalities.     

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

Study on a memory gradient method for the minimization of functions

A new accelerated gradient method for finding the minimum of a functionf(x) whose variables are unconstrained is investigated. The new algorithm can be stated as follows: where x is the change in the position vectorx, g(x) is the gradient of the functionf(x), and and are scalars chosen at each step so as to yield the greatest decrease in the function. The symbol denotes the change in the position vector for the iteration preceding that under consideration.For a nonquadratic function, initial convergence of the present method is faster than that of the Fletcher-Reeves method because of the extra degree of freedom available. For a test problem, the number of iterations was about 40–50% that of the Fletcher-Reeves method and the computing time about 60–75% that of the Fletcher-Reeves method, using comparable search techniques.This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensed version of the investigation described in Ref. 1. Portions of this paper were presented by the senior author at the International Symposium on Optimization Methods, Nice, France, 1969.