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Solving minimax problems by interval methods

Interval methods are used to compute the minimax problem of a twice continuously differentiable functionf(y, z),y ε ℝm,z ε ℝn ofm+n variables over anm+n-dimensional interval. The method provides bounds on both the minimax value of the function and the localizations of the minimax points. Numerical examples, arising in both mathematics and physics, show that the method works well.

On interval enclosures using slope arithmetic

This paper describes how a combination of interval derivative arithmetic and interval slope arithmetic can be used with the programming language Pascal-SC to obtain an enclosure of the range of a factorable function @?:R^n -> R^1 with @? @? C^2 (D) which is often narrower than that which is obtained by using slope arithmetic alone and which is always narrower than that which is obtained by using interval derivative arithmetic alone.

An interval maximum entropy method for a discrete minimax problem

An interval method for a discrete minimax problem is described, in which the constituent objective functions are Lipschitz continuous but not differentiable. A pseudocode Fortran 90 algorithm is described, and numerical results from a Fortran 90 implementation of the algorithm are presented.

The Krawczyk operator and Kantorovich's theorem

Abstract R. E. Moore showed that the Krawczyk operator can be used to verify the existence of a unique solution of F(x ∗ ) = 0 in a hypercube. When the derivative in the Krawczyk operator is replaced by a slope, a comparison shows that the resulting improved form of Moores existence test is at least as good as and, as shown by an example, sometimes better than that of Kantorovichs theorem with regard to sensitivity and precision.

On the existence of periodic solutions of periodically perturbed conservative systems

Consider the system of ordinary differential equations u(t)+ GH(u(t))=p(t), where p:R→R is a continuous function with period 2π and G:R n →R has a continuous second partial derivative. It is important to determine sufficient conditions for the existence of a unique 2π-periodic solution. In this paper, the above system is related to an initial value problem in terms of which a new set of sufficient conditions can be given

A note on the comparison of the Kantorovich and Moore theorems

F(x) = 0 (1.1) where F: D E R” --+ R” is continuously differentiable on the open convex set D. A theoretical comparison by Rall [6] of the theorems of Kantorovich [2] and of Moore [3, 41 which contain sufficient conditions for the existence a unique zero of F in a given subset of D shows that the Kantorovich theorem has only a slight advantage over the Moore theorem with regard to sensitivity and precision, while the latter requires ‘far less computational labour than the former. Rall’s comparison is based on the assumption that the interval extension F’: Z(D) + Z(R” ’ “) of the derivative F’: D -+ R” ’ n is defined by

A note on Moore's interval test for zeros of nonlinear systems

We study relations between Moores interval test and Mirandas theorem. As an application we combine the (real) Newton iteration with a computational test for Mirandas hypothesis by Moore and Kioustelidis to find an approximate solution to the systemf(x)=0 with specified error bounds.ZusammenfassungEs werden Beziehungen zwischen Moores Test und dem Zwischenwertsatz von Miranda hergeleitet. Als Anwendung wird ein konstruktiver Test von Moore und Kioustelidis mit dem reellen Newton-Verfahren kombiniert, um Nullstellennäherungen mit vorgegebener Genauigkeit zu konstruieren.

KANTOROVICH THEOREM FOR VARIATIONAL INEQUALITIES

Kantorovich theorem was extended to variational inequalities by which the convergence of Newton iteration, the existence and uniqueness of the solution of the problem can be tested via computational conditions at the initial point.

On the solution of nonlinear two-point boundary value problem

In this paper, a non-variational version of a max-min principle is proposed, and an existence and uniqueness result is obtained for the nonliner two-point boundary value problem u″+g(t,u)=f(t),u(0)=u(2π)=0

Slope tests for newton-type methods

It is shown that the computable test for the existence of a solution of a system of nonlinear algebraic equations in a given region due to Pandian [8] may be weakened, and that sharp componentwise error bounds can be obtained if derivatives are replaced with slopes [4], [12]. Illustrative numerical examples are presented.