On the η-Expansion Topology for the Co-Semi-Regularization and Mildly Hausdorff Spaces

The aim of this paper is to introduce the class of mildly Hausdorff spaces via the concept of -open sets and improve some results due to Császár, Noiri, Hamlett, Jankovi and Konstadilaki by proving that every weakly mildly Hausdorff locally rc-paracompact space is extremally disconnected. The class of -open sets forms a topology on every space (X, ) and contains its co-semi-regularization. We show that a space is weakly Hausdorff iff it is almost weakly Hausdorff and (weakly) mildly Hausdorff.     

Best interpolation in a strip II: Reduction to unconstrained convex optimization

In this paper, we study the problem of finding a real-valued function f on the interval [0, 1] with minimal L ² norm of the second derivative that interpolates the points (t _i, y _i) and satisfies e(t) f(t) d(t) for t [0, 1]. The functions e and d are continuous in each interval (t _i, t _i+1) and at t ₁ and t _nbut may be discontinuous at t _i. Based on an earlier paper by the first author [7] we characterize the solution in the case when e and d are linear in each interval (t _i, t _i+1). We present a method for the reduction of the problem to a convex finite-dimensional unconstrained minimization problem. When e and d are arbitrary continuous functions we approximate the problem by a sequence of finite-dimensional minimization problems and prove that the sequence of solutions to the approximating problems converges in the norm of W ^2,2 to the solution of the original problem. Numerical examples are reported.The first author was supported by National Science Foundation Grant Number DMS 9404431. The second author was supported by a François-Xavier Bagnoud doctoral fellowship and by National Science Foundation Grant Number MSS 9114630.     

On Compactness with Respect to Countable Extensions of Ideals and the Generalized Banach Category Theorem

We extend a theorem of Hamlett and Jankovi by proving that if a topological space (X, ) is compact with respect to the countable extension of I, then the local function A ^*(I) of every subset A of X with respect to and I is a compact subspace with respect to the extension in A ^* (I). We also give a generalized version of the Banach category theorem.     

Bartle-Graves Theorem Revisited

We take a fresh look at the Bartle-Graves theorem pointing out the main differences with the standard implicit function theorem. We then present a set-valued version of this theorem which generalizes some recent results. Applications to variational inequalities and differential inclusions are also given.

     

Unification Approach to the Separation Axioms Between T 0 and Completely Hausdorff

The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between T ₁ and completely Hausdorff. We call a topological space (X, τ) a T _κ,ξ-space if every compact subset of X with cardinality ≦ κ is ξ-closed, where ξ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and T _1/3-spaces.     

Convergence of Newton's method for convex best interpolation

Summary. In this paper, we consider the problem of finding a convex function which interpolates given points and has a minimal norm of the second derivative. This problem reduces to a system of equations involving semismooth functions. We study a Newton-type method utilizing Clarke's generalized Jacobian and prove that its local convergence is superlinear. For a special choice of a matrix in the generalized Jacobian, we obtain the Newton method proposed by Irvine et al. [17] and settle the question of its convergence. By using a line search strategy, we present a global extension of the Newton method considered. The efficiency of the proposed global strategy is confirmed with numerical experiments. Received October 26, 1998 / Revised version received October 20, 1999 / Published online August 2, 2000     

Quadratic Convergence of Newton's Method for Convex Interpolation and Smoothing

Abstract. In this paper, we prove that Newton's method for convex best interpolation is locally quadratically convergent, giving an answer to a question of Irvine, Marin, and Smith [7] and strengthening a result of Andersson and Elfving [1] and our previous work [5]. A damped Newton-type method is presented which has global quadratic convergence. Analogous results are obtained for the convex smoothing problem. Numerical examples are presented.     

State constraints in the linear regulator problem: Case study

In this paper, we consider the problem of minimum-norm control of the double integrator with bilateral inequality constraints for the output. We approximate the constraints by piecewise linear functions and prove that the Langrange multipliers associated with the state constraints of the approximating problem are discrete measures, concentrated in at most two points in every interval of discretization. This allows us to reduce the problem to a convex finite-dimensional optimization problem. An algorithm based on this reduction is proposed and its convergence is examined. Numerical examples illustrate our approach. We also discuss regularity properties of the optimal control for a higher-dimensional state-constrained linear regulator problem.The first author was supported by the National Science Foundation, Grant No. DMS-9404431. The second author was supported by a François-Xavier Bagnoud Doctoral Fellowship and by NSF Grants DMS-9404431 and MSS-9114630.