On the convergence of optimum stratifications for empiric distribution function in univariate case

Summary Let {X _n },n=1,2,..., be a sequence of independent random variables distributed according to a distribution functionF(x) with finite variance,F _n (x) be the empiric distribution function ofX ₁,...,X _n for eachn, andφ _(n) ^* andφ ^* be optimum stratifications corresponding toF _n (x) andF(x) respectively. It is shown in this paper thatφ _(a) ^* tends almost surely toφ ^* under a suitable criterion. Institute of Statistical Mathematics     

On superlinear convergence in univariate nonsmooth minimization

This note demonstrates a new result on superlinear convergence in nonsmooth univariate minimization. It also gives other concepts of rapid convergence for minimization of functions that may have discontinuous derivatives.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Numbers AFOSR-83-0210 and AFOSR-88-0180.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

On univariate slash distributions,continuous and discrete

In this article, I explore in a unified manner the structure of uniform slash and \(\alpha \)-slash distributions which, in the continuous case, are defined to be the distributions of Y / U and \( Y_\alpha /U^{1/\alpha }\) where Y and \(Y_\alpha \) follow any distribution on \(\mathbb {R}^+\) and, independently, U is uniform on (0, 1). The parallels with the monotone and \(\alpha \)-monotone distributions of \( Y \times U\) and \(Y_\alpha \times U^{1/\alpha }\), respectively, are striking. I also introduce discrete uniform slash and \(\alpha \)-slash distributions which arise from a notion of negative binomial thinning/fattening. Their specification, although apparently rather different from the continuous case, seems to be a good one because of the close way in which their properties mimic those of the continuous case.

     

On singular univariate specializations of bivariate hypergeometric functions

It is tempting to evaluate F₂(x,1) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x=0 straight away using the Gauss formula for ₂F₁(1). But this kind of naive evaluation can lead to errors as the ₂F₁(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications by concrete examples.     

On Bellman function for extremal problems in BMO

In this Note we describe our results on construction of the Bellman function solving an extremal problem for a large class of integral functionals on BMO.     

On the optimal stability of bases of univariate functions

Summary. This paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation. The only bases considered are those whose elements have no sign changes. Among these, an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval. A uniqueness result and many examples of such optimally stable bases are also provided. Received May 26, 2000 / Published online August 17, 2001     

On the natural merit function for solving complementarity problems

Complementarity problems may be formulated as nonlinear systems of equations with non-negativity constraints. The natural merit function is the sum of squares of the components of the system. Sufficient conditions are established which guarantee that stationary points are solutions of the complementarity problem. Algorithmic consequences are discussed.     

On identification in fuzzy systems and its applications in control problems

The problem of identification in fuzzy systems described by the use of fuzzy equation is considered. The identification method and its performance index is also presented. The formal procedure of the identification algorithm is illustrated by means of a numerical example. The possibility of using the proposed algorithm for the solution of a control problem is given as well.     

On the identification of coefficients of elliptic problems by asymptotic regularization

The problem of recovering coefficients of elliptic problems from measured data is considered. An algorithm is developed to identify the unknown coefficients without a minimization technique. The method is based on the construction of certain time-dependent problems which contain the original equation as asymptotic steady state. A Liapunovtype a-priori estimate is fundamental to prove that the solution of the time-dependent regularized equations approach a solution of the original problem as t →∞. A related behavior is proved for the solution of corresponding finite-dimensional Galerkin approximations. A stability result is proved for the Galerkin approximations.     

On the accurate identification of active set for constrained minimax problems

In this paper, the problem of identifying the active constraints for constrained nonlinear programming and minimax problems at an isolated local solution is discussed. The correct identification of active constraints can improve the local convergence behavior of algorithms and considerably simplify algorithms for inequality constrained problems, so it is a useful adjunct to nonlinear optimization algorithms. Facchinei et al. [F. Facchinei, A. Fischer, C. Kanzow, On the accurate identification of active constraints, SIAM J. Optim. 9 (1998) 14-32] introduced an effective technique which can identify the active set in a neighborhood of a solution for nonlinear programming. In this paper, we first improve this conclusion to be more suitable for infeasible algorithms such as the strongly sub-feasible direction method and the penalty function method. Then, we present the identification technique of active constraints for constrained minimax problems without strict complementarity and linear independence. Some numerical results illustrating the identification technique are reported.     

On Timonov's algorithm for global optimization of univariate Lipschitz functions

Timonov proposes an algorithm for global maximization of univariate Lipschitz functions in which successive evaluation points are chosen in order to ensure at each iteration a maximal expected reduction of the region of indeterminacy, which contains all globally optimal points. It is shown that such an algorithm does not necessarily converge to a global optimum.     

On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function

In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nehari manifold and by exploring some properties of the best Hardy-Sobolev constant together with an approach developed by Brezis and Nirenberg.     

On fourth-order elliptic boundary value problems with nonmonotone nonlinear function

This paper is concerned with the fourth-order elliptic boundary value problems with nonmonotone nonlinear function. The existence and uniqueness of a solution is proven by the method of upper and lower solutions. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution.     

On a Lagrangian penalty function method for nonlinear programming problems

We modify a Lagrangian penalty function method proposed in [4] for constrained convex mathematical programming problems in order to obtain a geometric rate of convergence. For nonconvex problems we show that a special case of the algorithm in the above paper is still convergent without coercivity and convexity assumptions.On leave from the Institute of Mathematics, Hanoi, by a grant from Alexander-von-Humboldt-Stiftung.