SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN Cn

Some previous results on convergence of Taylor series in Cn [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in Cn are constructed and the Taylor series expansion is deduced.     

An approximate subgradient algorithm for unconstrained nonsmooth,nonconvex optimization

In this paper a new algorithm for minimizing locally Lipschitz functions is developed. Descent directions in this algorithm are computed by solving a system of linear inequalities. The convergence of the algorithm is proved for quasidifferentiable semismooth functions. We present the results of numerical experiments with both regular and nonregular objective functions. We also compare the proposed algorithm with two different versions of the subgradient method using the results of numerical experiments. These results demonstrate the superiority of the proposed algorithm over the subgradient method.      

New results on input-to-state convergence for recurrent neural networks with variable inputs

In this paper, we study a class of recurrent neural networks (RNNs) arising from optimization problems. By constructing appropriate Lyapunov functions, we prove two new results on input-to-state convergence of RNNs with variable inputs. Numerical simulations are also given to demonstrate the convergence of the solutions.     

Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

Global solutions of mathematical programs with intrinsically concave functions

An implicit enumeration technique for solving a certain type of nonconvex program is described. The method can be used for solving signomial programs with constraint functions defined by sums of quasiconcave functions and other types of programs with constraint functions called intrinsically concave functions. A signomial-type example is solved by this method. The algorithm is described together with a convergence proof. No computational results are available at present.     

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

This paper extends the full convergence of the steepest descent method with a generalized Armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete Riemannian manifolds. Previous convergence results are obtained as particular cases and some examples in non-Euclidian spaces are given. In particular, our approach can be used to solve constrained minimization problems with nonconvex objective functions in Euclidian spaces if the set of constraints is a Riemannian manifold and the objective function is quasiconvex in this manifold.     

Local properties of inexact methods for minimizing nonsmooth composite functions

Methods for minimization of composite functions with a nondifferentiable polyhedral convex part are considered. This class includes problems involving minimax functions and norms. Local convergence results are given for “active set” methods, in which an equality-constrained quadratic programming subproblem is solved at each iteration. The active set consists of components of the polyhedral convex function which are active or near-active at the current iteration. The effects of solving the subproblem inexactly at each iteration are discussed; rate-of-convergence results which depend on the degree of inexactness are given.     

Continuity Properties of Walras Equilibrium Points

We explore convergence notions for bivariate functions that yield convergence and stability results for their max/inf points. The results are then applied to obtain continuity results for Walras equilibrium points under perturbations of the utility functions of the agents.     

New Version of the Newton Method for Nonsmooth Equations

In this paper, an inexact Newton scheme is presented which produces a sequence of iterates in which the problem functions are differentiable. It is shown that the use of the inexact Newton scheme does not reduce the convergence rate significantly. To improve the algorithm further, we use a classical finite-difference approximation technique in this context. Locally superlinear convergence results are obtained under reasonable assumptions. To globalize the algorithm, we incorporate features designed to improve convergence from an arbitrary starting point. Convergence results are presented under the condition that the generalized Jacobian of the problem function is nonsingular. Finally, implementations are discussed and numerical results are presented.     

On the rate of convergence of infinite horizon discounted optimal value functions

In this paper we investigate the rate of convergence of the optimal value function of an infinite horizon discounted optimal control problem as the discount rate tends to zero. Using the Integration Theorem for Laplace transformations we provide conditions on averaged functionals along suitable trajectories yielding quadratic pointwise convergence. From this we derive under appropriate controllability conditions criteria for linear uniform convergence of the value functions on control sets. Applications of these results are given and an example is discussed in which both linear and slower rates of convergence occur depending on the cost functional.