Strong uniqueness

Abstract

In the present work a common basis of convergence analysis is given for a large class of iterative procedures which we call general approximation methods. The concept of strong uniqueness is seen to play a fundamental role. The broad range of applications of this proposed classification will be made clear by means of examples from various areas of numerical mathematics. Included in this classification are methods for solving systems of equations, the Remes algorithm, methods for nonlinear Chebyshev-approximation, the classical Newton method along with its variants such as Newton's method for partially ordered spaces and for degenerate tangent spaces. As an example of the latter the approximation with exponential sums having coalescing frequencies is discussed, that is the case where the tangent space is degenerate.

Higher-Order Minimizers and Generalized (F,<i>ρ</i>)-Convexity in Nonsmooth Vector Optimization over Cones

“…Antczak gave the notion of a locally Lipschitz ( ) , F ρ -convex scalar function of order k [6] and a differentiable ( ) , F ρ -convex vector function of order 2 [7]. L. Cromme [8] defined the concept of a strict local minimizer of order k for a scalar optimization problem. This concept plays a fundamental role in convergence analysis of iterative numerical methods [8] and in stability S. K. Suneja et al 8 results [9].…”

“…L. Cromme [8] defined the concept of a strict local minimizer of order k for a scalar optimization problem. This concept plays a fundamental role in convergence analysis of iterative numerical methods [8] and in stability S. K. Suneja et al 8 results [9]. The definition of a strict local minimizer of order 2 is generalized to the vectorial case by Antczak [7].…”

“…With the idea of analyzing the convergence and stability of iterative numerical methods, L. Cromme [8] introduced the notion of a "strict local minimizer of order k". As a recent advancement on this platform, Bhatia and Sahay [10] defined the concept of a higher-order strict minimizer with respect to a nonlinear function for a differentiable multiobjective optimization problem.…”

Weak Sharp Minima on Riemannian Manifolds

“…To the best of our knowledge, it is the first work concerning the notions of this type for optimization problems on spaces with no linear structure. Recall that the notion of sharp minima is introduced by Polyak [53] in the case of finite-dimensional Euclidean spaces for the analysis of perturbation behavior of optimization problems and the convergence analysis of some numerical algorithms; a related notion of "strongly unique local minimum" can be found in the paper by Cromme [18]. Then Ferris [30] introduces in the same framework the notion of weak sharp minima to describe an extension of sharp minimizers in order to include the possibility of multiple solutions.…”

“…The later notion has been extensively studied by many authors in finite-dimensional and infinite-dimensional linear spaces. Primary motivations for these studies relate to sensitivity analysis [14,15,36,41,52,61,62,63] and to convergence analysis of a broad range of optimization algorithms [10,11,16,18,31,32,34]. In particular, Burke and Ferris [10] derive necessary optimality conditions for weak sharp minimizers, obtaining also their full characterizations in the case of convex problems of unconstrained minimization, with applications to convex programming and convergence analysis in finite-dimensional Euclidean spaces.…”