The subject of linear convergence of gradient-type methods on non-strongly convex optimization has been widely studied by introducing several notions as sufficient conditions. Influential examples include the error bound property, the restricted strongly convex property, the quadratic growth property, and the KurdykaLojasiewicz property. In this paper, we first define a group of error bound con...

The purpose of this thesis is to give a new construction for central extensions of certain classes of infinite dimensional Lie algebras which include multiloop Lie algebras as motivating examples. The key idea of this construction is to view multiloop Lie algebras as twisted forms. This perspective provides a beautiful bridge between infinite dimensional Lie theory and descent theory and is cru...

We present sufficient conditions under which effective descent morphisms in a quasivariety of universal algebras are the same as regular epimorphisms and examples for which they are the same as regular epimorphisms satisfying projectivity. 1. Preliminaries A variety is a full subcategory of the category of structures for a first order (one sorted) language, closed under substructures, products ...

In two recent papers regularization methods based on Taylor polynomial models for minimization were proposed that only rely on Hölder conditions on the higher order employed derivatives. Grapiglia and Nesterov considered cubic regularization with a sufficient descent condition that uses the current gradient and resembles the classical Armijo’s criterion. Cartis, Gould, and Toint used Taylor mod...

Sufficient number of linear and noisy measurements for exact and approximate sparsity pattern/support set recovery in the high dimensional setting is derived. Although this problem as been addressed in the recent literature, there is still considerable gaps between those results and the exact limits of the perfect support set recovery. To reduce this gap, in this paper, the sufficient con...

In this paper, We propose a new nonlinear conjugate gradient method (FRA) that satisfies sufficient descent condition and global convergence under the inexact line search of strong wolf powell. Our numerical experiment shaw efficiency in solving set problems from CUTEst package, proposed formula gives excellent results at CPU time, number iterations, ratings when compared to WYL, DY, PRP, FR me...

In this paper, we propose a new nonlinear conjugate gradient method for large-scale unconstrain optimization which possesses the following properties:(i)the sufficient descent condition −g k dk ≥ 7 8 ‖gk‖ 2 holds without any line searches;(ii)With exact line search, this method reduces to a nonlinear version of the Liu-Storey conjugate gradient scheme.(iii)Under some assumption, global converge...