This paper is a continuation of our previous paper were we presented generalizations of the Dennis-Mor\’e theorem to characterize q-superliner convergences of quasi-Newton methods for solving equations and variational inequalities in Banach spaces. Here we prove Dennis-Mor\’e type theorems for inexact quasi-Newton methods applied to variational inequalities in finite dimensions. We first consider variational inequalities … Read more
While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large consistent complex symmetric system, one may apply a non-Hermitian Krylov subspace method disregarding … Read more
In this paper, we consider the problem of general equilibrium in a finite-dimensional space on a closed convex set. For solving this problem, we developed an interior proximal point algorithm with phi-divergence. Under reasonable assumptions, we prove that the sequence generated by the algorithm converges to a solution of the Equilibrium Problem, when the regularization … Read more
An example is presented where Newton’s method for unconstrained minimization is applied to find an $\epsilon$-approximate first-order critical point of a smooth function and takes a multiple of $\epsilon^{-2}$ iterations and function evaluations to terminate, which is as many as the steepest-descent method in its worst-case. The novel feature of the proposed example is that … Read more
We describe the most recent evolution of our constrained and unconstrained testing environment and its accompanying SIF decoder. Code-named SIFDecode and CUTEst, these updated versions feature dynamic memory allocation, a modern thread-safe Fortran modular design, a new Matlab interface and a revised installation procedure integrated with GALAHAD. CitationTechnical Report Rutherford Appleton Laboratory Chilton, Oxfordshire, England, … Read more
We consider the problem of minimizing a convex objective which is the sum of a smooth part, with Lipschitz continuous gradient, and a nonsmooth part. Inspired by various applications, we focus on the case when the nonsmooth part is a composition of a proper closed convex function P and a nonzero affine map, with the … Read more
This paper presents four optimization models for solving nonlinear equation systems. The models accommodate both over-specified and under-specified systems. A variable endogenization technique that improves efficiency is introduced, and a basic comparative study shows one of the methods presented to be very effective. CitationSiwale, I. (2013). Solution of nonlinear equation systems via optimization. Technical Report … Read more
In this paper we analyze several new methods for solving nonconvex optimization problems with the objective function formed as a sum of two terms: one is nonconvex and smooth, and another is convex but simple and its structure is known. Further, we consider both cases: unconstrained and linearly constrained nonconvex problems. For optimization problems of … Read more
Kusuoka representations provide an important and useful characterization of law invariant coherent risk measures in atomless probability spaces. However, the applicability of these results is limited by the fact that such representations do not always exist in probability spaces with atoms, such as finite probability spaces. We introduce the class of functionally coherent risk measures, … Read more
We investigate constrained first order techniques for training Support Vector Machines (SVM) for online classification tasks. The methods exploit the structure of the SVM training problem and combine ideas of incremental gradient technique, gradient acceleration and successive simple calculations of Lagrange multipliers. Both primal and dual formulations are studied and compared. Experiments show that the … Read more