On Linear Convergence of Gradient-Type Minimization Algorithms

Abstract

Given the minimization problem of a real-valued function $f\left( x \right),x \in \Re ^n$ let A be any algorithm of type $x_{i + 1} + \lambda _i h_i$ with $\lambda _i\in \Re ,h_i \in \Re ^\mathfrak{n} ,$ $- h_i^T \nabla f\left( {x_i } \right) \geqslant \rho \left\| {h_i } \right\|\left\| {\nabla f\left( {x_i } \right)} \right\|,\rho\in \left( {{\text{0,1}}} \right)$ that converges to a local minimum $x^* \in f\left( x \right)$. In this note, new assumptions on f(x) under which A converges linearly to x* are established. These include the ones introduced in the literature which involve the uniform convexity of f(x).