A secant-based Nesterov method for convex functions

Abstract

A simple secant-based fast gradient method is developed for problems whose objective function is convex and well-defined. The proposed algorithm extends the classical Nesterov gradient method by updating the estimate-sequence parameter with secant information whenever possible. This is achieved by imposing a secant condition on the choice of search point. Furthermore, the proposed algorithm embodies an "update rule with reset" that parallels the restart rule recently suggested in O' Donoghue and Candes (2013). The proposed algorithm applies to a large class of problems including logistic and least-square losses commonly found in the machine learning literature. Numerical results demonstrating the efficiency of the proposed algorithm are analyzed with the aid of performance profiles.Keywords Convex optimization • Secant Methods • Fast gradient methods • Nesterov gradient method. IntroductionThis paper considers the unconstrained optimization of convex function f :where f : R n → R is a continuously differentiable convex function. The domain of f , dom f , is the convex set R n and x is a real vector. A necessary and sufficient condition for a point x * to be a minimizer of f is ∇f (x * ) = 0 [7]. Furthermore, it is assumed that f is bounded below and there exists a unique minimizer x * : f (x * ) = f * with f * ≤ f (x) ∀ x ∈ R n [27].

Maximal Oxygen Uptake in Breathing Exercises and Heart Rate Exercises Based on In-Depth Regression Equations

“…If the relative value is the target, the predictive significance is questioned. (2) e relative value is to exclude the interference of body weight on oxygen uptake, but according to the above statistical analysis, although the relative value has been used as the dependent variable, BMI still enters the prediction equation as a strongly correlated independent variable, which is contrary to the theoretical significance of prediction [24]. erefore, from a statistical point of view, the relative value is not suitable as a dependent variable.…”