# An Improved Gradient Method with Approximately Optimal Stepsize Based on Conic model for Unconstrained Optimization

## TL;DR

This paper introduces an improved gradient optimization method that adaptively constructs conic or quadratic models to determine nearly optimal stepsizes, enhancing convergence efficiency for unconstrained problems.

## Contribution

It presents a novel gradient method utilizing conic models for stepsize selection, extending previous work and demonstrating promising numerical performance.

## Key findings

- Method outperforms some well-known conjugate gradient algorithms
- Constructs conic models when the objective is not quadratic
- Numerical results show high efficiency and promising convergence

## Abstract

A new type of stepsize, which was recently introduced by Liu and Liu (Optimization, 67(3), 427-440, 2018), is called approximately optimal stepsize and is quit efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, based on the work (Numer. Algorithms 78(1), 21-39, 2018), we present an improved gradient method with approximately optimal stepsize based on conic model for unconstrained optimization. If the objective function $ f $ is not close to a quadratic on the line segment between the current and latest iterates, we construct a conic model to generate approximately optimal stepsize for gradient method if the conic model can be used; otherwise, we construct some quadratic models to generate approximately optimal stepsizes for gradient method. The convergence of the proposed method is analyzed under suitable conditions. Numerical comparisons with some well-known conjugate gradient software packages such as CG$ \_ $DESCENT (SIAM J. Optim. 16(1), 170-192, 2005) and CGOPT (SIAM J. Optim. 23(1), 296-320, 2013) indicate the proposed method is very promising.