q-Line Search Scheme for Optimization Problem

TL;DR

This paper introduces new q-derivative based descent line search schemes for unconstrained and constrained optimization, achieving superlinear convergence without requiring second-order differentiability.

Contribution

It develops a novel iterative scheme using q-derivatives that does not rely on exact Hessians or second derivatives, expanding optimization methods' applicability.

Findings

Scheme preserves Newton-like convergence properties.

Achieves superlinear convergence near minima.

Numerical results demonstrate effectiveness.

Abstract

In this paper new descent line search iterative schemes for unconstrained as well as constrained optimization problems are developed using q-derivative. At every iteration of the scheme, a positive definite matrix is provided which is neither exact Hessian of the objective function as in Newton scheme nor the positive definite matrix as generated in quasi-Newton scheme. Second order differentiablity property is not required in this process. Component of this matrix are constructed using q-derivative of the function. It is proved that the schemes preserve the property of Newton-like schemes in a local neighborhood of a minimum point which leads to the super linear rate of convergence. Numerical illustration of the scheme is also provided.