A Modified q-BFGS Algorithm for Unconstrained Optimization

Abstract

This paper presents a modification of the q-BFGS method for nonlinear unconstrained optimization problems. For this modification, we use a simple symmetric positive definite matrix and propose a new q-quasi-Newton equation, which is close to the ordinary q-quasi-Newton equation in the limiting case. This method uses only first order q-derivatives to build an approximate q-Hessian over a number of iterations. The q-Armijo-Wolfe line search condition is used to calculate step length, which guarantees that the objective function value is decreasing. This modified q-BFGS method preserves the global convergence properties of the q-BFGS method, without the convexity assumption on the objective function. Numerical results on some test problems are presented, which show that an improvement has been achieved. Moreover, we depict the numerical results through the performance profiles.

Keywords

• BFGS method
• q-calculus
• unconstrained optimization
• global convergence