Optimal Homotopy Analysis Method for Fractional-Order Gradient Based Dynamic System Generated by Optimization Problem

Abstract

We present optimal homotopy analysis method to generate an approximate analytic result for
fractional-order gradient based dynamic system generated by non-linear programming
constrained optimization problems. The constrained optimization problem is constructed in form
of a non-linear system of fractional differential equations and the solutions of the system,
modeled with conformable fractional derivative of steepest descent concept are investigated to
generate the minimizing point of the optimization problem. The construction extends the integer
order of the optimization problem to an arbitrary order which is very important to fill the gap
and create a path among two attracted research field, optimization and fractional calculus. We
determine the optimal values of the convergence-control parameters of the series solution by
using an optimization method of minimizing the squared residual error of the governing
equation. We show that optimal homotopy analysis method enables us to regulate and manage
the convergence domain of the series solution obtained by initiating convergence-control
parameters . Three exemplify examples were used to show the correctness and relevance of the
proposed techniques.