An Adaptive Order Caputo Fractional Gradient Descent Method for Multi-objective Optimization Problems

TL;DR

This paper presents a novel adaptive order Caputo fractional gradient descent algorithm for multi-objective optimization that is effective for both smooth and non-smooth problems without requiring prior parameter tuning.

Contribution

The paper introduces the MOAOCFGD algorithm, which adaptively uses fractional gradients for multi-objective problems without needing pre-set parameters or objective ordering.

Findings

Effective for smooth and non-smooth problems

No need for pre-specified parameters

Converges under mild assumptions

Abstract

This article introduces the multi-objective adaptive order Caputo fractional gradient descent (MOAOCFGD) algorithm for solving unconstrained multi-objective problems. The proposed method performs equally well for both smooth and non-smooth multi-objective optimization problems. Moreover, the proposed method does not require any a priori chosen parameters or ordering information of the objective functions. At every iteration of the proposed method, a subproblem is solved to identify a suitable descent direction toward an optimal solution. This subproblem involves an adaptive-order Caputo fractional gradient for each objective function. An Armijo-type line search is applied to determine a suitable step length. The convergence of this method for the Tikhonov-regularized solution is justified under mild assumptions. The proposed method is verified using different numerical problems, including neural networks.