A New Two-dimensional Model-based Subspace Method for Large-scale Unconstrained Derivative-free Optimization: 2D-MoSub

TL;DR

This paper introduces 2D-MoSub, a novel derivative-free optimization method that uses 2D quadratic models and trust-region techniques to efficiently solve large-scale unconstrained problems.

Contribution

The paper presents a new 2D model-based subspace method for large-scale DFO, including its framework, computational details, and convergence analysis.

Findings

Demonstrates effectiveness on various unconstrained problems

Shows improved efficiency over existing methods

Provides theoretical convergence guarantees

Abstract

This paper proposes the method 2D-MoSub (2-dimensional model-based subspace method), which is a novel derivative-free optimization (DFO) method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems. Our method combines 2-dimensional quadratic interpolation models and trust-region techniques to iteratively update the points and explore the 2-dimensional subspace. Its framework includes initialization, constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. We introduce the framework and computational details of 2D-MoSub, and discuss the poisedness and quality of the interpolation set in the corresponding 2-dimensional subspace. We also analyze some properties of our method, including the model's approximation error with projection property and the algorithm's convergence. Numerical results demonstrate the effectiveness and efficiency of 2D-MoSub for solving a variety of unconstrained optimization problems.