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

Pengcheng Xie, Ya-xiang Yuan

TL;DR

2D-MoSub is a novel derivative-free optimization method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems.

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.

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

TL;DR

2D-MoSub is a novel derivative-free optimization method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems.

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.
Paper Structure (14 sections, 8 theorems, 49 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 8 theorems, 49 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. 2D-MoSub
  3. Initialization
  4. Constructing the interpolation set
  5. Constructing the quadratic interpolation models
  6. Obtaining $Q_k$ based on $Q_k^{\text{sub}}$
  7. Obtaining $Q_{k+1}^{\text{sub}}$ based on $Q_{k}$
  8. Obtaining $Q_{k}^{\rm mod}$ based on $Q_{k}$
  9. Trust-region trial step
  10. Updating the trust-region radius and subspace
  11. Poisedness and quality of the interpolation set
  12. Some properties of 2D-MoSub
  13. Numerical results
  14. Conclusion

Key Result

Theorem 3.4

At the step of constructing the quadratic interpolation model at each iteration, 2D-MoSub has the following Lagrange basis function for computing. In the case where $f(\bm{x}_{k}) \le f(\bm{y}_1^{(k)})$, it holds that and in the case where $f(\bm{x}_{k}) > f(\bm{y}_1^{(k)})$, it holds that where $c_0^{(1)}=f(\bm{x}_k),\ c_0^{(2)}=f(\bm{x}_k),\ c_0^{(3)}=f(\bm{x}_k)+a^{(k)}\Delta_k+b^{(k)}\Delta_

Figures (5)

  • Figure 1: The initial case and the subspace $\bm{x}_{1}+\text{span}\{\bm{d}_1^{(1)},\bm{d}_2^{(1)}\}$
  • Figure 2: The iterative case at the $k$-th step and the subspace $\bm{x}_{k}+\text{span}\{\bm{d}_1^{(k)},\bm{d}_2^{(k)}\}$
  • Figure 3: Different cases for $\bm{y}_1^{(k)},\bm{y}_2^{(k)},\bm{y}_3^{(k)}$
  • Figure 4: Performance profile of solving test large-scale problems
  • Figure 5: Data profile of solving test large-scale problems

Theorems & Definitions (24)

  • Remark 1
  • Definition 2.1
  • Remark 2
  • Definition 3.1: $\Lambda$-poisedness
  • Definition 3.2: Basis function
  • Definition 3.3: $\Lambda$-poisedness for 2-dimensional case with 3 determined coefficients
  • Remark 3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • ...and 14 more