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A line search filter sequential adaptive cubic regularisation algorithm for nonlinearly constrained optimization

Yonggang Pei, Jingyi Wang, Shaofang Song, Qinghui Gao, Detong Zhu

TL;DR

The paper addresses nonlinear equality constrained optimization by proposing LsFSARC, a penalty-free line search filter sequential ARC method. It combines composite steps with a projective null-space projection $P(x) = I - A(x)^T(A(x)A(x)^T)^{-1}A(x)$ to decompose the search direction into a normal step $n_k$ that reduces constraint violation and a tangential step $t_k$ derived from a standard ARC subproblem in the null space. Feasibility and optimality are pursued via a backtracking line search with a filter, and the adaptive cubic regularization parameter is updated based on the ratio of objective to model decrease. Global convergence is established under mild assumptions, with numerical results on the CUTEst suite showing competitive performance against existing penalty-free and penalty-based methods, illustrating the practicality of a penalty-free, ARC-based approach for constrained problems.

Abstract

In this paper, a sequential adaptive regularization algorithm using cubics (ARC) is presented to solve nonlinear equality constrained optimization. It is motivated by the idea of handling constraints in sequential quadratic programming methods. In each iteration, we decompose the new step into the sum of the normal step and the tangential step by using composite step approaches. Using a projective matrix, we transform the constrained ARC subproblem into a standard ARC subproblem which generates the tangential step. After the new step is computed, we employ line search filter techniques to generate the next iteration point. Line search filter techniques enable the algorithm to avoid the difficulty of choosing an appropriate penalty parameter in merit functions and the possibility of solving ARC subproblem many times in one iteration in ARC framework. Global convergence is analyzed under some mild assumptions. Preliminary numerical results and comparison are reported.

A line search filter sequential adaptive cubic regularisation algorithm for nonlinearly constrained optimization

TL;DR

The paper addresses nonlinear equality constrained optimization by proposing LsFSARC, a penalty-free line search filter sequential ARC method. It combines composite steps with a projective null-space projection to decompose the search direction into a normal step that reduces constraint violation and a tangential step derived from a standard ARC subproblem in the null space. Feasibility and optimality are pursued via a backtracking line search with a filter, and the adaptive cubic regularization parameter is updated based on the ratio of objective to model decrease. Global convergence is established under mild assumptions, with numerical results on the CUTEst suite showing competitive performance against existing penalty-free and penalty-based methods, illustrating the practicality of a penalty-free, ARC-based approach for constrained problems.

Abstract

In this paper, a sequential adaptive regularization algorithm using cubics (ARC) is presented to solve nonlinear equality constrained optimization. It is motivated by the idea of handling constraints in sequential quadratic programming methods. In each iteration, we decompose the new step into the sum of the normal step and the tangential step by using composite step approaches. Using a projective matrix, we transform the constrained ARC subproblem into a standard ARC subproblem which generates the tangential step. After the new step is computed, we employ line search filter techniques to generate the next iteration point. Line search filter techniques enable the algorithm to avoid the difficulty of choosing an appropriate penalty parameter in merit functions and the possibility of solving ARC subproblem many times in one iteration in ARC framework. Global convergence is analyzed under some mild assumptions. Preliminary numerical results and comparison are reported.
Paper Structure (8 sections, 16 theorems, 109 equations, 2 figures, 1 algorithm)

This paper contains 8 sections, 16 theorems, 109 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Line search filter sequential ARC algorithm
  3. Global convergence
  4. Preliminary results
  5. Feasibility
  6. Optimality
  7. Numerical Results
  8. Conclusion and discussion

Key Result

Lemma 3.1

Suppose that $k\notin\mathcal{A}_{\rm inc}$, and the step $t_k^c$ is the Cauchy step for 99. Then for all $k\geq0$.

Figures (2)

  • Figure 4.1: Performance profiles based on NF (left) and NC (right)
  • Figure 4.2: Performance profiles based on NF (left) and NG (right)

Theorems & Definitions (16)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.1
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Lemma 3.9
  • ...and 6 more