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Enclosing minima in nonsmooth optimization via trust regions of higher-order cutting-plane models

Bennet Gebken, Michael Ulbrich

Abstract

We propose a globally convergent trust-region bundle method for minimizing lower-$C^2$ functions using higher-order cutting-plane models. Under certain growth assumptions on the objective around its minimum, the method is able to compute infinitely many trust regions of decreasing size that contain the minimum. We show that these growth assumptions are satisfied for certain finite max-type functions with sharp or quadratic growth. Enclosing the minimum in this way can be used to initialize local superlinearly convergent methods, which we demonstrate in numerical experiments.

Enclosing minima in nonsmooth optimization via trust regions of higher-order cutting-plane models

Abstract

We propose a globally convergent trust-region bundle method for minimizing lower- functions using higher-order cutting-plane models. Under certain growth assumptions on the objective around its minimum, the method is able to compute infinitely many trust regions of decreasing size that contain the minimum. We show that these growth assumptions are satisfied for certain finite max-type functions with sharp or quadratic growth. Enclosing the minimum in this way can be used to initialize local superlinearly convergent methods, which we demonstrate in numerical experiments.
Paper Structure (12 sections, 12 theorems, 64 equations, 4 figures, 2 algorithms)

This paper contains 12 sections, 12 theorems, 64 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonsmooth analysis
  4. Higher-order cutting-plane models
  5. Algorithm and global convergence
  6. Property (P) for finite max-type functions
  7. Sharp growth
  8. Quadratic growth
  9. Numerical experiments
  10. Trust regions containing the minimum
  11. Combination with local R-superlinear methods
  12. Discussion and outlook

Key Result

Lemma 2.1

Let $q \in \mathbb{N}$ and assume that $f$ satisfies assum:B1. Then for every bounded set $V \subseteq \mathbb{R}^n$ and every $\hat{\Delta} > 0$ there is some $K \geq 0$ such that for all $x \in V$, $\Delta \in [0,\hat{\Delta}]$, and finite, nonempty sets $W \subseteq \bar{B}_\Delta(x)$.

Figures (4)

  • Figure 1: The graph of $f$ in Ex. \ref{['example:polynomial_growth_not_sufficient']} for $p \in \{1,2,4\}$. For $p = 4$, the small additional plot shows a zoom for better visualization of the local minima (and maxima).
  • Figure 2: The distance $\| x^{j,i} - x^* \|$ in Ex. \ref{['example:LW2019_84_global']} for $n = 50$, $m = 100$ (left) and $n = 50$, $m = 40$ (right). The horizontal axis enumerates the indices $(j,i)$ of the iterates $x^{j,i}$ in the order they are encountered in Alg. \ref{['algo:global_method']}, i.e., $(1,0), (1,1), \dots, (1,N_1), (2,0), (2,1), \dots$, with the iterates corresponding to $(x^j)_j$ (cf. Alg. \ref{['algo:global_method']}, Step \ref{['state:global_method_change_j']}) marked as circles. The dashed, vertical lines indicate changes of $j$ (from $(j,N_j)$ to $(j+1,0)$), and the horizontal, red lines show the trust-region radius for the corresponding $j$. (This means that visually, $x^* \in \bar{B}_{\Delta_j}(x^j)$ is equivalent to $\| x^j - x^* \|$ lying below the corresponding red horizontal line.)
  • Figure 3: The distance $\| x^{j,i} - x^* \|$ in Ex. \ref{['example:LW2019_85_global']} for $n = 25$, $m = 100$ (left) and $n = 50$, $m = 40$ (right), in the same style as in Fig. \ref{['fig:Example_LW2019_84_global']}.
  • Figure 4: (a) The distance $\| x^{j,i} - x^* \|$ in Ex. \ref{['example:LW2019_eigval_global']}, in the same style as in Fig. \ref{['fig:Example_LW2019_84_global']}. (b) The distance $\| x^{j,i} - x^* \|$ for Alg. \ref{['algo:global_method']} in Ex. \ref{['example:LW2019_eigval_local']} and the distance $\| \hat{x}^j - x^* \|$ for the sequence $(\hat{x}^j)_j$ generated by the local method from GU2026a, with initial point $x^1$ and initial trust-region radius $\Delta_1$. The red, dotted line shows the trust-region radius of the local method. (c) Same as (b), but for Ex. \ref{['example:LW2019_84_local']} and for the sequence of $\hat{x}$ generated via the $k$-bundle Newton method ($k$-BN) from LW2019.

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Example 4.1
  • Definition 4.1
  • Lemma 4.1
  • proof
  • ...and 20 more