Quadratic Inner-Boundedness in Non-Smooth Analysis

• Quadratic inner-boundedness is a property that guarantees the finiteness of the Moreau envelope through sufficient quadratic regularization, key for nonconvex analysis.
• It employs the prox-threshold, derived from spectral properties of piecewise linear-quadratic functions, to determine the minimal regularization needed.
• Ensuring the quadratic parameter exceeds the prox-threshold leads to globally finite and smooth Moreau envelopes, enhancing algorithmic performance in variational analysis.

Quadratic inner-boundedness, also referred to as prox-boundedness in the context of non-smooth analysis and optimization, is a property of extended-real-valued functions ensuring the finiteness of the Moreau envelope under a quadratic penalization. This property, and the associated prox-threshold, play a central role in the analysis and algorithmic treatment of nonconvex functions—particularly, piecewise linear-quadratic (PLQ) functions that are ubiquitous in modern optimization and variational analysis. The prox-threshold effectively quantifies the minimal quadratic regularization required for the Moreau envelope of a function to be well-defined (i.e., not identically $-\infty$) at any point (Hare et al., 2016).

1. Definitions and Preliminaries

Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be proper and lower semicontinuous. For $r \geq 0$ and $\bar{x} \in \mathbb{R}^n$, the Moreau envelope of $f$ at $\bar{x}$ with prox-parameter $r$ is given by

$$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$

A function $f$ is prox-bounded (quadratic inner-bounded) if there exists some $r \geq 0$ and $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$0 such that $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$1. The prox-threshold $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$2 (also denoted $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$3) is defined as

$f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$4

The prox-threshold represents the minimal strength of the quadratic term that ensures the Moreau envelope is not everywhere $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$5.

2. Structure of Piecewise Linear-Quadratic Functions

A piecewise linear-quadratic (PLQ) function admits a finite partition of its domain into (closed) polyhedral sets $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$6, with the property that on each $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$7,

$f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$8

where $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$9 is symmetric. The overall function is the minimum (over the pieces) of these quadratic (or linear) forms, together with indicators for their respective domains. PLQ functions provide a tractable yet flexible class for variational analysis, encompassing both nonconvex and convex cases (Hare et al., 2016).

3. Computation of the Prox-Threshold

The computation of the prox-threshold $r \geq 0$0 for a PLQ function exploits the spectral properties of the quadratic forms on each piece:

• For each piece $r \geq 0$1, let $r \geq 0$2 be the local quadratic function, and let $r \geq 0$3 denote the smallest eigenvalue of $r \geq 0$4.
• The smallest prox-parameter $r \geq 0$5 for which the localized Moreau envelope $r \geq 0$6 is finite for some $r \geq 0$7 is $r \geq 0$8.
• The overall prox-threshold of the PLQ function is given by

$r \geq 0$9

This result follows from the spectral decomposition of the $\bar{x} \in \mathbb{R}^n$0 and the monotonicity of infimal convolution with respect to minimization over the pieces.

4. Behavior of the Moreau Envelope Near the Threshold

For $\bar{x} \in \mathbb{R}^n$1, the Moreau envelope $\bar{x} \in \mathbb{R}^n$2 is finite for all $\bar{x} \in \mathbb{R}^n$3, and its domain is the entire space. For $\bar{x} \in \mathbb{R}^n$4, $\bar{x} \in \mathbb{R}^n$5 for all $\bar{x} \in \mathbb{R}^n$6. At the threshold $\bar{x} \in \mathbb{R}^n$7, the behavior is more nuanced, with three possible scenarios:

• $\bar{x} \in \mathbb{R}^n$8 (e.g., when $\bar{x} \in \mathbb{R}^n$9 is convex and bounded below).
• $f$0 (e.g., a strictly linear PLQ piece).
• $f$1 is a proper, nonempty subset of $f$2 (e.g., a single quadratic piece with negative curvature).

If each active piece $f$3 has a unique minimizer, then $f$4. For convex $f$5, the map $f$6 is continuous and strictly decreasing in $f$7, and for $f$8, $f$9 is $\bar{x}$0-smooth (Hare et al., 2016).

5. Exemplary Case Study

Consider the nonconvex PLQ function on $\bar{x}$1:

$\bar{x}$2

For $\bar{x}$3: $\bar{x}$4, $\bar{x}$5, giving $\bar{x}$6 and $\bar{x}$7.

For $\bar{x}$8: $\bar{x}$9, $r$0, again $r$1 and $r$2.

Thus, $r$3.

Analysis by regime:

• For $r$4, $r$5 everywhere.
• For $r$6,

$r$7

which is finite for all $r$8 and continuous at $r$9.

• For $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$0, the envelope $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$1 is finite only at the vertex of each parabola: $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$2 at $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$3, $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$4 at $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$5; elsewhere, $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$6. Hence, $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$7 is not finite anywhere; $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$8.

A variant with $$e_r f(\bar{x}) = \inf_{y \in \mathbb{R}^n} \left\{f(y) + \frac{r}{2}\|y - \bar{x}\|^2\right\}.$$9 for all $f$0 yields $f$1, but $f$2 is finite only at $f$3.

6. Discussion and Context in Non-Smooth Analysis

Quadratic inner-boundedness governs the applicability of Moreau envelopes and, by extension, proximal point and splitting algorithms in non-smooth and nonconvex optimization. In particular, for PLQ functions, the prox-threshold provides both a theoretical bound and a computational tool for certifying regularization sufficiency. While $f$4 is always determined by the maximal negative curvature among the quadratic pieces, the finiteness domain of the Moreau envelope at the threshold can be highly sensitive to the alignment of envelope-minimizers across adjacent pieces. This sensitivity is especially relevant in algorithmic settings, where $f$5 secures global finiteness, but $f$6 may yield degeneracy or domain collapse (Hare et al., 2016).

• W. Hare and C. Planiden, "Thresholds of Prox-Boundedness of PLQ Functions," preprint (2014).
• R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Springer (1998).
• (Hare et al., 2016)