---
title: Quadratic Inner-Boundedness in Non-Smooth Analysis
url: https://www.emergentmind.com/topics/quadratic-inner-boundedness
type: topic
---

# Quadratic Inner-Boundedness in Non-Smooth 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 [1611.00996].

## 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 $\bar{x}$ such that $e_r f(\bar{x}) > -\infty$. The prox-threshold $r^*$ (also denoted $\bar{r}$) is defined as
$$
r^* = \inf\left\{ r \geq 0 : \exists\, \bar{x} \ \text{with} \ e_r f(\bar{x}) > -\infty \right\}.
$$
The prox-threshold represents the minimal strength of the quadratic term that ensures the Moreau envelope is not everywhere $-\infty$.

## 2. Structure of Piecewise Linear-Quadratic Functions

A piecewise linear-quadratic (PLQ) function admits a finite partition of its domain into (closed) polyhedral sets $S_1, \ldots, S_m$, with the property that on each $S_i$,
$$
f(x) = \frac{1}{2} x^\top A_i x + b_i^\top x + c_i,
$$
where $A_i$ 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 [1611.00996].

## 3. Computation of the Prox-Threshold

The computation of the prox-threshold $r^*$ for a PLQ function exploits the spectral properties of the quadratic forms on each piece:
- For each piece $S_i$, let $q_i(x) = \frac{1}{2} x^\top A_i x + b_i^\top x + c_i$ be the local quadratic function, and let $\lambda_{\text{min}}(A_i)$ denote the smallest eigenvalue of $A_i$.
- The smallest prox-parameter $r_i$ for which the localized Moreau envelope $e_r q_i(\bar{x})$ is finite for some $\bar{x}$ is $r_i = \max\{0, -\lambda_{\text{min}}(A_i)\}$.
- The overall prox-threshold of the PLQ function is given by
$$
r^* = \max_{i = 1, \ldots, m} r_i.
$$

This result follows from the spectral decomposition of the $A_i$ and the monotonicity of infimal convolution with respect to minimization over the pieces.

## 4. Behavior of the Moreau Envelope Near the Threshold

For $r > r^*$, the Moreau envelope $e_r f(\bar{x})$ is finite for all $\bar{x} \in \mathbb{R}^n$, and its domain is the entire space. For $0 \leq r < r^*$, $e_r f(\bar{x}) = -\infty$ for all $\bar{x}$. At the threshold $r = r^*$, the behavior is more nuanced, with three possible scenarios:
- $\operatorname{dom} e_{r^*} f = \mathbb{R}^n$ (e.g., when $f$ is convex and bounded below).
- $\operatorname{dom} e_{r^*} f = \emptyset$ (e.g., a strictly linear PLQ piece).
- $\operatorname{dom} e_{r^*} f$ is a proper, nonempty subset of $\mathbb{R}^n$ (e.g., a single quadratic piece with negative curvature).

If each active piece $i \in I^* = \{i : r_i = r^*\}$ has a unique minimizer, then $e_{r^*} f(\bar{x}) = \min_{i \in I^*} e_{r^*} f_i(\bar{x})$. For convex $f$, the map $r \mapsto e_r f$ is continuous and strictly decreasing in $r$, and for $r > r^*$, $e_r f$ is $C^1$-smooth [1611.00996].

## 5. Exemplary Case Study

Consider the nonconvex PLQ function on $\mathbb{R}$:
$$
f(x) =
\begin{cases}
 -x^2 & \text{if } x \leq 0, \\
 -(x - 1)^2 & \text{if } x \geq 0.
\end{cases}
$$
For $x \leq 0$: $A_1 = -2$, $b_1 = 0$, giving $\lambda_{\text{min}}(A_1) = -2$ and $r_1 = 2$.

For $x \geq 0$: $A_2 = -2$, $b_2 = 2$, again $\lambda_{\text{min}}(A_2) = -2$ and $r_2 = 2$.

Thus, $r^* = \max(r_1, r_2) = 2$.

Analysis by regime:
- For $r < 2$, $e_r f(\bar{x}) = -\infty$ everywhere.
- For $r > 2$,
  $$
  e_r f(\bar{x}) =
  \begin{cases}
    -\frac{1}{2(r-2)} \bar{x}^2 & \text{if } \bar{x} \leq 0, \\
    -\frac{1}{2(r-2)}(\bar{x} - 1)^2 & \text{if } \bar{x} \geq 0,
  \end{cases}
  $$
  which is finite for all $\bar{x}$ and continuous at $\bar{x} = 0$.
- For $r = 2$, the envelope $e_2 f_i$ is finite only at the vertex of each parabola: $e_2 f_1$ at $\bar{x} = 0$, $e_2 f_2$ at $\bar{x} = 1$; elsewhere, $e_2 f_i(\bar{x}) = -\infty$. Hence, $e_2 f(\bar{x})$ is not finite anywhere; $\operatorname{dom} e_2 f = \emptyset$.

A variant with $g(x) = -x^2$ for all $x$ yields $r^* = 2$, but $e_2 g(\bar{x})$ is finite only at $\bar{x} = 0$.

## 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 $r^*$ 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 $r > r^*$ secures global finiteness, but $r = r^*$ may yield degeneracy or domain collapse [1611.00996].

## 7. References

- 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).
- arXiv:1611.00996

Source: https://www.emergentmind.com/topics/quadratic-inner-boundedness