Capra Conjugacy in Convex Analysis

• Capra conjugacy is a generalized convex duality framework that normalizes functions constant along rays, revealing hidden convexity in intrinsically nonconvex settings.
• It overcomes limitations of classical Fenchel duality for 0-homogeneous functions like the ℓ0 pseudonorm, thereby supporting new approaches in sparse optimization.
• Its extension to matrix functions enables convex reformulations for rank minimization, offering novel algorithmic strategies in applied mathematics and signal processing.

Capra conjugacy is a generalized convex duality framework, introduced to address the limitations of classical Fenchel duality for functions exhibiting invariance along primal rays in normed vector spaces. The Capra conjugacy, defined via a coupling that is constant along rays (hence "Constant Along Primal Rays," or Capra), yields nontrivial duality results and reveals hidden convexity structures for intrinsically nonconvex, 0-homogeneous, or support-based functions such as the $\ell_0$ pseudonorm and rank function. This concept has led to new theoretical and algorithmic approaches for sparse and low-rank optimization in convex analysis and applied mathematics.

1. Definition of Capra Coupling and Capra Conjugacy

Let $X = \mathbb{R}^d$ be equipped with a source norm $\|\cdot\|$ and dual norm $\|\cdot\|_*$. The Capra coupling is defined for $x, y \in \mathbb{R}^d$ as

$$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$

This coupling is homogeneous of degree zero in $x$: $C(\lambda x, y) = C(x, y)$ for all $\lambda > 0$, making it constant along rays. The associated Capra conjugate and biconjugate of a function $f: \mathbb{R}^d \to \overline{\mathbb{R}}$ are

$X = \mathbb{R}^d$0

A function $X = \mathbb{R}^d$1 is called Capra-convex if $X = \mathbb{R}^d$2 (Chancelier et al., 2020, Chancelier et al., 2020, Franc et al., 8 Sep 2025).

2. Motivation: Failure of Fenchel Conjugacy for 0-Homogeneous and Support-Based Functions

For functions $X = \mathbb{R}^d$3 that are 0-homogeneous or constant along rays, such as the $X = \mathbb{R}^d$4 pseudonorm or indicator functions of level sets of support, the Fenchel conjugacy is degenerate. For instance, the Fenchel conjugate of the indicator of $X = \mathbb{R}^d$5 is the zero function, identifying all sparse-structure functions as zero. Similarly, the Fenchel biconjugate of $X = \mathbb{R}^d$6 is identically zero, which precludes classical convex-analytic tools for analyzing sparsity and support-based structures (Chancelier et al., 2020, Chancelier et al., 2019). The Capra conjugacy overcomes this by factoring out the ray-invariance intrinsic to 0-homogeneous functions via normalization.

3. Fundamental Results: Capra-Convexity, Biconjugacy, and Hidden Convexity

The key theorem asserts that if both the source norm $X = \mathbb{R}^d$7 and its dual $X = \mathbb{R}^d$8 are orthant-strictly monotonic (e.g., $X = \mathbb{R}^d$9 for $\|\cdot\|$0), then any nondecreasing, finite-valued function of the support, $\|\cdot\|$1, is Capra-convex: $\|\cdot\|$2 For the $\|\cdot\|$3 pseudonorm, this gives

$\|\cdot\|$4

with the Capra conjugate

$\|\cdot\|$5

where $\|\cdot\|$6 is a "coordinate-$\|\cdot\|$7" dual norm specific to the support size (Chancelier et al., 2020, Chancelier et al., 2020, Chancelier et al., 2020, Franc et al., 8 Sep 2025, Franc et al., 2021).

Capra-convexity implies that functions which are highly nonconvex in the classical sense (e.g., cardinality) become the exact representative of their own Capra biconjugate.

A striking consequence is hidden convexity: every such $\|\cdot\|$8 coincides with a proper convex lower semicontinuous function $\|\cdot\|$9 composed with the normalization mapping,

$\|\cdot\|_*$0

where $\|\cdot\|_*$1 is convex and lsc. On the unit sphere, $\|\cdot\|_*$2 and $\|\cdot\|_*$3 coincide (Chancelier et al., 2020, Chancelier et al., 2020, Chancelier et al., 2019).

4. Capra-Subdifferential and Variational Representations

The Capra-subdifferential of a function $\|\cdot\|_*$4 at $\|\cdot\|_*$5 is

$\|\cdot\|_*$6

For $\|\cdot\|_*$7 and source norm $\|\cdot\|_*$8 with $\|\cdot\|_*$9, the Capra-subdifferential at $x, y \in \mathbb{R}^d$0 consists of $x, y \in \mathbb{R}^d$1 such that:

• $x, y \in \mathbb{R}^d$2, where $x, y \in \mathbb{R}^d$3,
• certain monotonicity and ordering conditions on the dual coordinates, resulting in a convex, nonempty, but not generally linear set (Franc et al., 2021).

Capra-convexity allows for variational formulations: $x, y \in \mathbb{R}^d$4 where $x, y \in \mathbb{R}^d$5 are supported on $x, y \in \mathbb{R}^d$6, and $x, y \in \mathbb{R}^d$7 are generalized local $x, y \in \mathbb{R}^d$8-support dual norms. This representation is convex and finite-dimensional, making it suitable for optimization (Chancelier et al., 2020, Chancelier et al., 2020, Chancelier et al., 2020, Chancelier et al., 2019).

5. Capra-Convex Sets and Geometric Characterization

A set $x, y \in \mathbb{R}^d$9 is Capra-convex if its indicator is Capra-convex: $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$0. Any Capra-convex set is necessarily a cone (closed under positive scaling), though not necessarily classically convex or closed.

The main theorem is that $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$1 is Capra-convex if and only if:

• $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$2 is a cone,
• $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$3, where $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$4. The directions of $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$5 (its intersection with the unit sphere) form a closed convex subset. Closed convex cones are always Capra-convex; so are conical hulls of spherically-convex subsets of the unit sphere (Franc et al., 8 Sep 2025).

These properties offer a sharp geometric criterion for Capra-convexity, distinct from classical convex sets.

6. Extensions: Matrix Functions and Rank-Based Capra Conjugacy

Capra conjugacy extends to matrix spaces, replacing the scalar product with the trace. For a matrix norm $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$6, the Capra coupling is

$$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$7

For the rank function, the Capra conjugate is

$$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$8

with generalized $$C(x, y) = \begin{cases} \dfrac{\langle x, y \rangle}{\|x\|}, & x \neq 0, \ 0 & x = 0. \end{cases}$$9-rank dual norms. The Capra biconjugate gives a variational lower bound (tight for the Frobenius norm), providing a convex formulation for the rank: $x$0 when $x$1 is the Frobenius norm (Barbier et al., 2021).

7. Applications and Implications for Sparse and Low-Rank Optimization

The Capra framework supports exact convex-analytic reformulations for cardinality-constrained or penalized problems common in statistics, machine learning, and signal processing. For example, minimizing $x$2 over an affine set or the $x$3-sparse least-squares problem can be recast as minimizing a proper convex function over the unit sphere using hidden convexity: $x$4 where $x$5 is convex, enabling sphere-based or conic optimization techniques. Feature selection, compressive sensing, and low-rank matrix recovery now admit new algorithmic approaches based on convex minorants and variational formulations unlocked by Capra conjugacy (Franc et al., 8 Sep 2025, Chancelier et al., 2020, Chancelier et al., 2020).

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References:

• (Chancelier et al., 2020)
• (Chancelier et al., 2020)
• (Chancelier et al., 2020)
• (Franc et al., 2021)
• (Franc et al., 8 Sep 2025)
• (Barbier et al., 2021)
• (Chancelier et al., 2019)