Exactness of Infimal Postcomposition in Convex Optimization

• The paper establishes that under a dual strong relative interior condition, the infimal postcomposition equals (f*∘L*)*, eliminating the need for biconjugate relaxation.
• Exactness of infimal postcomposition is defined as the transfer of convexity, lower semicontinuity, and subdifferential information from the original space to the image space.
• The methodology enables exact computation of the proximity operator through resolvent and Moreau-type formulas, ensuring that proximal subproblems are attained.

Searching arXiv for the primary paper and closely related work on infimal postcomposition and exactness. Exactness of infimal postcomposition concerns when the value function induced by a linear map is a genuine proper lower semicontinuous convex function, when its dual representation requires no relaxation, and when its associated proximal subproblems are attained. In the Hilbert-space framework developed in "Resolvent of the parallel composition and proximity operator of the infimal postcomposition" (Briceño-Arias et al., 2021), if $f\in\Gamma_0(\mathcal H)$ and $L:\mathcal H\to\mathcal G$ is linear and bounded, the infimal postcomposition is

$$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$

The central issue is whether this infimum defines an element of $\Gamma_0(\mathcal G)$ that coincides with $(f^*\circ L^*)^*$, and whether the proximity operator of $L\triangleright f$ can be computed exactly from a proximal problem posed on $\mathcal H$ rather than through a closure or relaxed dual object.

1. Definition and operator-theoretic setting

In real Hilbert spaces $\mathcal H$ and $\mathcal G$, with $f\in\Gamma_0(\mathcal H)$ and $L:\mathcal H\to\mathcal G$0 linear and bounded, the infimal postcomposition is the constrained value function

$L:\mathcal H\to\mathcal G$1

with the convention that the infimum of an empty set is $L:\mathcal H\to\mathcal G$2. Thus, if $L:\mathcal H\to\mathcal G$3, then $L:\mathcal H\to\mathcal G$4 (Briceño-Arias et al., 2021).

Its convex-analytic significance comes from conjugacy. Under mild assumptions, one has

$L:\mathcal H\to\mathcal G$5

The paper also recalls the link with monotone operator theory: $L:\mathcal H\to\mathcal G$6 under the qualification

$L:\mathcal H\to\mathcal G$7

This identifies the subdifferential of the infimal postcomposition with the parallel composition of $L:\mathcal H\to\mathcal G$8 by $L:\mathcal H\to\mathcal G$9, thereby placing exactness simultaneously in Fenchel duality and maximally monotone operator theory (Briceño-Arias et al., 2021).

This setting clarifies that infimal postcomposition is not merely an elimination of variables. It is a structural operation that transfers convexity, lower semicontinuity, and subdifferential information from $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$0 to $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$1 only when an appropriate qualification rules out nonattainment and hidden closures.

2. Exactness and the qualification condition

In this framework, exactness means three things. First, the value function $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$2 coincides with $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$3, so no biconjugate relaxation is needed. Second, $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$4, hence it is proper, convex, and lower semicontinuous. Third, the proximal subproblems used to represent the proximity operator are attained (Briceño-Arias et al., 2021).

The key condition is the dual strong relative interior assumption

$$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$5

Under this hypothesis, the paper establishes

$$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$6

and

$$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$7

so the parallel composition is maximally monotone and is the subdifferential of a bona fide convex function (Briceño-Arias et al., 2021).

A common misconception is that exactness essentially requires surjectivity of $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$8 or strong monotonicity of $$L\triangleright f:\mathcal G\to[-\infty,+\infty],\qquad u\mapsto \inf\{\,f(x)\mid x\in\mathcal H,\ Lx=u\,\}.$$9. The Hilbert-space results show otherwise. The qualification above is explicitly weaker than full range of $\Gamma_0(\mathcal G)$0 or $\Gamma_0(\mathcal G)$1, and it is contrasted with stronger assumptions used in parts of the ADMM literature, including single-valuedness, full domain, or strong monotonicity of operators such as $\Gamma_0(\mathcal G)$2 or $\Gamma_0(\mathcal G)$3 (Briceño-Arias et al., 2021). A plausible implication is that exactness should be viewed primarily as an interiority phenomenon in the dual geometry of $\Gamma_0(\mathcal G)$4 and $\Gamma_0(\mathcal G)$5, rather than as a consequence of global surjectivity.

3. Exact proximity formulas and resolvent computation

The exactness theory becomes operational through a generalized proximity operator. For $\Gamma_0(\mathcal G)$6, linear $\Gamma_0(\mathcal G)$7, and strongly monotone self-adjoint $\Gamma_0(\mathcal G)$8,

$\Gamma_0(\mathcal G)$9

where $(f^*\circ L^*)^*$0. It satisfies

$(f^*\circ L^*)^*$1

(Briceño-Arias et al., 2021).

Under the qualification

$(f^*\circ L^*)^*$2

Proposition 4.2 gives the exact formula

$(f^*\circ L^*)^*$3

Equivalently,

$(f^*\circ L^*)^*$4

The paper emphasizes that this is the proximity operator of the genuine infimal postcomposition $(f^*\circ L^*)^*$5, not of some closure or biconjugate, because the qualification ensures $(f^*\circ L^*)^*$6 (Briceño-Arias et al., 2021).

The operator-theoretic backbone is the resolvent formula for parallel composition. For a maximally monotone $(f^*\circ L^*)^*$7, the parallel composition is

$(f^*\circ L^*)^*$8

When $(f^*\circ L^*)^*$9 is maximally monotone and $L\triangleright f$0 is strongly monotone and self-adjoint, Corollary 3.3 yields

$L\triangleright f$1

Setting $L\triangleright f$2 and using $L\triangleright f$3 recovers the proximal formula above (Briceño-Arias et al., 2021).

The same argument yields the composite Moreau-type identity

$L\triangleright f$4

hence

$L\triangleright f$5

This generalizes Moreau’s decomposition from ordinary convex functions to the composite pair $L\triangleright f$6 (Briceño-Arias et al., 2021).

4. Failure of exactness and nonattainment phenomena

Without the qualification

$L\triangleright f$7

exactness can fail in several ways. The infimal postcomposition may fail to belong to $L\triangleright f$8, the proximal subproblem may have no minimizer for some $L\triangleright f$9, and the operator $\mathcal H$0 may correspond only to a relaxed problem rather than to $\mathcal H$1 (Briceño-Arias et al., 2021).

Example 4.3 makes this explicit with $\mathcal H$2, $\mathcal H$3, and $\mathcal H$4 not closed. For $\mathcal H$5, the infimum

$\mathcal H$6

is $\mathcal H$7 but is not attained. Example 4.4 provides a two-dimensional exponential example where the same qualification fails. These examples show that nonattainment is not a marginal technicality; it is precisely what the strong relative interior condition excludes (Briceño-Arias et al., 2021).

Another common misconception is that strong convexity of $\mathcal H$8 resolves the entire issue. The paper notes that if $\mathcal H$9 is strongly convex, then $\mathcal H$0 is strongly monotone, so the proximal subproblem has a unique minimizer even without the qualification. However, the qualification remains crucial to ensure that

$\mathcal H$1

so that the parallel composition is truly a subdifferential and not merely an operator related to a relaxed closure (Briceño-Arias et al., 2021). This suggests that exactness is logically distinct from uniqueness.

5. Finite-dimensional exactness in constrained and penalized least squares

A finite-dimensional perspective appears in "Subspace decomposition in regularized least-squares: solution properties, restricted coercivity and beyond" (Xue et al., 28 Jul 2025). For $\mathcal H$2, $\mathcal H$3, and $\mathcal H$4, the paper studies the penalized problem

$\mathcal H$5

and the constrained problem

$\mathcal H$6

In Section 3 it uses the infimal postcomposition $\mathcal H$7, defined by

$\mathcal H$8

Here exactness at a point $\mathcal H$9 means that this infimum is attained, equivalently that the constrained problem admits a solution (Xue et al., 28 Jul 2025).

The operator

$\mathcal G$0

encodes solvability. For $\mathcal G$1,

$\mathcal G$2

so exactness of $\mathcal G$3 at $\mathcal G$4 is equivalent to $\mathcal G$5 (Xue et al., 28 Jul 2025).

When $\mathcal G$6, Corollary 3.7 gives a clean domain characterization: $\mathcal G$7 Moreover, the paper shows that the penalized least-squares problem has a solution for every $\mathcal G$8 if and only if $\mathcal G$9 is exact in $f\in\Gamma_0(\mathcal H)$0 and $f\in\Gamma_0(\mathcal H)$1 is maximally monotone (Xue et al., 28 Jul 2025).

This finite-dimensional theory reframes exactness as the bridge between a reduced problem in the image space and the original optimization problem in $f\in\Gamma_0(\mathcal H)$2. The reduction

$f\in\Gamma_0(\mathcal H)$3

is only fully faithful when minimizers of $f\in\Gamma_0(\mathcal H)$4 can be lifted back to minimizers in the original variable $f\in\Gamma_0(\mathcal H)$5. In that sense, exactness is the condition that legitimizes variable elimination rather than merely computing a lower semicontinuous envelope.

6. Broader exactness paradigms

The exactness of infimal postcomposition also appears in more general settings where the “postcomposition” is not just by a linear operator between Hilbert spaces. In "Interchange Rules for Integral Functions" (Bùi et al., 2023), the central equality

$f\in\Gamma_0(\mathcal H)$6

is interpreted as exactness of infimal postcomposition with the integration operator. Under Assumption 1.1, compliance of $f\in\Gamma_0(\mathcal H)$7, and normality of $f\in\Gamma_0(\mathcal H)$8, Theorem 5.1 guarantees this interchange. The same framework then brings conjugates, subdifferentials, recessions, Moreau envelopes, and proximity operators under the integral sign, for example

$f\in\Gamma_0(\mathcal H)$9

and

$L:\mathcal H\to\mathcal G$00

This suggests that exactness is a unifying principle for when pointwise convex-analytic operations commute with aggregation (Bùi et al., 2023).

A related finite-dimensional but nonlinear direction appears in "A study of convex convex-composite functions via infimal convolution with applications" (Burke et al., 2019). There, exactness concerns infimal-convolution representations of convex convex-composite functions. Under a Slater-type condition such as

$L:\mathcal H\to\mathcal G$01

the paper proves the exact conjugacy formula

$L:\mathcal H\to\mathcal G$02

and corresponding exact subdifferential identities. Although the terminology differs, the governing theme is the same: a value function defined through a composite or infimal construction admits an unrelexed dual formula with attainment under a verifiable interiority condition (Burke et al., 2019).

Across these settings, exactness consistently denotes the absence of hidden closure, the validity of a sharp dual representation, and the attainment needed to reconstruct primal objects from reduced or transformed formulations. In the linear Hilbert-space case, this culminates in the identity

$L:\mathcal H\to\mathcal G$03

which may be viewed as the canonical exactness statement for infimal postcomposition: the proximity operator of the reduced value function is computed exactly, not approximately, from the original-space proximal problem (Briceño-Arias et al., 2021).