Relaxation in infinite convex programming under Slater-type regularity conditions

The main purpose of this paper is to close the gap between the optimal values of an infinite convex program and that of its biconjugate relaxation. It is shown that Slater and continuity-type conditions guarantee such a zero-duality gap. The approach uses calculus rules for the conjugation and biconjugation of the sum and pointwise supremum operations. A second important objective of this work is to exploit these results on relaxation by applying them in the context of duality theory.

💡 Research Summary

The paper addresses a fundamental issue in infinite convex programming: the possible gap between the optimal value of the original problem and that of its biconjugate (Fenchel‑Moreau) relaxation. Consider a real Banach space X, its dual X* and bidual X**, and a family of proper convex functions f₀, fₜ (t∈T) defined on X. The primal problem (P) is
 inf f₀(x) subject to fₜ(x) ≤ 0 for all t∈T.
Because the Fenchel biconjugate satisfies fₜ** ≤ fₜ, the standard biconjugate relaxation (P**) – obtained by replacing each fₜ with fₜ** and allowing decision variables in X** – always yields a value v(P**) ≤ v(P). When X is reflexive and each fₜ is lower‑semicontinuous, the Fenchel‑Moreau–Rockafellar theorem guarantees fₜ** = fₜ, so the gap disappears. However, in non‑reflexive spaces or when the index set T is infinite, the inequality can be strict; the authors illustrate this with concrete counter‑examples (Section 6).

To close the gap, the authors propose a strengthened relaxation, denoted (P**∞). First they define the pointwise supremum of the constraint functions
 f := supₖ≥1 fₖ,
and its lim‑sup envelope
 f^∞(x) := lim supₖ→∞ fₖ(x).
The new relaxation adds the biconjugate of f^∞ as an extra constraint:
 inf f₀**(z) subject to fₖ**(z) ≤ 0 (k≥1), f^{∞}(z) ≤ 0, z∈dom f_{w*}.
When the number of constraints is finite, f^∞ ≡ −∞, so (P**∞) collapses to the ordinary (P**); this is shown in Proposition 2.

The central result (Theorem 6) states that under two mild regularity conditions the optimal values of (P) and (P**∞) coincide:

1. Slater condition – there exists x₀ ∈ dom f₀ with supₜ fₜ(x₀) < 0.
2. Continuity of the supremum – the function f = supₖ fₖ is continuous at some point of dom f₀.

The proof proceeds as follows. Lemma 4 (a variant of the classical Lagrange multiplier theorem for infinite constraints) guarantees the existence of a non‑negative sequence λ ∈ ℓ₊¹ and a scalar λ^∞ ≥ 0 such that
 v(P) = infₓ