# A strong duality for convex functional optimization that admits Lipschitz continuity constraints?

Problem Statement

I am looking for formal proof---hopefully textbook material---of two items:

1. an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; with
2. proof that it admits systems of Lipschitz continuity [2] and linear constraints.

By Lipschitz continuity "constraint" I mean a constraint of the form $${\rm Lip}(f) \leq K$$ where we conceptualize $${\rm Lip}$$ as an "extended Lipschitz functional" such that $${\rm Lip}(f)$$ is the Lipschitz constant of $$f$$ or else $$\infty$$ if the function is not Lipschitz.

A basic problem formulation would be something like

\begin{align*} \text{Infimum} & \int f(t) \ {\rm d}c(t) \\ \text{subject to: } & \\ & {\rm Lip}(f) \leq K \\ & \int f(t) \ {\rm d} {\bf a}(t) = {\bf b} \end{align*}

where $${\bf a}$$ is a vector function of fixed dimension and $${\bf b}$$ is an associated vector constant. (The linear constraint is given as a Lebesgue–Stieltjes integral [3] to admit point constraints such as bound endpoints, etc.)

A Slater's condition here would allow traction by basic dual problem analysis, which would be very helpful in some of my research.

I feel that my conjecture above must be true, and simple, and useful, so I am confident it is known and I am simply not using the right language to search. I have skimmed a number of functional analysis texts, expecting bits to appear as exercises for the reader, but Lipschitz constraints especially seem to be treated quite rarely.

Notes

The ordinary Slater's sufficient condition requires (i) convexity of the constraint set, and (ii) a non-empty relative interior.

Convexity seems obvious for linear constraints and for Lipschitz constraints by a superposition argument (at least in any sufficiently useful function space); it seems trivial by definition of the Lipschitz constant that

$${\rm Lip}(f_1) \leq K_1, \, {\rm Lip}(f_2) \leq K_2 \implies \alpha \ {\rm Lip}(f_1) + \beta \ {\rm Lip}(f_2) \leq \alpha K_1 + \beta K_2$$ for all $$(\alpha, \beta)$$ in the $$2$$-simplex.

What is less obvious is that the intersection of a "Lipschitz cone" with an affine functional subspace has non-empty relative interior; but it does seem true.

References