**Problem Statement**

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

- an analogue to Slater's condition [1] that obtains strong duality for optimization of convex
*functionals*; with - 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**