Normal cones to sublevel sets of convex and quasi-convex supremum functions

We provide sharp and explicit characterizations of the normal cone to sublevel sets of suprema of arbitrary functions, expressed exclusively in terms of subdifferentials of the data functions. In the convex case, the resulting formulas involve the approximate subdifferential of the individual data functions at the nominal point. In contrast, the quasi-convex framework requires the use of the Fréchet subdifferential of these data functions but evaluated at nearby points. These results are applied to derive optimality conditions for infinite convex and quasi-convex optimization problems.

💡 Research Summary

The paper addresses a fundamental problem in variational analysis and optimization: how to describe the normal cone to a sublevel set defined by a supremum (pointwise maximum) of possibly infinitely many functions, using only information from the individual data functions. The authors develop sharp, explicit characterizations for both convex and quasiconvex settings, and then apply these results to derive optimality conditions for infinite‑constraint optimization problems.

Motivation and background. In many optimization models the feasible set can be written as (C={x\in X\mid f(x)\le0}). First‑order optimality conditions, duality theory, sensitivity analysis, and higher‑order conditions are all expressed through the normal cone (N_C(x)). When the constraint is given by a family ({f_t}{t\in T}) of functions, it is convenient to replace the whole family by the pointwise supremum (f(x)=\sup{t\in T}f_t(x)). Consequently, a central question is how to compute (N_{{f\le0}}(x)) directly from the subdifferentials of the (f_t)’s. Earlier works either required convex combinations of the (f_t)’s or imposed restrictive continuity or qualification assumptions.

Main contributions – convex case. Let each (f_t:X\to\mathbb{R}) be convex (not necessarily lower semicontinuous) and define (f=\sup_{t\in T}f_t). Assuming the mild lower‑semicontinuity condition (