Qualification-free convex analysis via the joint supporting subspace
Matthew S. Scott
TL;DR
This work addresses the dependence on constraint qualifications in convex analysis by introducing a joint supporting subspace $T_a(C,D)$ that localizes analysis to an affine subspace where qualification conditions hold. The authors develop a geometric framework built on nested normals and supporting subspaces, establishing multiple characterizations of $T(C,D)$ and a constructive iterative bilateral facial reduction that terminates within $n$ steps. They derive qualification-free extensions of core convex-analysis results, including subdifferential calculus, the attained infimal convolution, exact Fenchel–Rockafellar duality, KKT conditions, and facial-reduction-like preprocessing for convex programs of the form $f(x)+g(Ax)$. The framework further introduces a local face lattice theory, with generated supporting subspaces and a tree-structured hierarchy of local constraints, enabling robust analysis and potential numerical benefits when classical CQs fail. Overall, the paper extends facial reduction concepts beyond conic programming to general convex programs, providing precise tools to study boundaries, intersections, and duality without traditional qualification assumptions.
Abstract
In convex analysis, qualification conditions (also termed constraint qualifications) help avoid pathological behavior at domain boundaries. In this work we remove the need for such conditions by localizing to an affine subspace -- the joint supporting subspace -- that contains the feasible region and ensures qualification conditions hold after localization. Our theory generalizes Borwein and Wolkowicz's facial reduction beyond conic programs to convex programs of the form $f(x) + g(Ax)$. Intuitively, the joint supporting subspace corresponds to a bilateral facial reduction between any two convex sets. It enables simple qualification-free generalizations for a host of central results of convex analysis. These include: an exact Fenchel-Rockafellar dual; Karush-Kuhn-Tucker (KKT) optimality conditions; attained infimal convolution for convex conjugates; subdifferential sum and chain rules; and a characterization of the normal cones of the intersection of two convex sets. All generalizations reduce seamlessly to their original formulations when qualification conditions hold. We offer a number of characterizations for the joint supporting subspace, one of which is constructive. Our proofs are self-contained, and introduce a novel theoretical framework consisting of nested normals and supporting subspaces, which simultaneously describe both the boundary of convex sets and the lattice of faces.