2025/10/14 by Scott, Matthew S.
#49J53 #52A41 #68Q25 #90C25 #FOS: Mathematics #Functional Analysis (math.FA) #G.1.6 #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2510.12244
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.