{"ID":2855606,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.12244","arxiv_id":"2510.12244","title":"Bilateral facial reduction: qualification-free subdifferential calculus and exact duality","abstract":"Qualification conditions (also termed constraint qualifications) help avoid pathological behavior at domain boundaries in convex analysis. By generalizing facial reduction from conic programming to general convex programs of the form $f(x) + g(Ax)$, we provide qualification-free generalizations of several key results: an exact Fenchel-Rockafellar dual, KKT optimality conditions, an attained infimal convolution for the conjugate of a sum, subdifferential sum and chain rules, and normal cones of intersections. All our results reduce seamlessly to their original formulations when qualification conditions hold. The core insight is that for a sum of two convex functions, there is an affine subspace$\\unicode{x2014}$the joint supporting subspace$\\unicode{x2014}$that contains the feasible region, and such that qualification conditions hold when restricting the effective domain of each function to it. We offer a number of characterizations for the joint supporting subspace, including one that obtains the affine subspace via iterative, bilateral reduction between the two domains. In our proofs, which are self-contained, we develop a structured induction on faces where inductive steps are associated with normal vectors nested in supporting subspaces (a generalization of supporting hyperplanes). With this tool, we characterize the facial structure of the difference of two convex sets from the facial structures of the individual convex sets.","short_abstract":"Qualification conditions (also termed constraint qualifications) help avoid pathological behavior at domain boundaries in convex analysis. By generalizing facial reduction from conic programming to general convex programs of the form $f(x) + g(Ax)$, we provide qualification-free generalizations of several key results:...","url_abs":"https://arxiv.org/abs/2510.12244","url_pdf":"https://arxiv.org/pdf/2510.12244v5","authors":"[\"Matthew S. Scott\"]","published":"2025-10-14T07:54:41Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.FA\"]","methods":"[]","has_code":false}