Attainment Boundaries and Escape Rates in Asymptotically Conic Optimization

Abstract

We study attainment boundaries for linear optimization over unbounded convex sets. For epigraphs of finite convex functions, convex conjugacy separates recession-cone copositivity, boundedness below, and attainment through three nested subsets of conjugate space. At a finite but unattained boundary value, we establish a facewise asymptotic selection theorem for inward perturbations of the objective. The theorem identifies the escaping directions of the perturbed minimizers, their precise blow-up scale, and the leading asymptotics of the optimal value; the critical set may be multidimensional, and no radial symmetry is assumed. For radial asymptotically conic epigraphs, we obtain a complete boundary trichotomy and universal scaling laws for objective tilting, hard truncation, and power regularization. For shifted ellipsoidal second-order cone programs, we derive explicit primal--dual formulas together with sharp escape, conditioning, and regularization rates. These models also admit an exact robust-optimization representation. At the attainment boundary, strict primal feasibility, zero duality gap, and dual attainment can coexist with failure of primal attainment.