渐近锥优化中的可达边界与逃逸率
Attainment Boundaries and Escape Rates in Asymptotically Conic Optimization
AI总结：
该研究针对无界凸集上的线性优化，建立了目标内扰动的逐面渐近选择定理，推导了渐近锥优化相关的边界三分法、标度律及移位椭球二阶锥规划的原始-对偶公式与逃逸、正则化率等结果。
AI中文摘要：
我们研究无界凸集上线性优化的可达边界。对于有限凸函数的上图，凸共轭性通过共轭空间的三个嵌套子集将衰退锥共正性、下方有界性与可达性区分开来。在有限但不可达的边界值处，我们建立了目标内扰动的逐面渐近选择定理，该定理确定了扰动极小化子的逃逸方向、精确的爆破尺度及最优值的主导渐近行为；临界集可为多维，且不假设径向对称性。对于径向渐近锥形上图，我们得到了目标倾斜、硬截断与幂正则化的完整边界三分法及通用标度律。对于移位椭球二阶锥规划，我们推导了显式原始-对偶公式，以及严格的逃逸、条件数与正则化率，这些模型还具有精确的鲁棒优化表示。在可达边界处，严格原始可行性、零对偶间隙与对偶可达性可与原始可达性的失效共存。
英文摘要：
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.
