Budget-Driven Polar Deformation of Cones under Structured Uncertainty: Geometric Collapse, Ordered Compression, and Adaptive Duality Stability

Traditional cone programming establishes an optimal structure on a closed convex cone K, while budget uncertainty theory typically achieves robust design by perturbing the parameter matrix or objective vector. This paper proposes a novel theoretical framework: budget-driven cone polarization deformation theory. This theory no longer applies uncertainty to linear operators but directly to the cone structure itself, constructing a budget-induced cone deformation operator and establishing its polarization equivalence representation. We prove that the budget uncertainty set is equivalent to an ordered compression structure on a polar cone, and further provide a geometric collapse threshold theorem and an adaptive stable duality consistency theorem. This theory geometrically unifies budget robust optimization and cone duality theory, revealing a deep connection between uncertainty and cone structure.