A Polar-Stable Optimization Framework with Uncertain Embedded Cones and Adaptive Dual Decomposition

This paper proposes a novel optimization framework: Uncertain Embedded Polar Conic Programming (UEPCP). Unlike traditional robust optimization which imposes uncertainty on constraints or coefficients, this paper embeds uncertainty directly into the cone geometry itself, constructing a parameterized family of cones K(u). Under the box uncertainty set U, we prove that: 1) The intersection of the uncertain family of cones forms a closed convex stable cone. 2) The polar cone of this stable cone can be represented as the closed convex hull of the union of polar cones. 3) A minimum polarization stable envelope cone exists. 4) Strong duality and uniqueness hold under appropriate regularization conditions. 5) An adaptive polar cone decomposition theorem is proposed. 6) A Lipschitz stability bound for the geometric elasticity exponent is given. This theory unifies cone programming, polar cone theory, and robust optimization at the geometric level, constructing a new structural hierarchy: cone structural uncertainty.