Cone-Induced Second-Order Risk Programming: A Spectral-Geometric Unification of Mean–Variance Optimization and Convex Cone Theory

This paper proposes a novel optimization theoretical framework—Cone-Induced Second-Order Risk Programming (CISRP). This theory unifies general cone programming and Mean-Variance optimization at the geometric level. Instead of treating risk as an external penalty term in the objective function, it constructs a cone-induced risk operator, embedding a stochastic second-order moment structure within the geometry of a convex cone, thus achieving intrinsic consistency between risk measurement and feasible region structure. This paper establishes the convexity, duality, spectral structure characterization, and projection expression of the cone-induced risk operator, proving that the model possesses structural simplification properties on generalized self-dual cones, and providing complete KKT conditions, weak duality theorems, and strong duality theorems. Furthermore, it proves the existence, uniqueness, and stability of the optimal solution, and establishes the theorem for the continuity of the parameter lambda and the monotonicity of the risk path. This theory provides a new unified framework for stochastic convex optimization, financial risk control, and robust cone programming.