Generalized Primal-Dual Conic Optimization: A Unified Framework for Conic Programming and Gap Analysis

This paper proposes a novel Generalized Primal-Dual Conic Optimization (GPDC) model that organically combines traditional general cone programming with primal-dual gap theory, achieving integrated optimization of the objective function, constraint structure, and dual convergence. By introducing an adaptive gap adjustment function, this model not only unifies the primal-dual analysis framework but also theoretically overcomes the limitations of traditional cone programming under non-strict feasibility and non-smooth objectives. The novel equations constructed in this paper can be applied to economic optimization, network flow optimization, and generalized equilibrium analysis, providing a unified analytical method for high-dimensional nonlinear constrained problems.