Reflexive Cone Closure Operator and a Unified Primal–Polar Coupled Optimization Framework for Stability Restoration in Conic Programming

This paper proposes a novel cone operator structure—the reflexive closure cone operator—and constructs a primordial-polar cone coupled optimization system. This theory embeds the primordial problem and the dual geometry into a unified framework by establishing a dynamic iterative closure mechanism between general cone programming and the polar cone structure. This paper proves that, under appropriate topological conditions, the cone sequence generated by the bipolar operator converges to a minimum reflexive closed cone, and further demonstrates that this closure structure induces a stable primordial-polar coupled optimal system. These results establish a new theoretical foundation for non-closed, non-self-dual cone programming.