Unified Dual Semidefinite Programming Walrasian Equilibrium Model: A Matrix-Structured Generalization of Economic Equilibrium Theory (UDS-WE)

This paper proposes a novel framework for economic equilibrium theory, unifying dual semidefinite programming (Dual SDP) and Walrasian general equilibrium theory into a "matrix variational inequality system." Unlike traditional Walrasian models that only deal with scalar price spaces, this paper introduces the positive semidefinite price operator P⪰0, enabling the economic system to possess a triple expression of risk, structure, and network. Simultaneously, by characterizing the resource-constrained shadow price system through the dual SDP structure, economic equilibrium is transformed into a matrix operator equation with complementary relaxation conditions.

This paper proves that the model has equilibrium solutions under appropriate conditions and can degenerate into classical Walrasian equilibrium and the standard dual SDP problem, thus achieving the unification of the two theoretical systems. This framework provides a unified mathematical foundation for complex economic systems, financial network stability, and AI resource allocation.