Algebraic optimization techniques for configuration design and performance enhancement in complex network systems

Abstract

Large-scale complex network systems, like communication networks, power grids, and syber-physical networks, demonstrate high operation efficiency, which depends heavily on structure, though nothing gives assurance that heuristic tuning of link capacities, weights of nodes, and routing structure leads to optimal solution. The study proposes algebraic optimization methods based on polynomial ideals, spectral graph theory, and semidefinite relaxations with the assistance of which structural guarantees do not come from the use of purely numerical heuristics. The object under consideration is represented algebraically by means of adjacency, incidence, and Laplacian matrices that distinguish the attributes of flow and capacity, hence configuration making is based on the constrained polynomial optimization problem with conservation and physical performance constraints. Roboust minimax performance enhancement formulation is proposed to protect against uncertainty coming from various sources of network disturbance, including link-capacity changes and alterations in topology, in addition to that, generalized Karush-Kuhn-Tucker conditions are acquired considering the generalized Mangasarian-Fromovitz qualification of constraints assuring boundedness, non emptiness, and convexity of multipliers sets given metric subregularity of algebraic mapping. The process of simplifying and dealing with difficult combinatorial problems is carried out through the use of Gröbner-basis reduction and moment-sum-of-squares hierarchies that transform non-convex network polynomials into manageable convex problems. It can also be claimed that strict second-order sufficiency conditions regarding the critical cone guarantee the uniqueness of solution.