A Moment-Sum-of-Squares Hierarchy for Robust Polynomial Matrix Inequality Optimization with Sum-of-Squares Convexity.

The article focuses on developing a moment-sum-of-squares (SOS) hierarchy for solving robust polynomial matrix inequality (PMI) optimization problems under SOS-convexity assumptions. These problems involve infinitely many PMI constraints defined by polynomial matrices with uncertainty sets also described by PMIs. The authors construct a hierarchy of moment-SOS relaxations that guarantee asymptotic convergence to the global optimum and provide a linear algebraic procedure to recover finitely atomic matrix-valued representing measures when a flat extension condition (FEC) holds, enabling extraction of global minimizers. Extensions to general convex and nonconvex settings are discussed, along with applications such as minimizing the smallest eigenvalue of a polynomial matrix subject to PMI constraints. The work leverages real algebraic geometry, matrix-valued measure theory, and convex optimization to generalize scalar polynomial optimization techniques to the matrix-valued robust setting.