  We study a class of polynomial optimization problems with a robust polynomial
matrix inequality (PMI) constraint where the uncertainty set itself is defined
also by a PMI. These can be viewed as matrix generalizations of semi-infinite
polynomial programs, since they involve actually infinitely many PMI
constraints in general. Under certain SOS-convexity assumptions, we construct a
hierarchy of increasingly tight moment-SOS relaxations for solving such
problems. Most of the nice features of the moment-SOS hierarchy for the usual
polynomial optimization are extended to this more complicated setting. In
particular, asymptotic convergence of the hierarchy is guaranteed and finite
convergence can be certified if some flat extension condition holds true. To
extract global minimizers, we provide a linear algebra procedure for recovering
a finitely atomic matrix-valued measure from truncated matrix-valued moments.
As an application, we are able to solve the problem of minimizing the smallest
eigenvalue of a polynomial matrix subject to a PMI constraint. If SOS-convexity
is replaced by convexity, we can still approximate the optimal value as closely
as desired by solving a sequence of semidefinite programs, and certify global
optimality in case that certain flat extension conditions hold true. Finally,
an extension to the non-convexity setting is provided under a rank one
condition. To obtain the above-mentioned results, techniques from real
algebraic geometry, matrix-valued measure theory, and convex optimization are
employed.
