A wide variety of problems in control system theory fall within the class of parameterized linear matrix inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. Such problems, though convex, involve an infinite set of LMI constraints and hence are inherently di cult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impact of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, mu analysis, and linear parameter-varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.