This paper presents a novel approach to efficiently solve parameter-dependent (PD) linear matrix inequality (LMI) problems for, amongst others, linear parameter-varying (LPV) control design. Typically, stability and performance is guaranteed by finding a PD Lyapunov function such that a PD LMI is feasible on a parameter domain. To solve the resulting semi-infinite problems, we propose a novel LMI relaxation technique relying on B-spline basis functions. This technique provides less conservative solutions and/or a reduced numerical burden compared to existing approaches. Moreover, an elegant generalization of worst-case optimization to the optimization of any signal norm is obtained by expressing performance bounds as a function of the system parameters. This generalization yields better performance bounds in a large part of the parameter domain. Numerical comparisons with the current state-of-the-art demonstrate the generality and effectiveness of our approach.