A linear programming approach is proposed to tune fixed-order linearly parameterized gain-scheduled controllers for stable SISO linear parameter varying (LPV) plants. The method is based on the shaping of the open-loop transfer functions in the Nyquist diagram with constraints on the robustness margins and on the lower bound of the crossover frequency. Two optimization problems are considered: optimization for robustness and optimization for performance. This method directly computes an LPV controller from a set of frequency domain models in different operating points or from an LPV model and no interpolation is needed. In terms of closed-loop performance, this approach leads to extremely good results. However, closed-loop stability is ensured only locally and for slow variations of the scheduling parameters. An application to a high-precision double-axis positioning system illustrates the effectiveness of the proposed approach.