We present a simple reconfigurable leaky-wave antenna (LWA) capable of fixed-frequency continuous beam-scanning, along with an efficient optimization procedure with which its radiation pattern can be shaped. To construct the LWA, an array of varactor-loaded unit-cells are connected in series over a grounded dielectric substrate, forming a reconfigurable leaky microstrip. The varactors are individually addressed, with a biasing scheme determined via optimization. To ensure the practicality of the converged solutions, several feasibility-based optimization constraints are derived. In contrast with conventional LWAs, the proposed device is synthesized based on an aperiodic non-local model which rigorously accounts for the mutual coupling between all radiating elements. As a result, it can be optimized to meet very demanding requirements, such as wide-angle continuous beam-scanning with strong side lobe suppression. Importantly, the antenna input impedance can be adaptively tuned to match that of the source, without the need for a complicated matching network. Furthermore, the optimization procedure can accommodate different end terminations, even when they cause strong backward reflections. These features of the proposed LWA are confirmed through numerical simulations. Measurements of a fabricated prototype working at 10 GHz demonstrated a 130° continuous scan range, and a peak gain of 8.9 dBi.