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An investor has to carefully select the location and size of new generation units it intends to build, since adding capacity in a market affects the profit from units this investor may already own. To capture this closed-loop characteristic, strategic investment (SI) can be posed as a bilevel optimization. By analytically studying a small market, we first show that its objective function can be non-convex and discontinuous. Realizing that existing mixed-integer problem formulations become impractical for larger markets and increasing number of scenarios, this work put forth two SI solvers: a grid search to handle setups where the candidate investment locations are few, and a stochastic gradient descent approach for otherwise. Both solvers leverage the powerful toolbox of multiparametric programming (MPP), each in a unique way. The grid search entails finding the primal/dual solutions for a large number of optimal power flow (OPF) problems, which nonetheless can be efficiently computed several at once thanks to the properties of MPP. The same properties facilitate the rapid calculation of gradients in a mini-batch fashion, thus accelerating the implementation of a stochastic gradient descent search. Tests on the IEEE 118-bus system using real-world data corroborate the advantages of the novel MPP-aided solvers.