This work addresses a cost optimization problem in facility location where customer demand is modeled using the cross-nested logit model, one of the most flexible demand models in the literature. The objective is to maximize a captured demand function by allocating a fixed investment budget across a set of facilities, where the investment directly influences the demand captured by each facility. The resulting optimization problem involves exponential and fractional terms, leading to a highly nonlinear structure. To the best of our knowledge, no existing methods can solve this problem to near-optimality. To address this, we propose a piecewise linear approximation technique and apply variable transformations to approximate the problem (to any desired precision) as a mixed-integer convex program, which can be solved to optimality using an outer-approximation method. Extensive experiments on generated instances of varying sizes demonstrate the effectiveness of our proposed approach compared to standard baselines.