The share-of-choice product design (SOCPD) problem is to find the product, as defined by its attributes, that maximizes market share arising from a collection of customer types or segments. When customers follow a logit model of choice, the market share is given by a weighted sum of logistic probabilities, leading to the logit-based share-of-choice product design problem. In this paper, we develop a methodology for solving this problem to provable optimality. We first analyze the complexity of this problem, and show that this problem is theoretically intractable: it is NP-Hard to solve exactly, even when there are only two customer types, and it is furthermore NP-Hard to approximate to within a non-trivial factor. Motivated by the difficulty of this problem, we propose three different mixed-integer exponential cone programs of increasing strength for solving the problem exactly, which allow us to leverage modern integer conic program solvers such as Mosek. Using both synthetic problem instances and instances derived from real conjoint data sets, we show that our methodology can solve large instances to provable optimality or near optimality in operationally feasible time frames and yields solutions that generally achieve higher market share than previously proposed heuristics.