  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.
