We consider assortment optimization over a continuous spectrum of products represented by the unit interval, where the sellers problem consists of determining the optimal subset of products to offer to potential customers. To describe the relation between assortment and customer choice, we propose a probabilistic choice model that forms the continuous counterpart of the widely studied discrete multinomial logit model. We consider the sellers problem under incomplete information, propose a stochastic-approximation type of policy, and show that its regret -- its performance loss compared to the optimal policy -- is only logarithmic in the time horizon. We complement this result by showing a matching lower bound on the regret of any policy, implying that our policy is asymptotically optimal. We then show that adding a capacity constraint significantly changes the structure of the problem: we construct a policy and show that its regret after $T$ time periods is bounded above by a constant times $T^{2/3}$ (up to a logarithmic term); in addition, we show that the regret of any policy is bounded from below by a positive constant times $T^{2/3}$, so that also in the capacitated case we obtain asymptotic optimality. Numerical illustrations show that our policies outperform or are on par with alternatives.
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