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Assortment optimization finds many important applications in both brick-and-mortar and online retailing. Decision makers select a subset of products to offer to customers from a universe of substitutable products, based on the assumption that customers purchase according to a Markov chain choice model, which is a very general choice model encompassing many popular models. The existing literature predominantly assumes that the customer arrival process and the Markov chain choice model parameters are given as input to the stochastic optimization model. However, in practice, decision makers may not have this information and must learn them while maximizing the total expected revenue on the fly. In “Online Learning for Constrained Assortment Optimization under the Markov Chain Choice Model,” S. Li, Q. Luo, Z. Huang, and C. Shi developed a series of online learning algorithms for Markov chain choice-based assortment optimization problems with efficiency, as well as provable performance guarantees. 

This paper develops a finite approximation approach to find a non-smooth solution of an integral equation of the second kind. The equation solutions with non-smooth kernel having a non-smooth solution have never been studied before. Such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available.