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The problem of optimal portfolio execution has become one of the most important problems in the area of financial mathematics. Over the past two decades, numerous researchers have developed a variety of different models to address this problem. In this dissertation, we extend the LOB (Limit Order Book) model proposed by Obizhaeva and Wang (2013) by incorporating a more realistic assumption on the order book depth; the amount of liquidity provided by a LOB market is finite at all times. We use an algorithmic approach to solve the problem of optimal execution under time-varying constraints on the depth of a LOB. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a one-dimensional root-finding problem which can be readily solved by standard numerical algorithms. When the depth of the LOB is monotone in time, we first apply the KKT (Karush-Kuhn-Tucker) conditions to narrow down the set of candidate strategies and then use a dichotomy-based search algorithm to pin down the optimal one. For the general case that the order book depth doesn't exhibit any particular pattern, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.