One of the fastest growing areas of scientific computing is in the financial industry. Many of the most basic problems in financial analysis are still unsolved, and are surprisingly resilient to the onslaught of legions of talented researchers from many diverse disciplines. In this article, we hope to give readers a sense of these challenges by describing a relatively simple problem that all investors face—managing a portfolio of financial securities over time to optimize a particular objective function—and showing how complex such a problem can become as real-world considerations such as taxes, preferences, and portfolio constraints are incorporated into its formulation.

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