Conic optimization problems are encountered in a wide array of fields including truss design, control and system theory, statistics, eigenvalue optimization, and antenna array weight design. Robust optimization formulations of many convex programming problems also lead to conic optimization problems, see, e.g., [8, 9]. Furthermore, conic optimization problems arise as relaxations of hard combinatorial optimization problems such as the max-cut problem. Finally, some of the most interesting applications of conic optimization are encountered in financial mathematics and we will address several examples in this chapter.

Tracking error and volatility constraints

In most quantitative asset management environments, portfolios are chosen with respect to a carefully selected benchmark. Typically, the benchmark is a market index, reflecting a particular market (e.g., domestic or foreign), or a segment of the market (e.g., large cap growth) the investor wants to invest in. Then, the portfolio manager's problem is to determine an index-tracking portfolio with certain desirable characteristics. An index-tracking portfolio intends to track the movements of the underlying index closely with the ultimate goal of adding value by beating the index. Since this goal requires departures from the underlying index, one needs to balance the expected excess return (i.e., expected return in excess of the benchmark return) with the variance of the excess returns.

The tracking error for a given portfolio with a given benchmark refers to the difference between the returns of the portfolio and the benchmark.