Using Lagrangian Relaxation to Obtain Small Portfolios

Abstract

Investors with small portfolios, or a limited number of securities in their portfolios, may benefit from a new portfolio optimization method. Placing a limit on the number of assets in a portfolio turns the ordinary mean variance portfolio optimization problem into a challenging puzzle, especially for larger investment universes. In response, practitioners typically employ either enumerative methods, such as branch-and-bound based on quadratic programming relaxation, or heuristic methods. Both approaches have their respective disadvantages in that quadratic programming–based branch-and-bound may fail to solve large problems in reasonable time and heuristics may produce solutions of unknown quality. The new method presented by the authors can be used to solve smaller problems to optimality. For larger problems, the method produces good heuristic solutions along with a useful estimate of their quality; that is, the distance from the optimum. The computational results are promising.

TOPICS: Portfolio construction, accounting and ratio analysis, statistical methods