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In the present paper, the primal-dual problem consisting of the investmentrisk minimization problem and the expected return maximization problem in themean-variance model is discussed using replica analysis. As a natural extensionof the investment risk minimization problem under only a budget constraint thatwe analyzed in a previous study, we herein consider a primal-dual problem inwhich the investment risk minimization problem with budget and expected returnconstraints is regarded as the primal problem, and the expected returnmaximization problem with budget and investment risk constraints is regarded asthe dual problem. With respect to these optimal problems, we analyze a quencheddisordered system involving both of these optimization problems using theapproach developed in statistical mechanical informatics, and confirm that bothoptimal portfolios can possess the primal-dual structure. Finally, the resultsof numerical simulations are shown to validate the effectiveness of theproposed method.

Replica Analysis for the Duality of the Portfolio Optimization Problem