Solving Risk-Return Tradeoff Model with Galois Connection

We propose a risk-return tradeoff problem that seeks to minimize risk while ensuring at least aprescribed level of return. We observe that this problem is equivalent to the inverse median problemon a tree under the Chebyshev (max) norm, which admits a parametric closed-form solution. Our maincontribution is to uncover the hidden structure underlying this solution through a new duality frameworkfundamentally different from the classical Lagrange and Fenchel-Rockafellar duality frameworks. Thekey idea is to exchange the roles of the risk and return functions to formulate a dual problem. Thisperspective yields two insights. First, the parametric solution to the original problem can be interpretedas a trivial solution to the dual problem. Second, we establish a Galois connection between the twoproblems, showing that solving the dual problem provides an immediate way to recover the solution ofthe original problem.