From Semi-Infinite Constraints to Structured Robust Policies: Optimal Gain Selection for Financial Systems

Abstract: This paper studies the robust optimal gain selection problem for financial trading systems, formulated within a \emph{double linear policy} framework, which allocates capital across long and short positions. The key objective is to guarantee \emph{robust positive expected} (RPE) profits uniformly across a range of uncertain market conditions while ensuring risk control. This problem leads to a robust optimization formulation with \emph{semi-infinite} constraints, where the uncertainty is modeled by a bounded set of possible return parameters. We address this by transforming semi-infinite constraints into structured policies -- the \emph{balanced} policy and the \emph{complementary} policy -- which enable explicit characterization of the optimal solution. Additionally, we propose a novel graphical approach to efficiently solve the robust gain selection problem, drastically reducing computational complexity. Empirical validation on historical stock price data demonstrates superior performance in terms of risk-adjusted returns and downside risk compared to conventional strategies. This framework generalizes classical mean-variance optimization by incorporating robustness considerations, offering a systematic and efficient solution for robust trading under uncertainty.