Unobserved confounding poses a central challenge to the credible estimation of causal effects. Proxy-based negative-control methods address this by using auxiliary outcome variables Formula: see text as proxies for latent confounders Formula: see text, but they typically rely on strong assumptions—such as reversibility and completeness—that are difficult to interpret empirically and to verify. As a result, their applicability in real-world settings is limited, especially when the transition matrix Formula: see text is noninvertible. We propose an optimization method: partial identification via sum-of-ratios fractional programming (PI-SFP). To our knowledge, this is the first optimization framework in causal inference that incorporates partial knowledge of the transition matrix Formula: see text. PI-SFP provides a general and flexible approach under weaker and more realistic assumptions. Concretely, it is a global branch-and-bound algorithm that addresses scenarios previously out of reach. We prove the global convergence of PI-SFP to valid bounds on causal effects and further show how mild bridge-function-based assumptions can tighten these bounds. Through both synthetic and empirical evaluations, PI-SFP delivers informative numerical results and addresses gaps in the literature by handling partial information about Formula: see text. Funding: This work is supported by the Fundamental Research Funds for the Central Universities Grant 2025110602 of the Shanghai University of Finance and Economics.
Zhiheng Zhang (Thu,) studied this question.