Regime-Adaptive Conformal Calibration of Entropic Soft-Min Relaxations for Heterogeneous Optimization Problems

Entropic soft-min relaxations are widely used to obtain smooth approximations of minimum operators in optimization, machine learning, and control. The accuracy of this approximation is governed by an inverse temperature (or sharpness) parameter that controls the trade-off between smoothness and fidelity, yet its principled selection is typically heuristic. This work studies the data-driven calibration of the inverse temperature parameter governing the entropic soft-min relaxation, with explicit guarantees on the relaxation error between the soft-min operator and the infimum of the cost function. After establishing monotonicity properties and approximation bounds for the relaxation error, we introduce a conformal calibration rule that selects the smallest inverse temperature ensuring that the approximation error satisfies a prescribed tolerance with distribution-free finite-sample validity. The resulting selector adapts to the distribution of candidate cost-vector geometries represented in the calibration sample, enabling regime-specific inverse temperature selection in heterogeneous settings. Numerical experiments, including an adaptive cruise control application with safety filtering, show that the proposed method accurately tracks oracle calibration inverse temperatures and achieves near-target coverage in the exchangeable setting covered by the theory, while an additional shifted evaluation illustrates the role of this assumption.