Parameter estimation in complementarity-constrained parabolic optimal control model: application to American option implied volatility

Abstract This paper addresses optimal control problems with parabolic complementarity constraints, characterized by non-convexity and nonlinearity. Motivated by the practical need for implied volatility calibration, we perform theoretical analysis and algorithmic design for this problem. Theoretically, we prove the existence of optimal controls and derive the first-order optimality conditions via a penalty relaxation approach. Numerically, we develop two bilevel optimization algorithms to handle state-control hybridity, decomposing the complex problem into tractable sub-problems. Numerical simulations demonstrate the correctness and efficiency of the proposed algorithms. Furthermore, an inexact stopping criterion is utilized to accelerate convergence while maintaining solution accuracy.