Mathematical Programs with Complementarity Constraints

• MPCC is an optimization framework where decision variables satisfy both standard constraints and complementarity conditions, leading to nonconvex and degenerate structures.
• It models real-world scenarios such as equilibrium problems, bilevel optimization, hybrid control, and machine learning hyperparameter tuning, demonstrating wide applicability.
• Specialized stationarity concepts, tailored regularization methods, and numerical algorithms are crucial for addressing the issues arising from violated standard constraint qualifications.

A Mathematical Program with Complementarity Constraints (MPCC) is an optimization problem where decision variables are subject not only to standard equality and inequality constraints but also to a collection of complementarity conditions, typically in the form $0 \leq y \perp w \geq 0$. MPCCs serve as canonical models for diverse applications, including equilibrium problems, nonsmooth dynamic systems, bilevel optimization, hybrid control, and machine learning hyperparameter tuning. The presence of complementarity constraints imparts a fundamentally disjunctive, nonconvex, and degenerate structure: standard constraint qualifications such as LICQ or MFCQ are violated at every feasible point, and classical KKT conditions do not apply directly. As a result, specialized stationarity concepts, constraint qualifications, regularization methods, and tailored algorithms are required for both theoretical analysis and computational treatment.

1. Formulation and Stationarity Theory

An MPCC is formulated in the general form: $$\begin{aligned} \min_{x} \quad & f(x) \ \text{s.t.}\quad & g(x) \leq 0, \ & h(x) = 0, \ & 0 \leq G(x) \perp H(x) \geq 0. \end{aligned}$$ Here $f,g,h,G,H$ are at least once continuously differentiable ($C^1$); the complementarity constraint $0 \leq G(x) \perp H(x) \geq 0$ means, for all (