First-Order Optimality Conditions for Mathematical Programming with Equilibrium Constraints
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913080054644736 |
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| author | Wang, Louis Shuo |
| author_facet | Wang, Louis Shuo |
| contents | We present a systematic introduction to first-order optimality conditions for mathematical programs with equilibrium constraints (MPECs), emphasizing the limitations of classical nonlinear programming techniques. The goal is twofold. First, we explain why a direct application of standard optimality conditions -- based on reformulating MPECs via KKT systems or differentiable exact penalty functions -- is often inadequate, as such approaches typically require strong and restrictive assumptions, including nondegeneracy and smoothness conditions.
Second, we develop a first-principles framework for analyzing MPECs by focusing on the geometric structure of the feasible region. In particular, we study stationarity concepts and provide a detailed characterization of the tangent cone at feasible points, which leads to appropriate constraint qualifications tailored to MPECs. These results form the foundation for rigorous first-order analysis and clarify the relationship between the original MPEC formulation and its KKT-based representation, offering practical guidance for handling these inherently challenging optimization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00388 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | First-Order Optimality Conditions for Mathematical Programming with Equilibrium Constraints Wang, Louis Shuo Optimization and Control We present a systematic introduction to first-order optimality conditions for mathematical programs with equilibrium constraints (MPECs), emphasizing the limitations of classical nonlinear programming techniques. The goal is twofold. First, we explain why a direct application of standard optimality conditions -- based on reformulating MPECs via KKT systems or differentiable exact penalty functions -- is often inadequate, as such approaches typically require strong and restrictive assumptions, including nondegeneracy and smoothness conditions. Second, we develop a first-principles framework for analyzing MPECs by focusing on the geometric structure of the feasible region. In particular, we study stationarity concepts and provide a detailed characterization of the tangent cone at feasible points, which leads to appropriate constraint qualifications tailored to MPECs. These results form the foundation for rigorous first-order analysis and clarify the relationship between the original MPEC formulation and its KKT-based representation, offering practical guidance for handling these inherently challenging optimization problems. |
| title | First-Order Optimality Conditions for Mathematical Programming with Equilibrium Constraints |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.00388 |