Operational Regimes in Non-Convex Optimization: A Multiplier-Based Taxonomy

• URL: http://arxiv.org/abs/2609.00471v1
• Date: Mon, 31 Aug 2026 23:22:05 GMT
• Title: Operational Regimes in Non-Convex Optimization: A Multiplier-Based Taxonomy
• Abstract summary: This paper introduces a structural taxonomy for constrained non-constrained optimization based on the signature Lagrange multipliers at KKT stationary points.<n>Four vector-free shape features this partition dual space into five operational regimes: Unconstrained, ResourceLimited, Saturation, Strongly--Coupled, and Hybrid.
• Score: 1.1907026010541673
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper introduces a structural taxonomy for constrained non-convex optimization based on the signature of Lagrange multipliers at KKT stationary points. Leveraging a unified game-theoretic interpretation of eight classical algorithm families--including block coordinate descent, ADMM, generalized Benders decomposition, successive convex approximation, interior-point methods, mirror descent, Frank-Wolfe, and Riemannian gradient descent--we show that the normalized multiplier vector carries an algorithm-independent structural fingerprint. Four scale-free shape features of this vector partition the dual space into five operational regimes: Unconstrained, Resource-Limited, Saturation, Strongly-Coupled, and Hybrid. We establish four structural theorems characterizing the partition: invariance under natural KKT symmetries, local stability under data perturbation with explicit Lipschitz margins from Robinson's strong regularity, codimension-one regime transitions, and the topological identification of the Hybrid regime as the Lebesgue-null boundary of the core regimes. A linear-time classifier is proposed with provable guarantees on correctness, iteration stabilization, sample complexity, and online tracking under data drift. Numerical experiments on 104 mixed-integer nonlinear programs and a downlink beamforming instance validate the theoretical predictions. The framework provides a foundational tool for regime-aware algorithm design and robustness analysis in non-convex optimization.

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