A Flow-based Method for Problems with Vanishing Constraints

Mathematical Programs with Vanishing Constraints (MPVCs) are a notoriously challenging class of problems owing to their lack of constraint qualification. Therefore, to tackle these problems, relaxation-based approaches are typically used. While often yielding satisfactory results, they generally require significant manual tuning and adjustment of the relaxation parameter. To circumvent these problems, we introduce a novel approach based on piecewise gradient flows leading to first-order stationary points. We demonstrate the effectiveness of our method on several real-world MPVC instances and compare it to a common relaxation approach.

💡 Research Summary

This paper addresses the challenging class of Mathematical Programs with Vanishing Constraints (MPVCs), which are difficult to solve because the product‑type constraints G_i(x)·H_i(x) ≥ 0 destroy standard constraint qualifications such as LICQ and MFCQ at bi‑active points. Traditional approaches mitigate this difficulty by introducing a regularization function Φ(G_i, H_i, σ) that smooths the complementarity condition for a positive parameter σ, then solving a sequence of NLPs while gradually decreasing σ. Although effective in many cases, this strategy requires careful, problem‑specific tuning of σ and only guarantees feasibility in the limit σ→0.

The authors propose a fundamentally different methodology that eliminates the need for an external relaxation parameter. First, they reformulate the MPVC in a “vertical” form by adding slack variables s∈ℝ^{2l} so that the original constraints become linear equalities involving s_i and s_{i+l}. For each pair (s_i, s_{i+l}) they define two convex branches: an upper branch C⁺(i) (s_i ≥ 0, s_{i+l} ≥ 0) and a lower branch C⁰(i) (s_i = 0, s_{i+l} ≤ 0). The Cartesian product of all branches yields 2^l convex pieces C_b(σ) parameterized by a binary signature σ∈{0,1}^l. Within any fixed piece the problem reduces to a standard nonlinear program (NLP_σ) without any vanishing constraints.

To solve each NLP_σ, the paper employs a projected primal‑dual gradient/anti‑gradient flow. Let z = (x, s) and y be the Lagrange multipliers for the equality constraints. The flow is defined by

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