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A Flow-based Method for Problems with Vanishing Constraints

Christoph Hansknecht, Julian Niederer, Andreas Potschka

TL;DR

A novel approach based on piecewise gradient flows leading to first-order stationary points is introduced based on piecewise gradient flows leading to first-order stationary points in Mathematical Programs with Vanishing Constraints.

Abstract

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.

A Flow-based Method for Problems with Vanishing Constraints

TL;DR

A novel approach based on piecewise gradient flows leading to first-order stationary points is introduced based on piecewise gradient flows leading to first-order stationary points in Mathematical Programs with Vanishing Constraints.

Abstract

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.
Paper Structure (23 sections, 2 theorems, 36 equations, 8 figures, 4 tables)

This paper contains 23 sections, 2 theorems, 36 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Constraint Qualifications
  4. A Practical Algorithm
  5. Gradient Flows
  6. Branch Decomposition
  7. Piecewise Gradient Flows
  8. Implementational Aspects
  9. Algorithmic Parameters
  10. Scaling
  11. Vanishing Constraints
  12. Initializing Branches
  13. Reduced Subsequent NLPs
  14. Numerical Experiments
  15. An Academic Example
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $x^{*}$ be a local minimum of eq:mpvc such that GCQ holds at $x^{*}$. Then there exist Lagrange multipliers $y^{*} \in \mathds{R}^{m}$, $\eta_{H}^{*}, \eta_{G}^{*} \in \mathds{R}^{l}$ such that where and

Figures (8)

  • Figure 1: The feasible region of a controlling / vanishing constraint in vertical form.
  • Figure 2: One example flow for the starting point (7,2) is shown in black. The feasible set is marked in grey with active controlling constraints in dark grey, showing the nonconvex structure. The global optimal point is $x^*$, the local optimal point is $\tilde{x}$, and a further convergence point for standard solver is $\hat{x}$.
  • Figure 3: Numerical results for the academic example. The feasible set is marked in grey with active controlling constraints in dark grey. The global optimal point is $x^*$, the local optimal point is $\tilde{x}$, and a further convergence point for standard solver is $\hat{x}$. Starting points where our method converges to the local optimum is shown with circles. Rectangles mark the convergence of our method to the global minimum. Figure (a) shows the convergence regions with the prescribed branch initialization strategy. Figure (b) shows the convergence regions, if every infeasible pair of slack variables is initialized in the lower branch.
  • Figure 4: Potential bars and solution of the TenBar instance, which visually coincide with e.g. diss_hoheiselHoheisel22. The load force $f_1$ is marked in dark grey. On the left the structure is fixed on the grey wall.
  • Figure 5: Potential structure of the cantilever problem in (a) and the solution of the Cant1 instance in (b). The load force $f_1$ is marked in dark grey. The structure is fixed to the wall on the left-hand side.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 2.1: Theorem 2.5 in mpvc_stat_cons
  • Theorem 5.1
  • proof