Tikhonov regularized exterior penalty dynamics for constrained variational inequalities

Abstract

Solving equilibrium problems under constraints is an important problem in optimization and optimal control. In this context an important practical challenge is the efficient incorporation of constraints. We develop a continuous-time method for solving constrained variational inequalities based on a new penalty regulated dynamical system in a general potentially infinite-dimensional Hilbert space. In order to obtain strong convergence of the issued trajectory of our method, we incorporate an explicit Tikhonov regularization parameter in our method, leading to a class of time-varying monotone inclusion problems featuring multiscale aspects. Besides strong convergence, we illustrate the practical efficiency of our developed method in solving constrained min-max problems.

Figures (2)

• Figure 1: $\|x(t)-p(t)\|$
• Figure 2: Feasibility Gap

Theorems & Definitions (15)

• Example 1: Generalized Nash equilibrium Problems
• Lemma 3
• Proposition 4
• Proposition 5
• Lemma 6
• Lemma 7
• Lemma 8
• proof
• Lemma 9
• proof