On the Arrow–Hurwicz differential system for linearly constrained convex minimization

ABSTRACT

In a real Hilbert space setting, we reconsider the classical Arrow–Hurwicz differential system in view of solving linearly constrained convex minimization problems. We investigate the asymptotic properties of the differential system and provide conditions for which its solutions converge towards a saddle point of the Lagrangian associated with the convex minimization problem. Our convergence analysis mainly relies on a ‘Lagrangian identity’ which naturally extends on the well-known descent property of the classical continuous steepest descent method. In addition, we present asymptotic estimates on the decay of the solutions and the primal-dual gap function measured in terms of the Lagrangian. These estimates are further refined to the ones of the classical damped harmonic oscillator provided that second-order information on the objective function of the convex minimization problem is available. Finally, we show that our results directly translate to the case of solving structured convex minimization problems. Numerical experiments further illustrate our theoretical findings.

KEYWORDS:

• Arrow–Hurwicz differential system
• Lyapunov analysis
• asymptotic properties
• convex minimization
• saddle-point problem

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37N40
• 46N10
• 49M30
• 65K05
• 90C25

Acknowledgments

The author expresses his gratitude to the two anonymous reviewers whose comments and suggestions led to a significant improvement of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We remark that, in the finite-dimensional case, the condition amounts to $b\in A(X)$ which is commonly referred to as Slater assumption; see, e.g. Hiriart-Urruty and Lemaréchal [37].

2 Given the above assumptions, it is easy to verify that $(x,\lambda )\mapsto T(x,\lambda )$ is Lipschitz continuous on the bounded subsets of $X\times Y$.

3 We recall that $A:X\to Y$ is bounded from below if and only if it is injective with closed range; see, e.g. Brézis [38].

4 We note that $A^{\ast }:Y\to X$ is bounded from below if and only if A is surjective; see, e.g. Brézis [38].

Additional information

Funding

Research supported by the German Research Foundation (DFG).