Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points

Abstract

Abstract. This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradientascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.

Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian

“…We choose b to be a vector of all ones, set T = I, and report results for (L f , m f ) = (1.24, 1.03) and (L f , m f ) = (27.81, 1.03). Figure 2a demonstrates the exponential convergence of dynamics (6) with (L f , m f ) = (1.24, 1.03) for different values of µ. We note that the convergence rate decreases when µ becomes larger than 2.…”

“…Recent reference [11] used a Lyapunov-based approach to show the global exponential stability for a class of problems with a strongly convex and smooth objective function f subject to either affine equality or inequality constraints. In our preliminary work [18], a similar quadratic Lyapunov function was used to prove global exponential stability of the primaldual gradient flow dynamics (6). In what follows, we employ the theory of IQCs in the time domain to obtain a quadratic Lyapunov function that establishes the global exponential stability of (6) and yields less conservative convergence rate estimates.…”

Distributed Algorithm for Economic Dispatch Problem With Separable Losses

“…However, note that the set E is the set of equilibrium points of (16). Thus, convergence to a point can be obtained by applying Lemma A.3 from [19]. For the feasibility of the converged solution, letλ = [λ 1 ; .…”

“…Then, the trajectory ofξ(t) approaches to Λ * δ1 with the speed of at least δ 1 /(3N ). For instance, ifξ(t) ≥Λ, it follows from (18a) and (19) thaṫ…”

Discounted Mirror Descent Dynamics in Concave Games

“…dual averaging [3] and follow-the-leader [4]. A continuous-time version of MDA, referred to as the mirror descent (MD) dynamics, [12], [15], captures many existing continuous-time dynamics as special cases, such as the gradient flow [12], [15], saddlepoint dynamics [9] and pseudo-gradient dynamics [10].…”