3 research outputs found
Primal-Dual Gradient Flow Algorithm for Distributed Support Vector Machines
In this paper, a primal-dual gradient flow algorithm for distributed support
vector machines (DSVM) is proposed. A network of computing nodes, each carrying
a subset of horizontally partitioned large dataset is considered. The nodes are
represented as dynamical systems with Arrow-Hurwicz-Uzawa gradient flow
dynamics, derived from the Lagrangian function of the DSVM problem. It is first
proved that the nodes are passive dynamical systems. Then, by employing the
Krasovskii type candidate Lyapunov functions, it is proved that the computing
nodes asymptotically converge to the optimal primal-dual solution
Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity
This paper studies the exponential stability of primal-dual gradient dynamics
(PDGD) for solving convex optimization problems where constraints are in the
form of Ax+By= d and the objective is min f(x)+g(y) with strongly convex smooth
f but only convex smooth g. We show that when g is a quadratic function or when
g and matrix B together satisfy an inequality condition, the PDGD can achieve
global exponential stability given that matrix A is of full row rank. These
results indicate that the PDGD is locally exponentially stable with respect to
any convex smooth g under a regularity condition. To prove the exponential
stability, two quadratic Lyapunov functions are designed. Lastly, numerical
experiments further complement the theoretical analysis.Comment: 8 page
Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian: A Lyapunov-based approach
For a class of nonsmooth composite optimization problems with linear equality
constraints, we utilize a Lyapunov-based approach to establish the global
exponential stability of the primal-dual gradient flow dynamics based on the
proximal augmented Lagrangian. The result holds when the differentiable part of
the objective function is strongly convex with a Lipschitz continuous gradient;
the non-differentiable part is proper, lower semi-continuous, and convex; and
the matrix in the linear constraint is full row rank. Our quadratic Lyapunov
function generalizes recent result from strongly convex problems with either
affine equality or inequality constraints to a broader class of composite
optimization problems with nonsmooth regularizers and it provides a worst-case
lower bound of the exponential decay rate. Finally, we use computational
experiments to demonstrate that our convergence rate estimate is less
conservative than the existing alternatives.Comment: 6 pages, 3 figure