  Equivalence of convex optimization, saddle-point problems, and variational
inequalities is a well-established concept. The variational inequality (VI) is
a static problem which is studied under dynamical settings using a framework
called the projected dynamical system, whose stationary points coincide with
the static solutions of the associated VI. VI has rich properties concerning
the monotonicity of its vector-valued map and the uniqueness of its solution,
which can be extended to convex optimization and saddle-point problems.
Moreover, these properties also extend to the representative projected
dynamical system. The objective of this paper is to harness rich monotonicity
properties of the representative projected dynamical system to develop the
solution concepts of the convex optimization problem and the associated
saddle-point problem. To this end, this paper studies a linear inequality
constrained convex optimization problem and models its equivalent saddle-point
problem as a VI. Further, the VI is studied as a projected dynamical
system\cite{friesz1994day} which is shown to converge to the saddle-point
solution. By considering the monotonicity of the gradient of Lagrangian
function as a key factor, this paper establishes exponential convergence and
stability results concerning the saddle-points. Our results show that the
gradient of the Lagrangian function is just monotone on the Euclidean space,
leading to only Lyapunov stability of stationary points of the projected
dynamical system. To remedy the situation, the underlying projected dynamical
system is formulated on a Riemannian manifold whose Riemannian metric is chosen
such that the gradient of the Lagrangian function becomes strongly monotone.
Using a suitable Lyapunov function, the stationary points of the projected
dynamical system are proved to be globally exponentially stable and convergent
to the unique saddle-point.
