Yosida-regularization based differential equation approach to generalized equations with applications to nonlinear convex programming

Abstract

The problem of generalized equation 0 ∈ T(z), for maximal monotone operator $T: {\mathscr{H}} \rightrightarrows {\mathscr{H}}$, covers many important optimization problems. The solution trajectory u_λ(t) of the classical Yosida-regularization based differential equation system together with numerical approximation trajectories ${u}_{\lambda }^{\alpha }(t)$, for finding solutions to the generalized equation, is investigated in this paper. The uniform convergence of the approximate trajectories ${u}_{\lambda }^{\alpha }(t) \rightarrow u_{\lambda }(t)$ on interval $[0,+\infty )$ is proved, as α tends to 0. Importantly, it is proved that the solution trajectory u_λ(t) has the exponential convergence rate under the upper Lipschitz continuity of T^− 1 at the origin. Moreover, the proposed differential equation approach is applied to the Karush-Kuhn-Tucker system for a smooth convex optimization problem. A numerical algorithm for getting an approximate solution trajectories of Yosida-regularization based differential equation of nonlinear programming is presented and an illustrative example is implemented by the numerical algorithm to verify the theory developed.