A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities

We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamica...

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Bibliographic Details
Published inOptimization Vol. 68; no. 7; pp. 1265 - 1277
Main Authors Boţ, Radu Ioan, Csetnek, Ernö Robert
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis 03.07.2019
Taylor & Francis LLC
Subjects
Convergence
convex optimization
Damping
Differential equations
Dynamical systems
Lyapunov analysis
Newton dynamics
nonautonomous systems
Optimization
Trajectories
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Summary:We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories.
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ISSN:0233-1934
1029-4945
1029-4945
DOI:10.1080/02331934.2018.1452922