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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| Published in | Optimization Vol. 68; no. 7; pp. 1265 - 1277 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia
Taylor & Francis
03.07.2019
Taylor & Francis LLC |
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| Online Access | Get full text |
| ISSN | 0233-1934 1029-4945 1029-4945 |
| DOI | 10.1080/02331934.2018.1452922 |
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| Abstract | 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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| AbstractList | 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. 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.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. |
| Author | Boţ, Radu Ioan Csetnek, Ernö Robert |
| Author_xml | – sequence: 1 givenname: Radu Ioan orcidid: 0000-0002-4469-314X surname: Boţ fullname: Boţ, Radu Ioan email: radu.bot@univie.ac.at organization: Faculty of Mathematics and Computer Science, Babeş-Bolyai University – sequence: 2 givenname: Ernö Robert surname: Csetnek fullname: Csetnek, Ernö Robert organization: Faculty of Mathematics, University of Vienna |
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| SubjectTerms | Convergence convex optimization Damping Differential equations Dynamical systems Lyapunov analysis Newton dynamics nonautonomous systems Optimization Trajectories |
| Title | A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities |
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