First-order methods and gradient dynamical systems for multiobjective optimization / Konstantin Heinrich Sonntag ; Prof. Dr. Sina Ober-Blöbaum, Prof. Dr. Sebastian Peitz, Univ.-Prof. Dr. Radu Ioan Boţ. Paderborn, 2025
Inhalt
- Introduction
- Theoretical background
- Functional analysis
- Differential equations and inclusions
- Differential equations
- Set-valued analysis and differential inclusions
- Differential and integral inequalities
- Multiobjective optimization
- A descent method for nonconvex locally Lipschitz continuous multiobjective optimization
- Generalized derivatives
- Derivation of the descent method
- Descent directions obtained from the multiobjective -subdifferential
- Computation of descent directions by adaptive subderivative sampling
- The final descent method
- Application in bicriterial optimal control of an obstacle problem
- Gradient dynamical systems for convex multiobjective optimization
- An introductory example from scalar optimization
- Review of existing gradient systems for multiobjective optimization
- The multiobjective steepest descent dynamical system (MSD)
- Adaption of (MSD) for constrained MOPs
- Adaption of (MSD) for constrained and nonsmooth MOPs
- The inertial multiobjective gradient system (IMOG)
- Existence results for a generalized differential equation
- The generalized differential equation (D)
- The associated differential inclusion (DI-D)
- Existence of solutions to (D)
- The inertial multiobjective gradient system (IMOG')
- The multiobjective gradient system with asymptotic vanishing damping (MAVD)
- Assumptions
- Discussion of existence and uniqueness of solutions
- Preparatory results
- Asymptotic analysis
- Numerical experiments
- The multiobjective Tikhonov regularized inertial gradient system (MTRIGS)
- An accelerated gradient method for convex multiobjective optimization
- Derivation of the accelerated multiobjective gradient method (MNAG)
- Nesterov's accelerated gradient method for scalar optimization
- Discretization of the system (MAVD)
- Relation to other existing methods
- Asymptotic analysis
- Assumptions
- Preparatory results
- Convergence of function values with rate O(k-2)
- Weak convergence of iterates
- Numerical experiments
- Conclusion
- References