Erweiterung spezifischer Techniken der konvexen Optimierung über die klassischen Anwendungsgebiete hinaus
Final Report Abstract
The addressed topics with a particular impact in the optimization community were those related to the formulation of numerical algorithms for nonsmooth convex and nonconvex optimization problems. For the provided iterative schemes, an exhaustive analysis with respect to the convergence of the generated sequences of iterates and of objective function values at these has been carried out. The numerical performances of the proposed algorithms have been validated in the context of treating practical problems in fields like image and signal processing, location theory, clustering, machine learning, optimal portfolio selection and network communication. To this end we first made use on the concept of smoothing through the Moreau-Yosida regularization of the functions occurring in the objective with both variable and constant smoothing parameters in combination with fast gradient methods. Further, we paid attention to the design of primal-dual splitting methods for monotone inclusion problems with complex and intricate structures in Hilbert spaces, having in mind that the systems of optimality conditions for convex optimization problems can be represented in this way. We formulated primal-dual algorithms of Douglas-Rachford type, algorithms suitable for dealing with sums of linearly composed maximally monotone operators and algorithms of penalty type. The latter proved to be appropriate when solving constrained minimization problems. Furthermore, we investigated the convergence rates of the primal-dual methods of forward-backward-forward and forward-backward type and proposed under the use of variable step sizes accelerated variants of these algorithms. Algorithmic schemes with inertial and memory effects came also into the focus of our research, being inspired by the implicit discretization of second-order differential systems in time. This idea led to the formulation of an inertial primal-dual Douglas-Rachford splitting algorithm for complexly monotone inclusion problems. The inertial pattern was employed to the forward-backward-forward algorithm and to its primal-dual extension, too. We also addressed the solving nonsmooth nonconvex optimization problems with algebraic features. For the minimization of a sum of a proper and lower semicontinuous function with a differentiable function with Lipschitz continuous gradient we proposed a forward-backwardforward and a forward-backward algorithm, both with inertial and memory effects, for which we were able to prove convergence results to a critical point of the objective. Last, but not least, by relying on Lyapunov analysis, we investigated the asymptotic behaviour of trajectories generated by first-order dynamical systems governed by a nonexpansive operator. The problems of finding the zeros of sums of maximally monotone operators and of solving unconstrained and constrained nonsmooth convex optimization problems, the latter via penalty techniques, have been approached from continuous perspective, too.
Publications
- (2013) - A Douglas-Rachford type primal-dual method for inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM Journal on Optimization 23(4), 2541-2565
Boţ RI, Hendrich C
(See online at https://doi.org/10.1137/120901106) - (2013) - A primal-dual splitting algorithm for finding zeros of sums of maximal monotone operators. SIAM Journal on Optimization 23(4), 2011-2036
Boţ RI, Csetnek ER, Heinrich A
(See online at https://doi.org/10.1137/12088255X) - (2014) - Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization. Journal of Mathematical Imaging and Vision 49(3), 551-568
Boţ RI, Hendrich C
(See online at https://doi.org/10.1007/s10851-013-0486-8) - (2014) - Forward-Backward and Tseng's type penalty schemes for monotone inclusion problems. Set-Valued and Variational Analysis 22(2), 313-331
Boţ RI, Csetnek ER
(See online at https://doi.org/10.1007/s11228-014-0274-7) - (2015) - Inertial Douglas-Rachford splitting for monotone inclusion problems. Applied Mathematics and Computation 256(1), 472-487
Boţ RI, Csetnek ER, Hendrich C
(See online at https://doi.org/10.1016/j.amc.2015.01.017) - (2015) - On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems. Mathematical Programming 150(2), 251-279
Boţ RI, Csetnek ER, Heinrich A, Hendrich C
(See online at https://doi.org/10.1007/s10107-014-0766-0) - (2015) - On the convergence rate of a forward-backward type primal-dual splitting algorithm for convex optimization problems, Optimization 64(1), 5-23
Boţ RI, Csetnek ER
(See online at https://doi.org/10.1080/02331934.2014.966306) - (2016) - An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO Journal on Computational Optimization 4(1), 3-2
Boţ RI, Csetnek ER, László S
(See online at https://doi.org/10.1007/s13675-015-0045-8)
