Convex Analysis and Monotone Operators: Forward and Backward
Final Report Abstract
New algorithmic methods for solving classes of nonsmooth (convex) optimization problems unapproachable by the existing proximal point type methods up to that moment were proposed and their efficacy was proven by illustrating applications consisting in iteratively solving real life problems in Finance Mathematics and Logistics, respectively, by implementing our new algorithms in matlab. On short, we provided the first ever inertial forward-backward proximal point type methods in the literature for solving nonsmooth convex vector optimization problems with composite objectives and the first proximal point method for solving nonlinear minmax location problems involving perturbed minimal time functions via conjugate duality. Potential applications of these results can be found, for instance, in the fields of Machine Learning or Image Processing, however the spectre of possible employment of our results is potentially wider. The first duality approach in conditional optimization is presented. Although this work is mainly theoretical, applications in Finance Mathematics can be foreseen. Further, we presented the first ever contribution towards inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure towards the minimum norm zero of the underlying monotone. The way we employed an ODE-solver in matlab for determining the mentioned zero should have a significant resonance in the literature, by opening the door for other iterative methods specific to differential equations to be employed in numerical optimization via proximal point methods and dynamical systems. Applications are possible in various fields, basically everywhere proximal point methods can be used for solving concrete problems, and not only. We identified an optimization problem whose conjugate dual turns out to be a portfolio optimization problem where the risk is assessed via the entropic value-at-risk and the optimal solutions of the latter are recovered via the corresponding optimality conditions. This is a novel approach that mirrors the method we proposed and makes us confident of the possibilities of extending them for iteratively solving general convex constrained optimization problems. Such problems appear in many fields, so again there are quite a large number of possible applications areas for our results. On the other hand, we proposed an extension of proximal point type algorithms beyond convexity by introducing a new class of generalized convex functions. A first application in solving variational inequalities is presented, too, and something similar happens in a paper, where forward-backward-forward methods are employed for solving pseudo-monotone variational inequalities, with applications in dynamic user equilibrium in traffic networks. Although they do not necessarily contain key findings in the framework of this research projects, our works contain solid results as well, with both theoretical and practical relevance as we included in them applications in game theory and entropy optimization. Last but not least, two survey papers had their role in this research project, too, despite the fact that they contain less new results than the other mentioned outcomes. On the first level they turned out to be practical ways of exploiting the extensive literature reviews necessary for writing some of the other articles mentioned above. On the other hand, they have the potential of attracting a large number of citations, also from other areas whose representatives need some tools from convex and numerical optimization, respectively, exposing thus our own results included in them to new audiences.
Publications
- A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality, Journal of Global Optimization 74(1):121–160, 2019
S.-M. Grad, O. Wilfer
(See online at https://doi.org/10.1007/s10898-019-00746-5) - Closedness type regularity conditions in convex optimization and beyond, Frontiers in Applied Mathematics and Statistics - Optimization 2:14, 2016
S.-M. Grad
(See online at https://doi.org/10.3389/fams.2016.00014) - Inertial forward-backward methods for solving vector optimization problems, Optimization 67(7):959–974, 2018
R. I. Bot, S.-M. Grad
(See online at https://doi.org/10.1080/02331934.2018.1440553) - A survey on proximal point type algorithms for solving vector optimization problems, in: H.H. Bauschke, R. Burachik, D.R. Luke (Eds.), “Splitting Algorithms, Modern Operator Theory, and Applications”, Springer-Verlag, Cham, 269–308, 2019
S.-M. Grad
(See online at https://doi.org/10.1007/978-3-030-25939-6_11)
