ODSGLT

Grant CNCS - UEFISCDI:
Optimization and dynamical systems: geometric and logic-based techniques

Project code: PN-III-P1-1.1-TE-2019-1306

Period: November 2020 - October 2022

Host: Babeş-Bolyai University

Research team: Adriana Nicolae (principal investigator), Adriana Buică, Adrian Viorel, Mihai-Radu Truşcă

Abstract: This project deals with problems in (convex and nonconvex) optimization and in dynamical systems relying on geometric-analytic tools and techniques suited with the structure of the spaces under consideration. In part, ideas stemming from an applied subfield of logic, called proof-mining, will also be used. More precisely, we will address specific algorithms in geodesic metric spaces of bounded curvature in the sense of Alexandrov, but will focus as well on other more subtle constructions and results with important applications in optimization theory, among others. In addition, we will apply proof mining techniques to obtain effective results for algorithms considering the context as general as possible. Besides, we will study other interesting properties in optimization and dynamical systems, including the finite length and periodicity of solutions of differential equations.

Summary: In this project we addressed research directions corresponding to the two parts of the project:

Optimization in Alexandrov spaces.
Asymptotic behavior of nonlinear algorithms and continuous dynamical system.

The obtained results fit within the following main objectives of the project:

Objective 1.1: Alternating projections for possibly nonconvex sets.
Objective 1.2: New tools for analysis in metric spaces.
Objective 1.3: Extension of mappings.
Objective 2.1: Regularity properties and rates of convergence.
Objective 2.2: Periodicity.

We highlight below some ideas from each stage of the project:

Stage 1: We addressed objectives 1.1, 2.1, and 2.2. We initiated the study of the alternating projection method for two sets that may not be convex in the setting of metric spaces with bounded curvature. Based on an interplay between ideas and techniques from proof mining and geometric analysis, we analyzed a pursuit-evasion game using a uniform betweenness property. We studied the numerical long-time integration of damped Hamiltonian systems by formulating and analyzing a numerical scheme that preserves the structure and, therefore, the main qualitative properties of the system. Bifurcations of periodic solutions from a resonant period manifold in forced systems are intensively studied in the literature, usually assuming that the period manifold is normally non-degenerate. We dealt with smooth systems in dimension two, considering bifurcations from a normally degenerate cycle.
Stage 2: We addressed objectives 1.1, 1.2, 2.1, and 2.2. We continued the study of nonconvex alternating projections by highlighting the two key geometric ingredients in a standard intuitive analysis of local linear convergence. The first is a transversality-like condition on the intersection; the second is a convexity-like condition on one set: "uniform approximation by geodesics". We developed several basic building blocks for extending the basics of classical convex analysis in Alexandrov spaces with curvature bounded above starting with a definition and a study of subgradients. We analyzed rates of convergence for some adaptive step discretization algorithms of gradient systems. We proved that uniformly exponentially stable abstract linear evolution equations are Ulam-Hyers stable on the unbounded time interval [0,∞). Then we studied the more general situation when the abstract linear evolution equation is uniformly exponentially dichotomic on the time interval (-∞,+∞). We also started to study deeper the finite-dimensional case.
Stage 3: We addressed all objectives. We further investigated classes of nonconvex sets that frequently appear in modern optimization problems, as well as other aspects of the method of alternating projections. We continued the study of subgradients in Alexandrov spaces by giving versions of classical results formulated in terms of our notion. We investigated a second-order dynamical system, relevant for optimization, which is obtained by scaling and averaging a convex gradient flow. After giving the concept of Ulam-Hyers stability with uniqueness, we showed that this concept is equivalent to the exponential dichotomy of abstract linear evolution equations which are exponentially bounded. For the finite-dimensional case we mainly solved an open problem for equations with periodic coefficients.

In short, this project addressed theoretical aspects of optimization and dynamical systems using techniques from different areas which resulted in an interdisciplinary approach to the considered problems.

Publications:

U. Kohlenbach, G. López-Acedo, A. Nicolae, A uniform betweenness property in metric spaces and its role in the quantitative analysis of the "Lion-Man" game, Pacific J. Math. 310 (2021), 181-212.
A. Viorel, C.D. Alecsa, T.O. Pinţa, Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems, Discrete Contin. Dyn. Syst. 41 (2021), 3319-3341.
A. Buică, Ulam-Hyers stability and exponentially stable evolution equations in Banach spaces, Carpathian J. Math. 37 (2021), 339-344.
A.S. Lewis, G. López-Acedo, A. Nicolae, Local linear convergence of alternating projections in metric spaces with bounded curvature, SIAM J. Optim. 32 (2022), 1094-1119.
A. Buică, Ulam-Hyers stability and exponentially dichotomic evolution equations in Banach spaces, Electron. J. Qual. Theory Differ. Equ. (2023), Paper No. 8, 10 pp.
A. Buică, Bifurcations from a normally degenerate cycle in forced planar differential equations, Nonlinear Differ. Equ. Appl. 30 (2023), Article nr. 63, 18 pp.
A. Buică, G. Tőtős, Characterization of Ulam-Hyers stability of linear differential equations with periodic coefficients, J. Math. Anal. Appl. 530 (2024), 127739, 20 pp.
A.S. Lewis, G. López-Acedo, A. Nicolae, Basic convex analysis in metric spaces with bounded curvature, SIAM J. Optim. 34 (2024), 366-388.
G. López-Acedo, A. Nicolae, Remarks on the continuity of convex functions in geodesic spaces, Topol. Methods Nonlinear Anal. 63 (2024), 299-307.
A.S. Lewis, A. Nicolae, T. Tian, Local geometry of feasible regions via smooth paths (submitted).
A. Viorel, Asymptotic behavior of a second order dynamical system obtained from time scaling and averaging a gradient flow (submitted).

Work in progress: Other research papers are close to completion. Details will be added once they are submitted for publication.

Conference talks:

Adriana Nicolae, A quantitative analysis of a discrete version of the lion and man game, Oberwolfach Workshop on Mathematical Logic: Proof Theory, Constructive Mathematics, Oberwolfach Research Institute for Mathematics, Germany, November 9-14, 2020 (online, invited).
Adrian Viorel, Adaptive gradient descent and slow manifolds, 3rd Romanian Itinerant Seminar on Mathematical Analysis and its Applications, "1 Decembrie 1918" University of Alba-Iulia, Romania, September 10-12, 2021 (onsite).
Adriana Nicolae, A discrete lion and man game: geometric and quantitative aspects, BIRS-IASM Workshop: New Frontiers in Proofs and Computation, Banff International Research Station, Canada, and Institute for Advanced Study in Mathematics, China, September 12-17, 2021 (online, invited).
Mihai-Radu Truşcă, Some local fixed point theorems and applications, Conference on Functional Equations and Inequalities, Mathematical Research and Conference Center of the Institute of Mathematics of the Polish Academy of Sciences, Bedlewo, Poland, September 12-18, 2021 (onsite).
Adriana Buică, Persistence and bifurcation of periodic solutions in nonautonomous systems, Second Virtual Workshop on Dynamical Systems, University of Sao Paulo, Brazil, November 10-12, 2021 (online, invited).
Adriana Nicolae, Local linear convergence of alternating projections, Workshop on Modern Nonsmooth Optimization, University of Washington, Seattle, USA, August 8-11, 2022 (onsite, invited).
Adriana Nicolae, Instances of proof mining in optimization theory, International Conference on Applied Proof Theory 2022, Università degli Studi "G. d'Annunzio", Pescara, Italy, August 29 - September 2, 2022 (onsite, invited).
Adriana Nicolae, The betweenness property, XIII International Symposium on Generalized Convexity and Monotonicity, Instituto de Matemática y Ciencias Afines, Lima, Peru, September 13-16, 2022 (online).

Seminar talks:

Adriana Buică, Ulam-Hyers stability and exponentially stable evolution equations in Banach spaces, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, March 4, 2021 (onsite).
Mihai-Radu Truşcă, Local fixed point theorems and open mapping principles, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, March 18, 2021 (onsite).
Adriana Buică, Ulam-Hyers stability and exponentially dichotomic evolution equations in Banach spaces, The International Online Seminar of GSD-UAB, Centre de Recerca Matematica, Barcelona, Spain, March 22, 2021 (online, invited).
Adrian Viorel, Geometric numerical integrator, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, April 8, 2021 (online).
Adriana Nicolae, Alternating projections and curvature, Virtual IMUS Seminar: Trends in Metric Fixed Point Theory and Related Areas, Instituto de Matemáticas Universidad de Sevilla, Spain, April 15, 2021 (online, invited).
Adriana Nicolae, Geometry of nonconvex alternating projections, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, April 22, 2021 (online).
Adriana Nicolae, Subdifferentials via normal cones in CAT(0) spaces, Virtual IMUS Seminar: Trends in Metric Fixed Point Theory and Related Areas, Instituto de Matemáticas Universidad de Sevilla, Spain, September 21, 2021 (onsite, invited).
Adriana Buică, Persistenţa şi continuarea soluţiilor periodice în sisteme periodice bidimensionale, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, November 4, 2021 (online).
Adriana Nicolae, Uniform approximation by geodesics and related concepts, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, November 25, 2021 (online).
Adriana Nicolae, Alternating projections and convexity-like properties, Seminar of the Research Center for Logic, Optimization & Security, Department of Computer Science, University of Bucharest, Romania, March 1, 2022 (online, invited).
Mihai-Radu Truşcă, Applications of some local fixed point theorems, Seminar on Nonlinear Operators and Differential Equations, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania, March 3, 2022 (onsite).
Adriana Nicolae, The method of alternating projections, Mini-course within the Activity Plan of the Mathematics Doctorate Programme 2022, Instituto de Matemáticas Universidad de Sevilla, Spain, June 13-15, 2022 (onsite, invited).