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.

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, J. Convex Anal. 32 (2025), 511-544.
A. Goodwin, A.S. Lewis, G. López-Acedo, A. Nicolae, Convex optimization on CAT(0) cubical complexes, Adv. in Appl. Math. 165 (2025), 102849, 20 pp.
A. Viorel, Asymptotic behavior of a second order dynamical system obtained from time scaling and averaging a gradient flow (submitted).

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).