Seminarof Analysis

Speaker:  Dr. Yashveer Kumar (INESC-ID, IST-ULisbon,Portugal)

Date/time: 02/04/2025 (Wednesday), from 14:15 to 15:15.

Location:  Room 112, Building IV.

Title:  Numerical wavelet approximation scheme fordistributed order fractional differential equations

Abstract:  In thistalk, we introduce a new method for solving distributed-order fractionaldifferential equations using Legendre wavelets. Our approach works for bothsingle-variable and two-variable cases. We combine the Legendre Gaussquadrature formula with the Tau technique to build an operational matrix forthese wavelets. This matrix helps convert complex differential problems intosimpler linear algebraic equations. To show how well our method works, wepresent several test examples.

Abstract:
In this work, we introduce two frameworks for derivative-free bilevel optimization. We consider both the upper and lower-level problems with bound constraints on the variables, as well as general nonlinear constraints, assuming that first-order information is either unavailable or impractical to obtain. Furthermore, we allow both the objective functions and constraints to be nonsmooth. The lower-level problem is solved with an accuracy that is progressively refined during the optimization process.We first propose a line-search-based method for problems where the upper-level is only bound-constrained, analyzing convergence to Clarke-Jahn stationary points when accuracy is allowed to reach its maximum. If a stricter bound is imposed on the refinement process, we prove convergence to approximate stationary points using an extended notion of Goldstein stationarity.We then extend this analysis to a MADS-type approach, initially for bound-constrained problems, investigating both cases: full accuracy refinement and bounded accuracy. For this framework, we provide a convergence analysis similar to that of the line-search-based method. Finally, we discuss how more complex constraints can be handled through an exact penalty function approach embedded in both frameworks, extending convergence results to Clarke-Jahn and approximate stationarity.

Short Bio:

Edoardo Cesaroni earned his Master’s degree in Management Engineering (Decision Models for Management Engineering curriculum) in July 2023 from Sapienza University of Rome. Since November of the same year, he has been a Ph.D. student in the ABRO doctoral program, specializing in Operations Research (MATH-06/A ex MAT/09), at the Department of Computer, Control, and Management Engineering Antonio Ruberti (DIAG) at Sapienza University of Rome, under the supervision of Professor Giampaolo Liuzzi. His main research interests focus on derivative-free optimization and the application of optimization techniques to machine learning problems. During his first year of doctoral studies, he focused on applying optimization techniques to real-world case-studies, specifically working on the optimal sizing of batteries in railway systems and the identification of risk factors for gastric neoplastic lesions in collaboration with medical researchers from Sant'Andrea Hospital.