Seminars

Abstract: In this talk, we consider a conic multiobjective linear optimization problem, where the objective and constraint functions are involving affinely parameterized data uncertainties. We examine necessary and sufficient optimality criteria for robust weak/efficient solutions of the conic robust multiobjective problem. It is shown that the obtained optimality conditions can be expressed in terms of linear matrix inequalities or second-order cone conditions for the corresponding subclasses of conic robust multiobjective programs. We also show how efficient solutions of conic robust multiobjective problems can be found by way of conic programming reformulation problems including semidefinite programming or second-order cone programming problems. In addition, an application to modeling a virtual power plant problem in electricity markets is given.

Abstract: In this talk, we introduce a new sliding mode observer for Lur’e set-valued dynamical systems, particularly addressing challenges posed by uncertainties not within the standard range of observation. Traditionally, most of Luenberger-like observers and sliding mode observer have been designed only for uncertainties in the range of observation. Central to our approach is the treatment of the uncertainty term which we decompose into two components: the first part in the observation subspace and the second part in its complemented subspace. We establish that when the second part converges to zero, an exact sliding mode observer for the system can be obtained In scenarios where this convergence does not occur, our methodology allows for the estimation of errors between the actual state and the observer state. This leads to a practical interval estimation technique, valuable in situations where part of the uncertainty lies outside the observable range. Finally, we show that our observer is also a T-observer as well as a strong H^infinity  observer.

Abstract: Solving linear equations with additional requirements/prior information on solutions can be modeled as nonsmooth optimization problems with regularized cost functions and linear constraints. This simple method has been successfully used in many different areas such as compressive sensing, imaging, and statistical/machine learning. In this talk, I will show that solving regularized linear inverse problems is highly stable and efficient. Connections to the theory of exact recovery will be storied. Several open questions simple enough for any undergraduate students to understand will be discussed.

Abstract: Mean Field Games (MFGs) provide a powerful framework for modeling complex real-world phenomena, including the evolution of public sentiment. The MFG system consists of two coupled partial differential equations (PDEs) governing two key unknowns: the density of individuals holding a particular opinion and their expected reward associated with that opinion.

A fundamental challenge in MFGs is the forecasting problem—predicting the evolution of both density and reward functions given their initial conditions. Due to the inherent instability of solutions, this problem remains notoriously difficult. Existing literature lacks a numerical method equipped with a rigorous convergence analysis for tackling this issue.

Convexification, using Carleman estimates as its principal tool, has proven effective in addressing a broad range of forward and inverse problems in PDEs. Recently, two new Carleman estimates have been established, enabling the application of Convexification to the MFG forecasting problem. This talk will present both the theoretical foundation and numerical validation of this method.

This is joint work with S. Chen, M. Klibanov, and K. McGoff.

Abstract:  First-order methods are the backbone of modern optimization, with applications spanning engineering, machine learning, and operations research. While classical methods rely on exact gradient information, many real-world problems introduce inexactness arising from stochasticity, smoothing techniques, or derivative-free settings. This talk presents methods designed to handle a broad class of nonconvex problems under various inexactness models together. We show how these methods unify and extend classical techniques such as proximal point, augmented Lagrangian, and recent gradient-based deep learning optimizers.

Abstract: Inverse source problems involve determining an unknown source from indirect measurements associated with the source. These problems, which arise in a variety of applications such as nondestructive testing, radar, and medical imaging, have been an active research topic in the mathematics, engineering, and physics communities for several decades. However, these problems are highly ill-posed, posing significant challenges in developing efficient numerical methods for their solution. In this talk, we will present our recent results in developing numerical methods with novel imaging functions for solving inverse source problems using boundary data at a single frequency. Theoretical justifications and numerical simulations of the proposed method will be discussed. Our approach is non-iterative, computationally efficient, and straightforward to implement. This talk is based on joint work with Isaac Harris and Thu Le.