An overview of 1 year research!
Greetings, and welcome to 2021! 😀💪 As we embark upon a new year, we are excited to set new goals and reflect upon the progress made in research during the past year. Over the course of the project's initial months, we have dedicated ourselves to exploring the concept of generalized Augmented Lagrangians for the solution of semidefinite programs, with a particular emphasis on the structures and difficulties that arise in the context of polynomial optimization. Two main aspects of our research have thus far been taken into account: ill-conditioning and a low-rank structure in the data. To address the ill-conditioning problem, we have proposed a general class of barrier functions, known as penalty-barrier-functions, which provide an upper bound for the eigenvalues of the Hessian of the Augmented Lagrangian. Although this approach has been reported in the literature previously, our project combines it with a regularization technique, providing a uniform lower bound for the eigenvalue