Research report + EUROPT presentation


During these months, we have been primarily working on the idea of Primal-Dual Augmented Lagrangian (PDAL), which is inspired by the work of Griva and Polyak in the NLP setting 'Primal-dual nonlinear rescaling method with dynamic scaling parameter update", Mathematical Programming, (2006)' for the semidefinite programming setting. Based on our numerical observations, Newton's system in the Penalty-Barrier Multipliers/Augmented Lagrangian (PBM/AL) approach could not be solved to the required precision, and as a consequence, in the inner iterations, we came up with many iterations without improvement in the value of dual feasibility which is the matter of ill-conditioning. By applying the PDAL approach, we avoid solving Newton's system to complete precision, and that is why we expect less trouble with ill-conditioning. 

The good news is that we could implement this approach in Matlab on three different classes of problems: 

  1. selected SDPLIB examples (by B. Borchers), 
  2. selected large-scale examples from the collection by K. Toh, 
  3. medium to significant problems from truss topology design (by M. Kocvara). 
  • As a summary of our numerical results:primal-dual updates improve the performance and stability of PBM/AL methods, 
  • it can be applied to large-scale SDPs, high-precision results obtained, 
  • if a reasonable preconditioner available. 

Details about the PDAL approach and its application on truss topology design have been discussed in this archive paper on "Barrier and penalty methods for low-rank semidefinite programming with application to truss topology"


I could also get the chance to have a presentation on "A Primal-Dual Augmented Lagrangian Algorithm for SDPs in truss topology design" in Europt 2021 (the 18th international workshop on continuous optimization). The workshop is the annual event of the EUROPT continuous optimization working group of EURO (The Association of European Operational Research Societies), taking place virtually in Toulouse-France on the 7th-9th of July. 

Comments

Post a Comment

Popular posts from this blog

Summary of my research's theoretical part

Image
During the last year of my study, we have been exploring the convergence analysis of the developed algorithm (PBM-AL) . We developed a dual convergence analysis for the penalty barrier multiplier augmented Lagrangian (PBM-AL) method using a hyperbolic penalty function. Our analysis showed that PBM-AL can be characterized as a proximal point algorithm based on Fenchel conjugates. This allows for a more efficient scheme without sacrificing convergence results. Notably, our analysis relaxes the assumption of exact subproblem solutions, introducing an early stopping condition based on Fenchel duality. We also proposed a primal-dual update scheme ( PDAL ) to ensure dually feasible iterates and allowed controlled violation of equality constraints for the use of iterative solvers. A relaxation of Fenchel duality using an exact penalty function was utilized for the early stopping criterion. Below are the key literature sources that were utilized for the analysis and derivation of the conve...
Read more

POEMA Final Workshop

Image
 From September 5th to 9th, 2022, the final POEMA workshop was held in the romantic city of Paris😍. The Sorbonne University Pierre and Marie Curie Campus provided a fitting location for this open international conference on Polynomial Optimization, which aimed to disseminate the network's results. Almost all ESRs planned to present their final results, making it exciting and significant to see everyone from the project in person again. The daily presentations were informative, providing opportunities to connect with other researchers and expand our network. We had many friendly conversations with the other ESRs, discussing our challenges and life😊. On September 7th, I was pleased to present " A Primal-Dual Augmented Lagrangian Method for the solution of Convex Semidefinite Programming Problems ." The feedback and questions I received were invaluable, allowing me to delve deeper into certain parts of my results. ( The YouTube link of my presentation ). On September 8th,...
Read more

First day of working

Image
Hello everyone, I am Arefeh (ESR12) my working title of the thesis is “On convex nonlinear semidefinite programming concepts for the solution of p olynomial optimization problems”. I am hosted by the Mathematical Optimization group in the Mathematics Department of the Friedrich-Alexander-University of Erlangen-Nuremberg. The team, led by Michael Stingl, has strong expertise in algorithmic optimization as well as optimization with (partial) differential equations. I am deeply honored that I would be a part of the POEMA from now on. In this weblog, I will be posting  articles describing my research life, my secondment or any topic related to my research in the POEMA. Today is the first day of my working at FAU. I am really really excited about this new chapter of my study life and I am also a little bit stressed out! I have met my supervisor and our team in the mathematics department. the senior researcher of the team, Dr. Fabian Wein, helped me a lot to learn some formaliti...
Read more