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Summary of my research's theoretical part

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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...
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POEMA Final Workshop

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 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,...
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My third (final) research stay at Tilburg University

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Fortunately, as COVID-19  😫  restrictions gradually lifted, I received confirmation that I could undertake my research stay on-site at Tilburg University in the Netherlands, commencing on April 1, 2022, and continuing until the end of May 2022 😇 (a duration of 2 months). My initial impression of the Netherlands was overwhelmingly positive 😍. The entire country was teeming with bicycles, which was entirely logical given its remarkably flat terrain. Moreover, I was fortunate to be there during the most enchanting time of the year—springtime  ⚘😍 !  Undoubtedly, everyone is familiar with the renowned Dutch tulips, correct? Throughout my stay, I endeavoured to maximize my experience by exploring the famed cities, all easily accessible and budget-friendly by train. I even had the opportunity to venture to Brussels over the weekend, a mere 90-minute train ride away! During my time in the Netherlands, I had the privilege of attending the celebrated Dutch Flower Para...
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My second research stay at NAG (Oxford-UK)

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    My secondment in  NAG   took place online in November-December 2021 in collaboration with Jan Fiala and Shuanghua Bai. As an introduction,  NAG  provides industry-leading numerical software and technical services to banking and finance, energy, engineering, and market research, as well as academic and government institutions. NAG also offers Automatic Differentiation, Machine Learning, and Mathematical Optimization products, as well as world-class technical consultancy across HPC and Cloud HPC, code porting and optimization, and other areas of numerical computing. Founded over 50 years ago from a multi-university venture, NAG is headquartered in Oxford, UK, with offices in the UK, US, EU, and Asia. At FAU (my host university), my supervisor (Prof. Michael Stingl) and I could design a Primal-Dual Penalty/Barrier Multiplier Augmented Lagrangian ( PBM-AL ) algorithm for the solution of Semidefinite Programs (SDP), the efficiency of the overall algorithm is...
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POEMA Learning Week 2 (Toulouse-France, on-site)

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Finally, after almost two years, we could have another on-site event !😍 I have had the chance to be enrolled in and participate in the  POEMA Learning Week 2  (13th-17th of September, Toulouse-France). It was a great learning and networking opportunity with many informative and entertaining talks and activities. I also could give a talk on " A Primal-Dual Augmented Lagrangian Method for the Solution of Convex Semidefinite Programming Problems with Applications in Structural Optimization  ." In addition, my supervisor "Michael Stingl," gave a Lecture:  On the solution of Semidefinite Programs ," which was an excellent tutorial on the solution of semidefinite programs (SDP), including interior point as well as augmented Lagrangian-type methods. It feels great to see all the partners, including Ph.D. students and supervisors, again in person. We could discuss and share the progress of our research and get the chance to hang out and have so much fun after every s...
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Research report + EUROPT presentation

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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:  selec...
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An overview of 1 year research!

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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...
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