by Arefeh Kavand, University of Erlangen-Nuremberg
My secondment in NAG took place completely 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 more than 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 demonstrated by numerical expriments for different class of SDPs, including examples from SDPLIB, medium to large graph problems as well as truss topology optimization problems. The purpose of this secondment was to computationally study and explore how to effectively solve certain class of SDP problems, including SDP relaxations coming from optimal power flow (OPF) problems by using our developed algorithm at FAU. The classical OPF problem is a nonconvex NLP, on the other hand the SDP belongs to convex optimization and can guarantee global optimal solution using a desired algorithm so it is worth to study how to properly reformulate the classical OPF to a SDP model and benefits from the SDP techniques .
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Related literature as well as a package called "MATPOWER"-an open source Matlab-language M-files for solving steady-state power system simulation and optimization problems such as power flow  - has been introduced by NAG, as it was a totally new subject for me, in our weekly online meetings with Shuanghau Bai, we mostly discussed on the structure and modelling of the SDP relaxation for these specific problems. In this open source MATPOWER package, another software package for optimization modelling tool YALMIP as well as a semidefinite programming solver compatible with YALMIP, such as SEDUMI or SDPT3 are required. The plan was to do the required changes in modeling OPF problems in order to be able to plug in the Primal-Dual PBM-AL solver instead of so called solvers, so I
 Bai, Xiaoqing, et al. "Semidefinite programming for optimal power flow problems." International Journal of Electrical Power and Energy Systems 30.6-7 (2008): 383-392