2017/09/13 by Cédric Josz, Cedric Josz, Josz, Cedric +2 · 5 citations
Decision Sciences · Mathematics · #14Q99 #28A99 #47N10 #90C06 #90C22 #90C26 #Advanced Optimization Algorithms Research #FOS: Mathematics #Numerical methods for differential equations #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design #math.OC #msc:14Q99 #msc:28A99 #msc:47N10 #msc:90C06 #msc:90C22 #msc:90C26
paper · pdf · doi:10.48550/arxiv.1709.04376
30 pages, 6 figures
openalex publication_date 2017/09/13 · arxiv created 2018/05/22 · arxiv updated 2018/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the twenty first century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex to numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto-Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: 1) to use a different relaxation order for each constraint, and 2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries. To the best of our knowledge, the Lasserre hierarchy was previously limited to small scale problems, while we solve a large scale industrial problem with thousands of variables and constraints to global optimality.