Lasserre hierarchy for large scale polynomial optimization in real and complex variables

TL;DR

This paper extends the Lasserre hierarchy to large-scale polynomial optimization in real and complex variables, demonstrating its effectiveness on an industrial power flow problem with thousands of variables.

Contribution

It introduces a complex number generalization, a multi-ordered hierarchy exploiting sparsity, and a block diagonal structure to handle large-scale problems efficiently.

Findings

Successfully optimized a power flow problem with 4,500 variables.

Generalized Lasserre hierarchy to complex variables for better tractability.

Developed a multi-ordered hierarchy that exploits problem sparsity.

Abstract

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.