Checking non-negativity of polynomials using sum-of-squares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. Additionally, they often become increasingly ill-conditioned. To alleviate these problems, it is important to exploit properties of the analyzed polynomial, and post-process the obtained solution. This technical note describes how the sum-of-squares module in the MATLAB toolbox YALMIP handles these issues.

/pdf/pre-and-post-processing-sum-of-squares-programs-in-practice-2i4e2uwhy6.pdf

Pre- and Post-Processing Sum-of-Squares Programs in Practice

In the last years many results in the area of semidefinite programming were obtained for invariant (finite dimensional, or infinite dimensional) semidefinite programs - SDPs which have symmetry. This was done for a variety of problems and applications. The purpose of this handbook chapter is to give the reader the necessary background for dealing with semidefinite programs which have symmetry. Here the basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.

Invariant semidefinite programs

The field of network synchronization has seen tremendous growth following the introduction of the master stability function (MSF) formalism, which enables the efficient stability analysis of synchr...

https://epubs.siam.org/doi/pdf/10.1137/19M127358X

Symmetry-Independent Stability Analysis of Synchronization Patterns

The multireference alignment problem consists of estimating a signal from multiple noisy shifted observations. Inspired by existing Unique-Games approximation algorithms, we provide a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment problem. Although we show that the MLE problem is Unique-Games hard to approximate within any constant, we observe that our poly-time approximation algorithm for the MLE appears to perform quite well in typical instances, outperforming existing methods. In an attempt to explain this behavior we provide stability guarantees for our SDP under a random noise model on the observations. This case is more challenging to analyze than traditional semi-random instances of Unique-Games: the noise model is on vertices of a graph and translates into dependent noise on the edges. Interestingly, we show that if certain positivity constraints in the SDP are dropped, its solution becomes equivalent to performing phase correlation, a popular method used for pairwise alignment in imaging applications. Finally, we show how symmetry reduction techniques from matrix representation theory can simplify the analysis and computation of the SDP, greatly decreasing its computational cost.

Multireference Alignment using Semidefinite Programming

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.

/pdf/sparse-noncommutative-polynomial-optimization-tkg7mosugv.pdf

Sparse noncommutative polynomial optimization

Motivated by recent interest in group-symmetry in the area of semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of symmetric matrices, or equivalently the irreducible decomposition of the matrix ⁄-algebra generated by symmetric matrices. The method does not require any algebraic structure to be known in advance, whereas its validity relies on matrix ⁄-algebra theory. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes the full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. Numerical examples of truss design are also presented.

/pdf/a-numerical-algorithm-for-block-diagonal-decomposition-of-3lmhs2k0iw.pdf

A Numerical Algorithm for Block-Diagonal Decomposition of Matrix ⁄-Algebras ⁄

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A Numerical Algorithm for Block-Diagonal Decomposition of Matrix ⁄-Algebras ⁄