TSSOS: a Julia library to exploit sparsity for large-scale polynomial optimization

The Julia library TSSOS aims at helping polynomial optimizers to solve large-scale problems with sparse input data. The underlying algorithmic framework is based on exploiting correlative and term sparsity to obtain a new moment-SOS hierarchy involving potentially much smaller positive semidefinite matrices. TSSOS can be applied to numerous problems ranging from power networks to eigenvalue and trace optimization of noncommutative polynomials, involving up to tens of thousands of variables and constraints.

[1]  Victor Magron,et al.  Chordal-TSSOS: A Moment-SOS Hierarchy That Exploits Term Sparsity with Chordal Extension , 2020, SIAM J. Optim..

[2]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[3]  Victor Magron,et al.  Semialgebraic Optimization for Lipschitz Constants of ReLU Networks , 2020, NeurIPS.

[4]  J. Lasserre,et al.  A Sparse Version of Reznick's Positivstellensatz , 2020, Math. Oper. Res..

[5]  Jie Wang,et al.  TSSOS: A Moment-SOS hierarchy that exploits term sparsity , 2020 .

[6]  Jie Wang,et al.  Exploiting term sparsity in noncommutative polynomial optimization , 2020, Computational Optimization and Applications.

[7]  Masakazu Muramatsu,et al.  SparsePOP: a Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems , 2005 .

[8]  Jakub Mareček,et al.  Proper Learning of Linear Dynamical Systems as a Non-Commutative Polynomial Optimisation Problem , 2020, ArXiv.

[9]  Jakub Marecek,et al.  Fairness in Forecasting and Learning Linear Dynamical Systems , 2020, AAAI.

[10]  Victor Magron,et al.  The Constant Trace Property in Noncommutative Optimization , 2021, ISSAC.

[11]  J. Lasserre,et al.  Exploiting Constant Trace Property in Large-scale Polynomial Optimization , 2020, ACM Trans. Math. Softw..

[12]  M. Ferris,et al.  The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms , 2019, ArXiv.

[13]  D. Henrion,et al.  Exploiting Sparsity for Semi-Algebraic Set Volume Computation , 2019, Foundations of Computational Mathematics.

[14]  Victor Magron,et al.  Interval Enclosures of Upper Bounds of Roundoff Errors Using Semidefinite Programming , 2016, ACM Trans. Math. Softw..

[15]  Daniel K. Molzahn,et al.  Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables , 2017, SIAM J. Optim..

[16]  Jie Wang,et al.  CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization , 2020, ArXiv.

[17]  George A. Constantinides,et al.  Certified Roundoff Error Bounds Using Semidefinite Programming , 2015, ACM Trans. Math. Softw..

[18]  Victor Magron,et al.  A sublevel moment-SOS hierarchy for polynomial optimization , 2021, Computational Optimization and Applications.

[19]  WakiHayato,et al.  Sums of Squares and Semidefinite Program Relaxations for Polynomial Optimization Problems with Structured Sparsity , 2006 .

[20]  Arie M. C. A. Koster,et al.  Treewidth computations I. Upper bounds , 2010, Inf. Comput..

[21]  Iain Dunning,et al.  JuMP: A Modeling Language for Mathematical Optimization , 2015, SIAM Rev..

[22]  Jean B. Lasserre,et al.  Convergent SDP-Relaxations for Polynomial Optimization with Sparsity , 2006, ICMS.

[23]  Milan Korda,et al.  Sparse moment-sum-of-squares relaxations for nonlinear dynamical systems with guaranteed convergence , 2020, 2012.05572.

[24]  Mark Cannon,et al.  COSMO: A conic operator splitting method for large convex problems , 2019, 2019 18th European Control Conference (ECC).

[25]  Igor Klep,et al.  Sparse noncommutative polynomial optimization , 2019 .

[26]  Jie Wang,et al.  SparseJSR: A Fast Algorithm to Compute Joint Spectral Radius via Sparse SOS Decompositions , 2020, 2021 American Control Conference (ACC).

[27]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .