TY - JOUR

T1 - Efficient Data Structures for Representation of Polynomial Optimization Problems

T2 - Implementation in SOSTOOLS

AU - Jagt, Declan

AU - Shivakumar, Sachin

AU - Seiler, Peter

AU - Peet, Matthew

N1 - Publisher Copyright:
© 2017 IEEE.

PY - 2022

Y1 - 2022

N2 - We present a new data structure for representation of polynomial variables in the parsing of sum-of-squares (SOS) programs. In SOS programs, the variables s(x;P) are polynomial in the independent variables x, but linear in the decision variables P. Current SOS parsers, however, fail to exploit the semi-linear structure of the polynomial variables, treating the decision variables as independent variables in their representation. This results in unnecessary overhead in storage and manipulation of the polynomial variables. To reduce this computational overhead, we introduce a new representation of polynomial variables, the dpvar structure, which allows the parser to exploit the structure of the decision variables. We show that use of the dpvar structure significantly reduces the computational complexity of the polynomial operations required for parsing SOS programs. We further show that the memory complexity required to store polynomial variables is significantly reduced when using the dpvar structure, particularly when combined with the MATLAB Compressed Sparse Column (CSC) matrix representation. Finally, we incorporate the dpvar structure into SOSTOOLS 4.00, and test performance for several polynomial optimization problems.

AB - We present a new data structure for representation of polynomial variables in the parsing of sum-of-squares (SOS) programs. In SOS programs, the variables s(x;P) are polynomial in the independent variables x, but linear in the decision variables P. Current SOS parsers, however, fail to exploit the semi-linear structure of the polynomial variables, treating the decision variables as independent variables in their representation. This results in unnecessary overhead in storage and manipulation of the polynomial variables. To reduce this computational overhead, we introduce a new representation of polynomial variables, the dpvar structure, which allows the parser to exploit the structure of the decision variables. We show that use of the dpvar structure significantly reduces the computational complexity of the polynomial operations required for parsing SOS programs. We further show that the memory complexity required to store polynomial variables is significantly reduced when using the dpvar structure, particularly when combined with the MATLAB Compressed Sparse Column (CSC) matrix representation. Finally, we incorporate the dpvar structure into SOSTOOLS 4.00, and test performance for several polynomial optimization problems.

KW - Computational methods

KW - LMIs

KW - Large-scale systems

KW - Stability of nonlinear systems

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U2 - 10.1109/LCSYS.2022.3183650

DO - 10.1109/LCSYS.2022.3183650

M3 - Article

AN - SCOPUS:85132777775

SN - 2475-1456

VL - 6

SP - 3493

EP - 3498

JO - IEEE Control Systems Letters

JF - IEEE Control Systems Letters

ER -