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 -