Efficient Data Structures for Representation of Polynomial Optimization Problems: Implementation in SOSTOOLS

Declan Jagt, Sachin Shivakumar, Peter Seiler, Matthew M. Peet

Abstract: 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s(x;P)$ </tex-math></inline-formula> are polynomial in the independent variables <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> , but linear in the decision variables <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> . 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dpvar</i> 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.