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 $s(x;Q)$ are polynomial in the independent variables $x$, but linear in the decision variables $Q$. 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, prohibiting the parser from addressing larger-scale optimization problems. To eliminate this computational overhead, we introduce a new representation of polynomial variables, the "dpvar" structure, that is affine in the decision variables. We show that the complexity of operations on variables in the dpvar representation scales favorably with the number of decision variables. We further show that the required memory for storing polynomial variables is relatively small using the dpvar structure, particularly when exploiting the MATLAB sparse storage structure. Finally, we incorporate the dpvar data structure into SOSTOOLS 4.00, and test the performance of the parser for several polynomial optimization problems.
翻译:我们展示了一种新的数据结构,用于在对方数总和(SOS)方案中代表多元变量。在SOS方案中,变量$(x;Q)是独立的变量的多元值 $x$,但在决定变量中是线性 $Q美元。然而,目前的SOS分析者未能利用多元变量的半线性结构,将决定变量作为独立的变量处理。这导致在多边变量的存储和操作中不必要的间接间接处理,禁止分析器处理大范围的优化问题。为了消除这一计算性管理,我们引入了多元变量的新代表,即“dpvar”结构,这是决定变量中的直线性。我们显示,dpvar代表尺度中变量操作的复杂性优于决定变量的数量。我们进一步表明,使用 dpvar结构存储多边变量所需的记忆相对较小,特别是在利用 MATLAB 分散的存储结构时,我们将多位数个多位数数据测试问题纳入SOS.00OL。