General Algorithm for Analytical Calculations of Jacobi Matrix Elements in Sparse Nonlinear Programming Problems

• Published in: Moscow University computational mathematics and cybernetics Vol. 46; no. 4; pp. 183 - 196
• Main Author: Zlobin, D. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2022; Springer Nature B.V
• Subjects: Algorithms; Jacobi matrix method; Jacobian matrix; Linear programming; Mathematics; Mathematics and Statistics; Nonlinear programming; numerical solution; Jacobian matrix of constraints; symbolic mathematics; nonlinear programming; speed-efficient code; variables with indices
• ISSN: 0278-6419; 1934-8428
• DOI: 10.3103/S0278641922040082

The author deals with sparse non-linear programming problems of high dimension. When solving such problems numerically by gradient means, we must calculate the Jacobi constraint matrix, which contains a significant number of elements that are identically equal to zero. Reducing the number of calculations by eliminating operations for calculating obviously zero elements results in a considerable gain in the speed of solving the problem. The author proposes a general algorithm for analyzing the structure of the Jacobian matrix and calculating the values of its elements for a class of problems distinguished by variables with indices in the symbolic notation of the condition. Constraints in such problems are parameterized by sets of indices and form several families in which the constraint expressions have the same structure. The algorithm allows the use of symbolic mathematics to analyze the Jacobian matrix and obtain formulas that do not require the calculating of zero elements. The main ideas of the algorithm are illustrated with an example.