(118h) A Second-Order Linesearch Procedure within Newton’s Method for Highly Nonlinear Steady-State Systems Simulation

Newton’s method is widely acknowledged as the primary approach for tackling coupled systems of Nonlinear Algebraic Equations (NLAEs), particularly within Chemical Engineering and Process Systems Engineering (Kang et al., 2022). Pantelides' seminal work (Pantelides, 1988a), forming the cornerstone of NLAE solution methodology, has been instrumental in the development of industry-standard process simulation tools like SpeedUp (Pantelides, 1988b) and gPROMS (PSE, 2011).

Linesearch, a crucial component of Newton’s method, ensures global convergence, guaranteeing convergence to a local solution from any starting point while satisfying all simultaneous nonlinear equations (Bellavia and Morini, 2003). Despite Newton’s method being considered established both theoretically and algorithmically, leaving little room for further improvements, this contribution focuses on enhancing the linesearch procedure and revealing significant advancements over existing methods.

Specifically, this study aims to incorporate second-order information in a computationally efficient manner to improve the performance of the linesearch procedure, especially for highly nonlinear equation systems. Nonlinearity, particularly near the starting point, can substantially hinder algorithmic efficiency, necessitating frequent step reductions at the expense of function evaluations and major iterations involving Jacobian evaluations and factorizations (Gill and Zhang, 2024).

The proposed approach leverages a a higher-order Taylor series expansion around the operating point of a major iteration in Newton’s algorithm, coupled with a custom Jacobian vector product finite difference scheme. This combination requires only one additional Jacobian evaluation to construct a locally accurate fourth-degree polynomial approximating the merit function along the search direction.

In addition to the theoretical advancements, this contribution provides computational evidence supporting the claim that for highly nonlinear systems, significant computational savings and enhanced solution procedure stability can be achieved. Utilizing a Python implementation, linear subsets of equations are treated separately to boost the efficiency of function and Jacobian evaluations, aligning with standard practices in professional software development. While Python may not be a high-performance language, its suitability for rapid algorithm prototyping and validation precedes potential transfer to higher-performance languages like C++.

Moreover, given Newton’s method central roles in various iterative solution tools, such as its repeated use within a Differential-Algebraic Equations (DAEs) integrators and potentially Partial Differential-Algebraic Equations (PDAEs) solvers, the significance of this work extends even further. Future research endeavors will explore these areas, building upon the foundations laid by this study.

References

Bellavia, S., Morini, B., 2003, Global convergence of classical linesearch interior point methods for MCPs, Journal of Computational and Applied Mathematics 151 (1), 171-199, https://doi.org/10.1016/S0377-0427(02)00745-8.

Gill, P.E., Zhang, M., 2024, A projected-search interior-point method for nonlinearly constrained optimization, Computational Optimization and Applications, https://doi.org/10.1007/s10589-023-00549-1.

Kang, Y., Luo, Y., Yuan, X., 2022, Recent progress on equation-oriented optimization of complex chemical processes, Chinese Journal of Chemical Engineering 41, 162-169, https://doi.org/10.1016/j.cjche.2021.10.018.

Pantelides, C.C., 1988, The consistent initialization of differential-algebraic systems, SIAM Journal on Scientific and Statistical Computing 9 (2), https://doi.org/10.1137/0909014.

Pantelides, C.C., 1988, SpeedUp – recent advances in process simulation, Computer and Chemical Engineering 12 (7), 745-755, https://doi.org/10.1016/0098-1354(88)80012-7.

Process Systems Enterprise Ltd., 2011, gPROMS Advanced User Guide, Process Systems Enterprise Ltd., London, United Kingdom, http://www.psenterprise.com/.