Two methods based on second-order dynamical systems for solving a special class of nonlinear optimization problems

Juhe Sun; Guolin Huang; Li Wang; Ning Ma; Juhe Sun; Guolin Huang; Li Wang; Ning Ma

doi:10.3934/era.2026088

Electronic Research Archive

2026, Volume 34, Issue 3: 1957-1987. doi: 10.3934/era.2026088

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Research article

Two methods based on second-order dynamical systems for solving a special class of nonlinear optimization problems

1.
School of Science, Shenyang Aerospace University, Shenyang 110136, China
2.
School of Mechatronics Engineering, Shenyang Aerospace University, Shenyang 110136, China

Received: 08 November 2025 Revised: 15 February 2026 Accepted: 25 February 2026 Published: 05 March 2026

Symbolic functions, as an important class of nonsmooth functions, play a key role in many fields such as system control and optimization theory. In this study, the special case of nonlinear optimization problems with sign function constraints is discussed in depth. Based on the smooth approximation theory, the study first adopts an approximate substitution method to smooth the nonsmooth constraints and then transforms them into differentiable optimization problems, thus effectively avoiding the numerical computational difficulties caused by the sign function. Second, the constrained optimization problem is transformed into an unconstrained optimization problem by constructing an exact penalty function. Third, two forms of second-order dynamical systems are established to solve the problem, and a rigorous theoretical analysis of the stability of these systems is conducted. Finally, the convergence and computational efficiency of the proposed method are verified by numerical simulation experiments of the system, and the simulation results fully prove the effectiveness and practicality of the algorithm.
- symbolic functions,
- nonlinear optimization problems,
- smooth approximation,
- penalty function method,
- second-order dynamical systems
Citation: Juhe Sun, Guolin Huang, Li Wang, Ning Ma. Two methods based on second-order dynamical systems for solving a special class of nonlinear optimization problems[J]. Electronic Research Archive, 2026, 34(3): 1957-1987. doi: 10.3934/era.2026088

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Abstract

Symbolic functions, as an important class of nonsmooth functions, play a key role in many fields such as system control and optimization theory. In this study, the special case of nonlinear optimization problems with sign function constraints is discussed in depth. Based on the smooth approximation theory, the study first adopts an approximate substitution method to smooth the nonsmooth constraints and then transforms them into differentiable optimization problems, thus effectively avoiding the numerical computational difficulties caused by the sign function. Second, the constrained optimization problem is transformed into an unconstrained optimization problem by constructing an exact penalty function. Third, two forms of second-order dynamical systems are established to solve the problem, and a rigorous theoretical analysis of the stability of these systems is conducted. Finally, the convergence and computational efficiency of the proposed method are verified by numerical simulation experiments of the system, and the simulation results fully prove the effectiveness and practicality of the algorithm.

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