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<p>System of nonlinear equations under some kind of constraints is a major problem that appears in numerical analysis which often raises interesting challenges. Its applications can be seen in engineering, economic sciences, biology and other applied sciences. In some applications, the derivative of the nonlinear equations are not available or the dimensions is so huge that Jacobian evaluations directly are prohibitive. In this paper we deal with nonlinear systems of equations with box constraints, that is, the solution must respect some bounds (lower and upper bounds). We propose an algorithm based on an affine-scaling matrix (devised by Coleman and Li in Math. Programm. <b>67</b>, 189–224, <CitationRef CitationID="CR7">1994</CitationRef>) combined with Quasi-Newton-secant update schemes, which leads to a Jacobian-free approach for this interesting and challenging mathematical problem. Under some usual assumptions we assure that the algorithm either converges to a solution or to a stationary point of an auxiliary minimization problem, regardless the initial guess taken. Numerical results in a class of problems from the literature show that the computational performance of our method is competitive with a known algorithm based on the discrete Newton approach.</p>

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Quasi-Newton interior point method for solving nonlinear system of equations with box constraints

  • 期刊论文

  • 发表日期：

    2025年5月8日

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  • Juliano B. Francisco

摘要

System of nonlinear equations under some kind of constraints is a major problem that appears in numerical analysis which often raises interesting challenges. Its applications can be seen in engineering, economic sciences, biology and other applied sciences. In some applications, the derivative of the nonlinear equations are not available or the dimensions is so huge that Jacobian evaluations directly are prohibitive. In this paper we deal with nonlinear systems of equations with box constraints, that is, the solution must respect some bounds (lower and upper bounds). We propose an algorithm based on an affine-scaling matrix (devised by Coleman and Li in Math. Programm. 67, 189–224, 1994) combined with Quasi-Newton-secant update schemes, which leads to a Jacobian-free approach for this interesting and challenging mathematical problem. Under some usual assumptions we assure that the algorithm either converges to a solution or to a stationary point of an auxiliary minimization problem, regardless the initial guess taken. Numerical results in a class of problems from the literature show that the computational performance of our method is competitive with a known algorithm based on the discrete Newton approach.