The focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

本文的重点是用于边界约束非线性优化的内点方法，其中使用牛顿法求解非线性方程组。在直接求解提供高质量解决方案的牛顿系统与求解许多计算上较便宜但给出较低质量解决方案的近似牛顿系统之间需要权衡取舍。我们为牛顿系统提出部分和完全近似的解决方案。具体的近似解决方案取决于解决方案上活动约束和非活动约束的估计。这些集合在每次迭代时都由基本启发式算法估算。局部近似解在计算上不昂贵，而完整的近似解需要求解线性方程组。系统的大小取决于解决方案中非活动约束的估计。此外，我们提出并建议了两种基于牛顿类方法，它们是基于由部分近似解组成的中间步骤的。介绍了理论设置并给出了渐近误差范围。我们还给出了数值结果，以研究理论框架内外的近似解的性能。