论文标题
在内点方法中产生的方程系统的近似解决方案，以进行边界约束优化
Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization
论文作者
论文摘要
本文的重点是用于界限的非线性优化的内点方法，其中使用牛顿的方法求解了出现的非线性方程系统。直接解决牛顿系统（提供高质量的解决方案）和解决许多近似牛顿系统的牛顿系统之间的权衡，这些牛顿系统在计算上便宜较低，但提供了较低的质量解决方案。我们建议对牛顿系统的部分和完全近似的解决方案。特定的近似解决方案取决于解决方案处的活动和非活动约束的估计。这些集合在基本启发式方法估计的每次迭代中。部分近似解决方案在计算上是便宜的，而线性方程系统则需要用于完整的近似解决方案。系统的大小取决于解决方案处的非活性约束的估计。此外，我们激励并提出了两种基于由部分近似解决方案组成的中间步骤的类似牛顿的方法。引入理论设置，并给出渐近误差界。我们还提供数值结果，以研究理论框架内外的近似解决方案的性能。
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.
