Convex Optimization and Numerical Issues

P. Garoche
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Abstract

This chapter aims at providing the intuition behind convex optimization algorithms and addresses their effective use with floating-point implementation. It first briefly presents the algorithms, assuming a real semantics. As outlined in Chapter 4, convex conic programming is supported by different methods depending on the cone considered. The most known approach for linear constraints is the simplex method by Dantzig. While having an exponential-time complexity with respect to the number of constraints, the simplex method performs well in general. Another method is the set of interior point methods, initially proposed by Karmarkar and made popular by Nesterov and Nemirovski. They can be characterized as path-following methods in which a sequence of local linear problems are solved, typically by Newton's method. After these algorithms are considered, the chapter discusses approaches to obtain sound results.
凸优化与数值问题
本章旨在提供凸优化算法背后的直觉,并解决它们在浮点实现中的有效使用。它首先简要介绍了算法,假设一个真实的语义。如第4章所述,凸二次规划有不同的方法支持,这取决于所考虑的圆锥。最著名的线性约束方法是丹齐格的单纯形法。虽然相对于约束的数量具有指数级的时间复杂度,但单纯形方法通常表现良好。另一种方法是内点法集,最初由Karmarkar提出,由Nesterov和Nemirovski推广。它们可以被描述为路径跟踪方法,其中求解一系列局部线性问题,通常采用牛顿方法。在考虑了这些算法之后,本章讨论了获得可靠结果的方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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