Homogeneous Analytic Center Cutting Plane Methods with Approximate Centers

IF 1.4 3区数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Optimization Methods & Software Pub Date : 1999-01-01 DOI:10.1080/10556789908805753

Y. Nesterov, Olivier Péton, J. Vial

引用次数: 11

Abstract

In this paper we consider a homogeneous analytic center cutting plane method in a projective space. We describe a general scheme that uses a homogeneous oracle and computes an approximate analytic center at each iteration. This technique is applied to a convex feasibility problem, to variational inequalities, and to convex constrained minimization. We prove that these problems can be solved with the same order of complexity as in the case of exact analytic centers. For the feasibility and the minimization problems rough approximations suffice, but very high precision is required for the variational inequalities. We give an example of variational inequality where even the first analytic center needs to be computed with a precision matching the precision required for the solution.

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具有近似中心的齐次解析中心切割平面方法

本文研究了射影空间中的齐次解析中心切割平面方法。我们描述了一种使用齐次oracle并在每次迭代中计算近似解析中心的一般方案。该技术适用于一个凸可行性问题，变分不等式，和凸约束最小化。我们证明了这些问题可以用与精确解析中心相同的复杂度来解决。对于可行性和最小化问题，粗略的近似就足够了，但对于变分不等式，则需要很高的精度。我们给出了一个变分不等式的例子，其中即使第一个分析中心也需要以与解所需的精度相匹配的精度计算。

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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来源期刊

Optimization Methods & Software 工程技术-计算机：软件工程

CiteScore

4.50

自引率

0.00%

发文量

审稿时长

7 months

期刊介绍： Optimization Methods and Software publishes refereed papers on the latest developments in the theory and realization of optimization methods, with particular emphasis on the interface between software development and algorithm design. Topics include: Theory, implementation and performance evaluation of algorithms and computer codes for linear, nonlinear, discrete, stochastic optimization and optimal control. This includes in particular conic, semi-definite, mixed integer, network, non-smooth, multi-objective and global optimization by deterministic or nondeterministic algorithms. Algorithms and software for complementarity, variational inequalities and equilibrium problems, and also for solving inverse problems, systems of nonlinear equations and the numerical study of parameter dependent operators. Various aspects of efficient and user-friendly implementations: e.g. automatic differentiation, massively parallel optimization, distributed computing, on-line algorithms, error sensitivity and validity analysis, problem scaling, stopping criteria and symbolic numeric interfaces. Theoretical studies with clear potential for applications and successful applications of specially adapted optimization methods and software to fields like engineering, machine learning, data mining, economics, finance, biology, or medicine. These submissions should not consist solely of the straightforward use of standard optimization techniques.