多面体上的切比雪夫投影

V. Zorkaltsev
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引用次数: 0

摘要

定义为线性不等式系统的解集的凸多面体上的加权切比雪夫范数的最小化问题可能有一个非唯一解。而且,在这个问题的解中,可能明显存在多面体上最接近零向量的点不适合担当这个角色的点。它使切比雪夫近似变得特别复杂。为了克服由此产生的问题,使用了哈尔条件,即对所指示问题解的唯一性的要求。这个要求并不总是容易验证的,如果它不是真的,我们也不清楚该怎么办。提出了一种基于对有限序列线性规划问题的内点搜索的算法,该算法对给定的问题总是产生唯一解。所提出的解称为原点在多面体上的切比雪夫投影。证明了该解是一个分量绝对值为帕累托极小的多面体的向量。证明了坐标原点在多面体上的切比雪夫投影集(根据所介绍的算法)和欧几里得投影集是一致的,这些投影集是通过改变最小化欧几里得范数和切比雪夫范数的正权系数而形成的。
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THE CHEBYSHEV PROJECTIONS ON POLYHEDRON
The problem of minimizing weighted Chebyshev norm on a convex polyhedron defined as the set of solutions to a system of linear inequalities may have a non-unique solution. Moreover, among the solutions to this problem, there may be clearly not suitable points of the polyhedron for the role of the closest points to the zero vector. It complicates, in particular, the Chebyshev approximation. In order to overcome the problems arising from this, the Haar condition is used, which means the requirement for the uniqueness of the solution of the indicated problem. This requirement is not always easy to verify and it is not clear what to do if it is not true. An algorithm is presented that always generates a unique solution to the indicated problem, based on the search with respect to interior points for optimal solutions of a finite sequence of linear programming problems. The solution developed is called the Chebyshev projection of the origin onto the polyhedron. It is proved that this solution is a vector of a polyhedron with Pareto-minimal absolute values of the components. It is proved that the sets of Chebyshev (according to the introduced algorithm) and Euclidean projections of the origin of coordinates onto the polyhedron, formed by varying the positive weight coefficients in the minimized Euclidean and Chebyshev norms, coincide.
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