We consider linear problems in the worst-case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the operator uniformly on a convex and balanced set by means of algorithms using at most such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show here, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest for two reasons. First, the homogeneity allows us to extend any error bound on the unit ball to the full input space. Second, homogeneous algorithms are better suited to tackle problems on cones, a scenario far less understood than the classical situation of balls. We use the optimality of homogeneous algorithms to prove solvability for a family of problems defined on cones. We illustrate our results by several examples.

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锥体上的齐次算法和可解问题

我们考虑最坏情况下的线性问题。也就是说，给定一个线性算子和一组可接受的线性测量值，我们希望通过最多使用此类测量值的算法在凸且平衡的集合上均匀地近似该算子。众所周知，一般来说，线性算法不会产生最佳近似值。然而，正如我们在此所示，始终可以使用齐次算法获得最佳近似值。这之所以有趣有两个原因。首先，同质性允许我们将单位球上的任何误差范围扩展到整个输入空间。其次，同构算法更适合解决锥体上的问题，这种情况比球的经典情况更难理解。我们使用齐次算法的最优性来证明锥体上定义的一系列问题的可解性。我们通过几个例子来说明我们的结果。