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 values of the operator uniformly on a convex and balanced set by means of algorithms that use at most $n$ such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show in this paper, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest to us 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 that is 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.

翻译：考虑最坏情形下的线性问题。即给定线性算子及一组容许的线性测量，我们希望通过最多使用n次此类测量的算法，在凸平衡集上一致逼近该算子的值。众所周知，线性算法通常无法实现最优逼近。然而本文证明，通过齐次算法总能获得最优逼近。这一发现具有双重意义：其一，齐次性使得我们能将单位球上的任意误差界推广至整个输入空间；其二，齐次算法更适用于处理锥集上的问题——此类场景的研究远不如经典球集情形透彻。我们利用齐次算法的最优性，证明了一族定义在锥集上的问题具有可解性，并通过多个算例展示所得结论。