Homogeneous algorithms and solvable problems on cones

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 n 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.