Journal of Complexity ( IF 1.338 ) Pub Date : 2019-08-24 , DOI: 10.1016/j.jco.2019.101427
Leszek Plaskota; Paweł Siedlecki; Henryk Woźniakowski

Two classes of information have been mainly considered in Information-Based Complexity (IBC) for approximate solutions of continuous problems. The first class is ${\Lambda }^{\mathrm{all}}$ and consists of all linear functionals, whereas the second class is ${\Lambda }^{\mathrm{std}}$ and consists of only function evaluations. A different class of information has been studied in the context of phase retrieval, where it is assumed that only absolute values of linear functionals from $\Lambda \subseteq {\Lambda }^{\mathrm{all}}$ are available. We denote this class $|\Lambda |$ and call it the absolute value information class. For $|\Lambda |$ we need to modify the algorithm error to compensate the missing phase in information values.

The purpose of this paper is to establish the powers of $|{\Lambda }^{\mathrm{all}}|$ and $|{\Lambda }^{\mathrm{std}}|$ in comparison to ${\Lambda }^{\mathrm{all}}$ and ${\Lambda }^{\mathrm{std}}$ for various IBC problems in the worst case setting. Our main result is that $|{\Lambda }^{\mathrm{all}}|$ is roughly of the same power as ${\Lambda }^{\mathrm{all}}$ for linear IBC problems. On the other hand, $|{\Lambda }^{\mathrm{std}}|$ is usually too weak to solve linear problems.

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