Approximate Real Function Maximization and Query Complexity

Consider the problem of approximate function maximization: Given a continuous real function with domain [0, 1], what is its approximate maximum of absolute error 2−n? More precisely, we are interested in the complexity of approximating the real functional MAX : C[0,1] 3 f 7→max0≤x≤1 ∈ R up to a prescribed absolute error 2−n. Regarding the complexity, we examine quantitative and parameterized bounds; that is, the complexity is measured in parameters of the input function f , e. g., in its Lipschitz constant. For that, we model the approximation of real functionals by using methods from recursive analysis where computations of functions are black-boxed via function oracles. As an extension, the retrievable information about a function is defined by protocols. At first, we analyze the mutual simulation of these protocols, their computability and their computational complexity. Second, we determine the query complexity of approximating the MAX-functional relative to selected protocols. As a result, the simulation complexity often depends on the function’s Lipschitz constant and the notion of a modulus of strong unicity. Furthermore, it is necessary to give the Lipschitz constant to any simulation algorithm as long as non-adaptive information is concerned. However, no such limitation holds for adaptive information and access to a range approximating protocol.