Relaxed verification for continuous problems

Abstract We analyze the complexity of verifying whether a given element is close to a solution element. Closeness is measured by two nonnegative parameters, e and α. If α = 0, then we get the strong verification problem which usually cannot be solved in the worst-case setting, see Woźniakowski, J. Complexity, 8, 1992. For α > 0, we have the relaxed verification problem which is studied in this paper in the worst-case setting. We show that the (e, α)-verification complexity is roughly equal to the η-computation complexity, where η = η(e, α). Under certain general assumptions we prove that η(e, α) = Θ(eαr), where r depends on the problem norm, r ϵ [0, 1]. We also show that adaptation sometimes helps significantly.