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A parametric class of trust-region algorithms for unconstrained nonconvexoptimization is considered where the value of the objective function is nevercomputed. The class contains a deterministic version of the first-order Adagradmethod typically used for minimization of noisy function, but also allows theuse of (possibly approximate) second-order information when available. The rateof convergence of methods in the class is analyzed and is shown to be identicalto that known for first-order optimization methods using both function andgradients values, recovering existing results for purely-first order variantsand improving the explicit dependence on problem dimension. This rate is shownto be essentially sharp. A new class of methods is also presented, for which aslightly worse and essentially sharp complexity result holds. Limited numericalexperiments show that the new methods' performance may be comparable to that ofstandard steepest descent, despite using significantly less information, andthat this performance is relatively insensitive to noise.

Complexity of a Class of First-Order Objective-Function-Free Optimization Algorithms