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A class of algorithms for unconstrained nonconvex optimization is consideredwhere the value of the objective function is never computed. The class containsa deterministic version of the first-order Adagrad method typically used forminimization of noisy function, but also allows the use of second-orderinformation when available. The rate of convergence of methods in the class isanalyzed and is shown to be identical to that known for first-orderoptimization methods using both function and gradients values. The result isessentially sharp and improves on previously known complexity bounds (in thestochastic context) by Defossez et al. (2020) and Gratton et al. (2022). A newclass of methods is designed, for which a slightly worse and essentially sharpcomplexity result holds. Limited numerical experiments show that the newmethods' performance may be comparable to that of standard steepest descent,despite using significantly less information, and that this performance isrelatively insensitive to noise.

First-Order Objective-Function-Free Optimization Algorithms and Their Complexity