In this paper, we introduce a new variant of the BFGS method designed to perform well when gradient measurements are corrupted by noise. We show that by treating the secant condition using a penalty method approach  , one can smoothly interpolate between updating the inverse Hessian approximation with the original BFGS update formula and not updating the inverse Hessian approximation. Furthermore, we find the curvature condition is smoothly relaxed as the interpolation moves towards not updating the inverse Hessian approximation, disappearing entirely when the inverse Hessian approximation is not updated. These developments allow us to develop an algorithm we refer to as secant penalized BFGS (SP-BFGS) that allows one to relax the secant condition based on the amount of noise in the gradient measurements. Mathematically, SP-BFGS provides a means of incrementally updating the new inverse Hessian approximation with a controlled amount of bias towards the previous inverse Hessian approximation . Practically speaking, this can be used to help replace the overwriting nature of the  BFGS update with an averaging nature that resists the destructive effects of noise . We provide a convergence analysis of SP-BFGS, and present numerical results illustrating the performance of SP-BFGS  in the presence of noisy gradients .
In this paper, we introduce a new variant of the BFGS method designed to perform well when gradient measurements are corrupted by noise. We show that by treating the secant condition with a penalty method approach motivated by regularized least squares estimation , one can smoothly interpolate between updating the inverse Hessian approximation with the original BFGS update formula and not updating the inverse Hessian approximation. Furthermore, we find the curvature condition is smoothly relaxed as the interpolation moves towards not updating the inverse Hessian approximation, disappearing entirely when the inverse Hessian approximation is not updated. These developments allow us to develop a method we refer to as secant penalized BFGS (SP-BFGS) that allows one to relax the secant condition based on the amount of noise in the gradient measurements. SP-BFGS provides a means of incrementally updating the new inverse Hessian approximation with a controlled amount of bias towards the previous inverse Hessian approximation , which allows one to replace the overwriting nature of the original BFGS update with an averaging nature that resists the destructive effects of noise and can cope with negative curvature measurements. We discuss the theoretical properties of SP-BFGS, including convergence when minimizing strongly convex functions in the presence of uniformly bounded noise. Finally, we present extensive numerical experiments using over 30 problems from the CUTEst test problem set that demonstrate the superior performance of SP-BFGS compared to BFGS in the presence of both noisy function and gradient evaluations .
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