Jorge Nocedal This paper introduces a modified Byrd-Omojokun (BO) trust region algorithm to address the challenges posed by noisy function and gradient evaluations. The original BO method was designed to solve equality constrained problems and it forms the backbone of some interior point methods for general large-scale constrained optimization. A key strength of the BO method is … Read more
The development of nonlinear optimization algorithms capable of performing reliably in the presence of noise has garnered considerable attention lately. This paper advocates for strategies to create noise-tolerant nonlinear optimization algorithms by adapting classical deterministic methods. These adaptations follow certain design guidelines described here, which make use of estimates of the noise level in the … Read more
Classical trust region methods were designed to solve problems in which function and gradient information are exact. This paper considers the case when there are bounded errors (or noise) in the above computations and proposes a simple modification of the trust region method to cope with these errors. The new algorithm only requires information about … Read more
A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is noisy, the optimal choice requires information about the noise level and higher-order derivatives of the function, which is often … Read more
The problem of interest is the minimization of a nonlinear function subject to nonlinear equality constraints using a sequential quadratic programming (SQP) method. The minimization must be performed while observing only noisy evaluations of the objective and constraint functions. In order to obtain stability, the classical SQP method is modified by relaxing the standard Armijo … Read more
The goal of this paper is to investigate an approach for derivative-free optimization that has not received sufficient attention in the literature and is yet one of the simplest to implement and parallelize. It consists of computing gradients of a smoothed approximation of the objective function (and constraints), and employing them within established codes. These … Read more
The motivation for this paper stems from the desire to develop an adaptive sampling method for solving constrained optimization problems in which the objective function is stochastic and the constraints are deterministic. The method proposed in this paper is a proximal gradient method that can also be applied to the composite optimization problem min f(x) … Read more
This paper describes an extension of the BFGS and L-BFGS methods for the minimization of a nonlinear function subject to errors. This work is motivated by applications that contain computational noise, employ low-precision arithmetic, or are subject to statistical noise. The classical BFGS and L-BFGS methods can fail in such circumstances because the updating procedure … Read more
The classical convergence analysis of quasi-Newton methods assumes that the function and gradients employed at each iteration are exact. In this paper, we consider the case when there are (bounded) errors in both computations and establish conditions under which a slight modification of the BFGS algorithm with an Armijo-Wolfe line search converges to a neighborhood … Read more
This paper presents a finite difference quasi-Newton method for the minimization of noisy functions. The method takes advantage of the scalability and power of BFGS updating, and employs an adaptive procedure for choosing the differencing interval h based on the noise estimation techniques of Hamming (2012) and Moré and Wild (2011). This noise estimation procedure … Read more