An algorithm is proposed, analyzed, and tested for minimizing locally Lipschitz objective functions that may be nonconvex and/or nonsmooth. The algorithm, which is built upon the gradient-sampling methodology, is designed specifically for cases when objective and generalized gradient values might be subject to bounded, uncontrollable errors. A novel assumption pertaining to such errors in nonsmooth … Read more
Gradient sampling (GS) has proved to be an effective methodology for the minimization of objective functions that may be nonconvex and/or nonsmooth. The most computationally expensive component of a contemporary GS method is the need to solve a convex quadratic subproblem in each iteration. In this paper, a strategy is proposed that allows the use … Read more
We present a new algorithm, called manifold sampling, for the unconstrained minimization of a nonsmooth composite function $h\circ F$ when $h$ has known structure. In particular, by classifying points in the domain of the nonsmooth function $h$ into manifolds, we adapt search directions within a trust-region framework based on knowledge of manifolds intersecting the current … Read more
Recently, optimization problems involving nonsmooth and locally Lipschitz functions have been subject of investigation, and an innovative method known as Gradient Sampling has gained attention. Although the method has shown good results for important real problems, some drawbacks still remain unexplored. This study suggests modifications to the gradient sampling class of methods in order to … Read more
A line search algorithm for minimizing nonconvex and/or nonsmooth objective functions is presented. The algorithm is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed because it typically yields fast convergence to the vicinity of a stationary point, and together with the adaptive GS strategy … Read more
We present an algorithm for the minimization of f : Rn → R, assumed to be locally Lipschitz and continuously differentiable in an open dense subset D of Rn. The objective f may be non-smooth and/or non-convex. The method is based on the gradient sampling (GS) algorithm of Burke et al. [A robust gradient sampling … Read more
We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on … Read more
We give a detailed numerical and theoretical analysis of a stabilization problem posed by V. Blondel in 1994. Our approach illustrates the effectiveness of a new gradient sampling algorithm for finding local optimizers of nonsmooth, nonconvex optimization problems arising in control, as well as the power of nonsmooth analysis for understanding variational problems involving polynomial … Read more