Let f be a continuous function on R n , and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but whose gradient is easily computed where it is defined. We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. No subgradient information is required by the algorithm.When f is locally Lipschitz and has bounded level sets, and the sampling radius ǫ is fixed, we show that, with probability one, the algorithm generates a sequence with a cluster point that is Clarke ǫ-stationary. Furthermore, we show that if f has a unique Clarke stationary pointx, then the set of all cluster points generated by the algorithm converges tox as ǫ is reduced to zero.

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Optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in NASA's Hubble Space Telescope (HST). NASA's success with reconstruction algorithms on the HST has lead to an effort to develop software that can aid and in some cases replace complicated, expensive and error-prone hardware. Among the many applications is HST's replacement, the Next Generation Space Telescope (NGST).This work details the theory of optical wavefront reconstruction, reviews some numerical methods for this problem, and presents a novel numerical technique which we call extended least squares. We compare the performance of these numerical methods for potential inclusion in prototype NGST optical wavefront reconstruction software. We begin with a tutorial of Rayleigh-Sommerfeld diffraction theory.

H ∞ controller design for linear systems is a difficult, nonconvex and typically nonsmooth (nondifferentiable) optimization problem when the order of the controller is fixed to be less than that of the open-loop plant, a typical requirement in e.g. embedded aerospace control systems. In this paper we describe a new matlab package called hifoo, aimed at solving fixed-order stabilization and local optimization problems. It depends on a new hybrid algorithm for nonsmooth, nonconvex optimization based on several techniques, namely quasiNewton updating, bundling and gradient sampling. The user may request hifoo to optimize one of several objectives, including H ∞ norm, which requires either the Control System Toolbox for matlab or, for much better performance, the linorm function in the slicot package. No other external package is required, but the quadratic programming code quadprog from either mosek or the Optimization Toolbox for matlab is recommended. Numerical experiments on benchmark problem instances from the COMPl e ib database indicate that hifoo could be an efficient and reliable computer-aided control system design (CACSD) tool, with a potential for realistic industrial applications.

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