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This paper contains a new convergence analysis for the Lewis and Torczon generalized pattern search (GPS) class of methods for unconstrained and linearly constrained optimization. This analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems. Specifically, even if the objective function is discontinuous or extended-valued, the methods find a limit point with some minimizing properties. Simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm, but also on the directions it uses and on the smoothness of the objective at the limit point in question. The contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm, and it gives previous results as corollaries. [ABSTRACT FROM AUTHOR]

In this paper we discuss the polynomiality of a feasible version of Mehrotra's predictor-corrector algorithm whose variants have been widely used in several interior point method (IPM)-based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important for proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrotra's algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most O(n2log(x°}T s° /e) iterations. By modestly modifying the corrector direction, we reduce the iteration complexity to O(nlog(x°}T s° /e). To ensure fast asymptotic convergence of the algorithm, we changed Mehrotra's updating scheme of the centering parameter slightly while keeping the safeguard. The new algorithms have the same order of iteration complexity as the safeguarded algorithms but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported. [ABSTRACT FROM AUTHOR]