Pattern search in the presence of degeneracy

This paper deals with generalized pattern search (GPS) algorithms for linearly constrained optimization. At each iteration, the GPS algorithm generates a set of directions that conforms to the geometry of any nearby linear constraints. This set is then used to construct trial points to be evaluated during the iteration. In previous work, Lewis and Torczon developed a scheme for computing the conforming directions; however, the issue of degeneracy merits further investigation. The contribution of this paper is to provide a detailed algorithm for constructing the set of directions whether or not the constraints are degenerate. One difficulty in the degenerate case is in classifying constraints as redundant or nonredundant. We give a short survey of the main definitions and methods for treating redundancy and propose an approach to identify nonredundant ε-active constraints, which may be useful for other active set algorithms. We also introduce a new approach for handling nonredundant linearly dependent constraints, which maintains GPS convergence properties without significantly increasing computational cost. Some simple numerical tests illustrate the effectiveness of the algorithm. We conclude by briefly considering the extension of our ideas to nonlinear constrained optimization in which constraint gradients are linearly dependent.