Exploiting Explicit and Implicit Structure in Complex Optimization Problems

Abstract

Abstract : A quasi-Newton version of a VU-bundle algorithm for minimizing a convex function, with knowledge of only one subgradient value at each point, was perfected to the point where numerical superlinear convergence could be observed. The algorithm is important, because it is the type needed for minimizing implicitly defined functions resulting from applying decomposition, relaxation and/or dualization techniques to complex real-world optimization problems. Also, valuable research was carried out for nonconvex objective functions. This included a non-VU bundle method for composite functions where the outer function is a positively homogeneous convex function and the inner vector function is a smooth mapping. Such an explicitly known structure separates the two difficulties of nonconvexity and nonsmoothness by allowing only the components of the inner mapping to be nonconvex and only the outer function to be nonsmooth. This new algorithm was shown to be convergent to stationary points and judged to be the best performer out of four methods tested on many examples. Also, significant progress was made on designing a VU algorithm to run on general semismooth functions. This entailed making a V-model based bundle method subalgorithm with convergence to stationary points.