Lennart Frimannslund

A generating set search method using curvature information

AbstractnDirect search methods have been an area of active research in recent years. On many real-world problems involving computationally expensive and often noisy functions, they are one of the few applicable alternatives. However, although these methods are usually easy to implement, robust and provably convergent in many cases, they suffer from a slow rate of convergence.nnUsually these methods do not take the local topography of the objective function into account. We present a new algorithm for unconstrained optimisation which is a modification to a basic generating set search method. The new algorithm tries to adapt its search directions to the local topography by accumulating curvature information about the objective function as the search progresses.nnThe curvature information is accumulated over a region thus smoothing out noise and minor discontinuities. We present some theory regarding its properties, as well as numerical results. Preliminary numerical testing shows that the new algorithm outperforms the basic method most of the time, sometimes by significant relative margins, on noisy as well as smooth problems.n

A subclass of generating set search with convergence to second-order stationary points

In generalized pattern search algorithms for linearly constrained optimization, certain limit points have been shown to satisfy a pseudo-second-order necessary condition, in which positive semidefiniteness of the Hessian is achieved only with respect to an orthonormal basis. Satisfaction of the full second-order condition can only be ensured if the finite set of positive spanning directions includes the eigenvectors of the Hessian at the limit point. Of course, this is generally not possible in practice, due to the unavailability of derivatives. The present work takes advantage of a chief feature of the more general parent class of generating set search algorithms – that of allowing the use of an infinite (non-dense) set of directions in the limit. If approximate second derivative information is gathered during the iteration process, then the normalized eigenvectors of the approximate Hessian can be used to construct the set of 2n orthonormal directions used by the algorithm. If the Hessian approximation converges in the limit to the true Hessian (or a matrix in the generalized Hessian), then the full second-order optimality conditions can be achieved. Two approaches for generating such a convergent approximate Hessian are described. Numerical examples show the effectiveness of the approach.

Parallel solution of the pooling problem with application to the Cell Broadband Engine Architecture

In this paper we apply an existing simple optimization algorithm to the pooling problem, which is a hard global optimization problem occurring among other places in the petroleum industry. We propose a technique for generating initial solutions and show that the global optimum is found on many standard problems using relatively few starting points. We also show that the proposed solution methodology can be implemented efficiently on the Cell Broadband Engine Architecture, and consequently that it can be solved on the Sony PlayStation 3 gaming console, which can perform a very large number of floating point operations per second compared to its price.

A Class of Methods Combining L-BFGS and Truncated Newton

We present a class of hybrid methods for large-scale nonlinear optimization which allows for a smooth transition between the limited memory BFGS and truncated Newton methods. Numerical experiments suggest that the resulting methods in the class are more robust than the parent methods, in the sense that the methods solve more of the problems tested, at a moderate computational cost.