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Nonmonotone line searches for optimization algorithms

Sachs E. W., Sachs S. M.

Control and Cybernetics

2011

Vol. 40, no 4

1059-1075

In this paper we develop a general convergence theory for nonmonotone line searches in optimization algorithms. The advantage of this theory is that it is applicable to various step size rules that have been published in the past decades. This gives more insight into the structure of these step size rules and points to several relaxations of the hypotheses. Furthermore, it can be used in the framework of discretized infinite-dimensional optimization problems like optimal control problems and ties the discretized problems to the original problem formulation.

Reduced order models in PIDE constrained optimization

Sachs E. W., Schu M.

Control and Cybernetics

2010

Vol. 39, no 3

661-675

Mathematical models for option pricing often result in partial differential equations originally starting with the Black-Scholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one solves the problems mentioned last numerically, this yields large linear systems of equations with dense matrices. However, by using the special structure and an iterative solver the problem can be handled efficiently. To further reduce the computational cost in the calibration phase we implement a reduced order model, like proper orthogonal decomposition (POD), which proves to be very efficient. In this paper we use a special multi-level trust region POD algorithm to calibrate the option pricing model and give numerical results supporting the gain in efficiency.