APPLICATIONS OF STOCHASTIC OPTIMISATION IN NON-LINEAR AND DISCONTINUOUS DESIGN SPACES Lorenz Drack Hossein S. Zadeh Maritime Platform Division School of Business Information Technology Defence Science and Technology Organisation Royal Melbourne Institute of Technology Department of Defence GPO Box 2497V PO Box 4331, Melbourne, VIC 3001, Australia Melbourne VIC 3001, Australia lorenz.drack@dsto.defence.gov.au hossein.zadeh@rmit.edu.au KEYWORDS: Stochastic, Multi-objective, Optimal Design, Optimization, Simulation ABSTRACT The application of a stochastic optimiser to two problems in engineering design is presented. The benefits of using such an optimiser in conjunction with a calculus based method are discussed, and its ability to succeed in non-linear and discontinuous design spaces is shown in light of two aerospace design optimisation problems: the design of quiet and efficient propellers and the design of a manoeuvre controller for a satellite structure. INTRODUCTION Successful engineering design processes of complex sys- tems require that numerous design variables and constraints be taken into account across multiple disciplines. The systematic modification of design parameters relying on judgement in a manual design process is often ineffective, and the benefits of computer-aided design optimisation can reduce design time, improve design through improved methodology, solve complex interactions and ultimately reduce the cost of design. Multidisciplinary design problems require an optimiser ca- pable of efficiently handling local minima in non-linear and discontinuous design spaces of high dimensionality. Tradition- ally optimisers rely on a good starting point to obtain a solution or even to converge, thus an additional requirement is that an acceptable solution be found without a good initial starting point to the optimisation. Furthermore, the optimiser must be robust, as the computational expense of objective func- tion calculation makes convergence on non-optimal solutions unacceptable. The application of a two-stage optimisation process that meets these requirements is discussed here. The first stage uses the unconstrained stochastic optimisation method Simulated Annealing (SA) (Ingber, 1989) to obtain a good solution. Once the region of an acceptable minimum has been found, a con- strained non-linear programming method is used to converge on the final solution. Two very different aerospace design problems to which this optimiser was successfully applied are discussed, these being the design of high performance propellers subject to noise constraints, and the design of a manoeuvre controller for a satellite. OPTIMISATION METHODOLOGY SA amounts to a stochastic search over a cost landscape, being directed by noise that is gradually reduced during convergence. The origins of SA lie in the statistical mechanics of condensed matter physics. A cooling liquid will solidify into an optimal crystalline structure when the lowest atomic energy state is attained. The optimal state is achieved through specific cooling schedules - the progress of temperatures known as annealing. The reader is referred to Ingber (Ingber, 1989) and Drack, Zadeh et al (Drack, Zadeh, Wharington, Herszberg and Wood, 1999) for details of the implementation used in the applications discussed here. There are several reasons for not adopting a more tradi- tional, calculus-based approach, to the design optimisation problem. Firstly, they rely on gradient information that is supplied either analytically or calculated numerically. For en- gineering applications, analytical gradients are rarely available, thus numerical methods must be employed introducing the possibility of numerical error. Furthermore, the cost function must be continuous to the first or second derivative, depending on the method used. These optimisers are not suitable for design variables of integer value, as gradients become infinite. SA does not calculate or estimate gradients, thus it is free from these restrictions. Another problem that afflicts calculus-based methods is the need for a good initial guess to the solution (Arora, 1989), in which convergence is often not possible or reliable if the starting point is poor. This becomes a serious issue in design spaces of higher dimensionality, where an unsuitable initial guess made by the designer may not seem unrealistic, or intuition fails due to the large number of variables. In addition, an unsuitable starting point combined with a design space of high dimensionality compounds the problem by lessening the possibility of convergence, slowing the optimisation process considerably. The tendency of gradient-based optimisers to become trapped in local minima is well known (Gage, 1994). One of the most attractive features of SA is that it is less susceptible to becoming trapped in local minima, since escape from these minima is still possible at non-zero temperature, thus affording great robustness to the optimiser in finding a good solution. Nevertheless, it must be said that in non-linear programming problems SA does not always result in a global minimum (Van Laarhoven, 1987), something that is offset by the fact that for most practical engineering applications, a global minimum is not required and a near global solution is sufficient. SA is well suited to problems of high dimensionality with performance improvements over calculus-based methods becoming more pronounced as the dimensionality increases. The stochastic nature of the algorithm ensures it is suitable for discontinuous design spaces. For this reason, it also does not require a good initial guess for successful convergence. A Proceedings 12th International Conference ASMTA 2005 Khalid Al-Begain, Gunter Bolch, Miklos Telek © ECMS, 2005 ISBN 1-84233-112-4 (Set) / ISBN 1-84233-113-2 (CD)