Abstract
In this paper a new capability is introduced in the Multipoint Approximation Method (MAM) focusing on the development of metamodels for the objective and constraint functions as well as solving an optimization problem within a trust region when all, or some of, design variables are defined on a discrete set of values. This development targets an important class of industrial problems where it is allowed to perform a response function evaluation only for points that have discrete values of the design variables. The main aim is to develop an optimization technique applicable to large design optimization problems in which the response functions are computationally expensive, could be affected by numerical noise, and occasionally are impossible to evaluate at some points. The new discrete optimization capability is demonstrated on a well established benchmark problem and compared to published results. I. Introduction he multipoint approximation method (MAM) is an optimization technique utilising high quality explicit approximations in order to reduce the total number of calls for analysis needed to solve large-scale optimization problems. This approach has been influenced by the previous work 1-3 on two-point approximation methods. This was later generalized to multi-point approximations 4-6 . The present approach is based on the assembly of multiple metamodels. Such an approach was used in Refs. 7 and 8 where a metamodel assembly was based on the weighted sum formulation. In the current work, following the previous research for a case of continuous variables 6 , a metamodel assembly is built using linear regression. The regression coefficients of the assembly model are not scaled weights but tuning parameters determined by the least squares method. Therefore, the tuning parameters of the assembly model are not restricted to a positive range but may have negative values as well. This technique is utilized in the multipoint approximation method within the mid-range approximation framework 4-6 . In this paper a special attention is paid to a case when some of, or all, design variables are discrete. It is assumed that it is allowed to perform a response function evaluation only for points that have discrete values of the design variables 9 . This makes it impossible to initially ignore the discrete nature of the design variables, solve a continuous problem and adjust the result to the given set of the discrete values, as sometimes suggested 10 . Thus, new procedures for sampling, metamodel building, their use for solving an optimization problem with discrete properties within a rust region, and the trust region adaptation strategy is required. In this paper, a discrete form of the coordinate search algorithm 9 is implemented within the MAM to search for the solution in the sub-space of the discrete variables only starting from the optimal continuous values obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. This is compared to the use of a simple rounding-off method for the discrete variables. The new mixed discrete-continuous capability was implemented within the MAM and tested on well established benchmark problems including the ten-bar truss problem 11 . The obtained results are compared with the solution obtained by a binary GA and with a continuous case to demonstrate the efficiency of the technique.
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