Mixture Surrogate Models for Multi- Objective Optimization

Multi-objective optimization problems (MOOP) involve minimization of more than one objective functions and all of them are to be simultaneously minimized. The solution of these problems involves a large number of iterations. The multi- objective optimization problems related structural optimization of complex engineering structures is usually solved with finite element analysis (FEA). The solution time required to solve these FEA based solutions are very high. So surrogate models or meta- models are used to approximate the finite element solution during the optimization process. These surrogate assisted multi- objective optimization techniques are very commonly used in the current literature. These optimization techniques use evolutionary algorithm and it is very difficult to guarantee the convergence of the final solution, especially in the cases where the budget of costly function evaluations is low. In such cases, it is required to increase the efficiency of surrogate models in terms of accuracy and total efforts required to find the final solutions.In this paper, an advanced surrogate assisted multi- objective optimization algorithm (ASMO) is developed. This algorithm can handle linear, equality and non- linear constraints and can be applied to both benchmark and engineering application problems. This algorithm does not require any prior knowledge for the selection of surrogate models. During the optimization process, best single and mixture surrogate models are automatically selected. The advanced surrogate models are created by MATSuMoTo, the MATLAB based tool box. These mixture models are built by Dempster- Shafer theory (DST). This theory has a capacity to handle multiple model characteristics for the selection of best models. By adopting this strategy, it is ensured that most accurate surrogate models are selected. There can be different kind of surrogate models for objective and constraint functions. Multi-objective optimization of machine tool spindle is studied as the test problem for this algorithm and it is observed that the proposed strategy is able to find the non- dominated solutions with minimum number of costly function evaluations. The developed method can be applied to other benchmark and engineering applications.

Integration of Normative Decision-Making and Batch Sampling for Global Metamodeling

The cost of adaptive sampling for global metamodeling depends on the total number of costly function evaluations and to which degree these evaluations are performed in parallel. Conventionally, samples are taken through a greedy sampling strategy that is optimal for either a single sample or a handful of samples. The limitation of such an approach is that they compromise optimality when more samples are taken. In this paper, we propose a thrifty adaptive batch sampling (TABS) approach that maximizes a multistage reward function to find an optimal sampling policy containing the total number of sampling stages, the number of samples per stage, and the spatial location of each sample. Consequently, the first batch identified by TABS is optimal with respect to all potential future samples, the available resources, and is consistent with a modeler’s preference and risk attitude. Moreover, we propose two heuristic-based strategies that reduce numerical complexity with a minimal reduction in optimality. Through numerical examples, we show that TABS outperforms or is comparable with greedy sampling strategies. In short, TABS provides modelers with a flexible adaptive sampling tool for global metamodeling that effectively reduces sampling costs while maintaining prediction accuracy.

Engineering Design Optimization by Dynamic Cluster based Framework using Mixture Surrogates

The real-world engineering optimization problems utilize complex computational methods like finite element frameworks. These approaches are computationally costly and need high solution time. The work pays attention on finding the optimal solution to these complex engineering problems by using Surrogate Models (SM). SMs are mathematical models, which are utilized to minimize the required number of such costly function evaluations at the time of the optimization cycles. Instead of optimizing the Design Space (DS) as a whole, subregion based strategies are found to be effectual, especially in the cases where prior knowledge of optimal solution is unavailable. In the present work, a surrogate centered optimization scheme is presented for local search, which dynamically sub-divides the DS into an optimum number of sub-regions by choosing the best cluster evaluation techniques as followed by the selection of best mixture SMs for each optimization cycle. For all objective functions and constraint functions in every sub-region, the mixture SMs are created by a combination of two or more single SMs. The MATSuMoTo, the Matlab based SM Toolbox by Juliane Muller and Robert Piché has been adapted for the creation and selection of best mixture SM. In this method, an individual surrogate is combined by utilizing the DempsterShafer theory (DST). Besides the above local search, a global search module is also introduced for ensuring faster convergence. This approach is tested on a constrained optimization benchmark test problem with smaller, disconnected feasible regions. It is perceived that the proposed algorithm accurately located all the local and global optima points with minimum function evaluations. The approach is applied to engineering problems like optimization of Machine Tool Spindle (MTS) design and frontal crash simulation on a full car body. For these engineering application problems also, mixture SMbased sub-region based search strategy is utilized to attain most accurate global optimum solution with a minimal number of costly function evaluations.