# Adaptive Multi-Level Search for Global Optimization:
## 1 Introduction  
With tremendous advances in computational sciences, information technology, and artificial intelligence, design optimization becomes increasingly popular in many engineering subjects, such as mechanical, civil, structural, aerospace, automotive, and energy engineering. It helps to shorten the design-cycle time and to identify creative designs that are not only feasible but also progressively optimal, given predetermined design criteria. At the outset of design optimization, running a gradient-based algorithm with a multi-start process proves to be very successful in finding the global optimum of simple problems when gradient information is available [1]. While under the pressure of being faced with increasingly complex optimization problems in which derivative information is unreliable or unavailable, researchers gradually focus on the development of derivative-free optimization methods [2] and metaheuristic methods to address this issue. Followed by [PERSON]'s convention [3], modern metaheuristic algorithms such as simulated annealing(SA) [4], genetic algorithms (GA) [5; 6], particle swarm optimization (PSO) [7], and ant colony optimization (ACO) [8] have been applied with good success in solving complex nonlinear optimization problems [9; 10]. The popularity of these nature-inspired algorithms lies in their ease of implementation and the capability to obtain a solution close to the global optimum. However, for many real-life design problems, more than thousands of calls for high-fidelity simulations (for example, computational fluid dynamics simulation) may be executed to seek a near-optimal solution. This is the overwhelming part of the total run time required in the design cycle. Thus, it is desirable to retain the appeal of metaheuristic algorithms on a global search while replacing as many as possible calls to the solver with evaluations on metamodels for the purpose of less computational cost [11]. The typical techniques for metamodel building include Kriging [12], polynomial response surface (PRS) [13], radial basis function (RBF) [14], artificial network (ANN) [15], etc. Among them, PRS and ANN are regression methods that have advantages in dealing with convex problems; Kriging and RBF belong to interpolation methods that are more appropriate for nonconvex or multi-modal problems [16]. Therefore, metamodels have been successfully employed to assist evolutionary optimizations [17; 18; 19] and the PSO method. For example, [PERSON] et al. [20] proposed a hybrid surrogate model formed from a quadratic polynomial and a RBF model to develop a surrogate-based PSO method and applied it to solve mostly low-dimensional test problems and engineering design problems. [PERSON] [21] used RBF surrogates on PSO to identify the most promising trail position surrounding the current overall global best position for solving a 36-dimensional bioremediation problem. However, the inherent nature of the PSO method leads to an extremely large number of calls for function evaluations, which might be prohibitive in simulation-based optimization. In this paper, an adaptively integrated swarm intelligence-metamodelling technique (ASIM) is proposed, which combines multi-level search and model management during the entire optimization process. It orients the solution of the approximate model to the global optimum with a smaller number of iterations of analyses and achieves a higher level of efficiency than conventional approximation methods. Meanwhile, model management in the optimization process has been established, which integrates an adaptive trust-region strategy with a space reduction scheme implemented in the multipoint approximation method (MAM) framework. The model management has been able to facilitate the optimization process and to improve robustness during iterations. It especially has allowed a small perturbation to be assigned to the current position in case of no updates for the optimal position.