## 1 Introduction  
Centrifugal compressors have been widely used in small and medium aero-engines due to their apparent advantages in size, pressurization capability, and operating range. With the increasing demand for higher aerodynamic efficiency, broader operating range, and better structural integrity, the optimization of centrifugal compressors has been extensively investigated. Many researchers have focused on a single discipline design optimization for centrifugal compressors, such as the aerodynamic optimization or the structural optimization [1; 2; 3]. With the upgrading of the design requirements, the multidisciplinary design optimization (MDO) methodology has been carried out so as to consider the couplings of different disciplines and the trade-offs among different performance indexes. In recent years, the MDO for centrifugal compressors has drawn an increasing amount of attention. The studies involve the aerodynamic optimizations with stress or eigenfrequency constraints [4; 5] and the structural optimizations with aerodynamic performance constraints [6]. In the optimization of a centrifugal compressor, there may be dozens or even hundreds of design variables. With the increase in design variables, the iteration number of the optimization grows exponentially. The MDO for the centrifugal compressor is a typical multi-objective and multi-point optimization problem. The compressor has to meet the requirements from multiple disciplines and improve the performance at both the design point and off-design points, which may lead to high computational costs. It is inefficient to apply global optimization algorithms, such as evolutionary methods, to find the global optimum of the high dimensionality optimization problem due to the high computational costs of the multidisciplinary design analysis (MDA) simulations. Therefore, the surrogate models have been widely used to reduce searching time. The results have proved that the methods have the potential to satisfy industrial design needs. The commonly used surrogate models in these studies include Kriging [7; 8], radial basis function (RBF) [9], artificial neural networks (ANNs) [10], polynomial response surface (PRS) [11], and support vector regression (SVR) [12]. The accuracy of surrogate models has a great influence on optimization results and may lead to the failure of optimization. To ensure the accuracy of the surrogate models, a large number of initial sampling points should be chosen for a high-dimensional optimization problem, which may be still plagued by the curse of dimensionality. A great deal of literature has utilized the inverse design technique, which is characterized by the parameterization of blade geometry with very few design parameters to overcome this issue [13; 14]. However, the technique could not directly control the parameters that indicate the improvement of the performance. In addition, it is not clear if the global optimum could be achieved by taking the aerodynamic-performance-related parameters (such as the distribution of the blade loading) as local variables. An alternative way is to use the proper orthogonal decomposition (POD), which has been widely used in the field of computational fluid dynamics, such as the flow field reconstruction [15; 16], the airfoil inverse design [17], and the flow characteristics prediction [18]. In the field of optimization, POD is mainly used to construct the reduced order model (ROM) of the high-fidelity simulations to reduce the computational costs [19; 20; 21; 22]. The technique could also be used for the design space reduction [23; 24]. Traditional variable screening techniques reduce the dimensionality of the optimization problem by removing variables that seem to be irrelevant to the problem. They may fail when all variables are equally relevant in the problem or when some variables are relevant only in part of the space [25]. Additionally, the traditional variable screening techniques may also reduce the flexibility of the geometry parameterization and the ability to achieve a better design.