# A Surrogate Model Based Multi-Objective Optimization Method for Optical Imaging System
### Overview of Optical Imaging System Design  
The optical imaging system usually consists of a series of well-designed sequential lenses with constraints in manufacturing, physical size, tolerances, and cost. The excellent performance of the system is typically realized through a careful iterative process, including the definition of performance objectives and optical constraints, construction and minimization of an appropriate merit function comprising these objectives, and constraints to realize the optimum design of the optical system, and then a prediction of the realized performance with a tolerance analysis of the design [21]. The aim of the optimum design of the optical lens under several physical and system constraints is to obtain a series of optimal lens variables with a satisfactory optical performance, such as a low aberration. Optimal variables in lens design include targets of the lens, such as element material, surface curvatures, surface aspherical coefficients, element thicknesses, and spacings. A merit function in the optical design procedure is defined as the measure of optical quality, typically with zero indicating \"perfection\" of the optical system. The value of the merit function is calculated through the process of ray tracing and optical analyses in an optical system. Computers became widely used in optical design because of the high computational complexity of ray tracing [22; 23]. However, the guidance and intervention of competent users is critical in achieving an optimized and well-balanced design solution; even modern high-speed computers with extreme processing power can be applied in the design process [24]. With high-order aspherical surfaces or more optimization variables implemented in modern lens designs processes, the optimization process is becoming further sophisticated with new techniques, such as integrating manufacturing tolerances into optimization in order to achieve minimal performance degradation with as-built lenses [25; 26] or incorporating computational photography steps into the lens design stage [27; 28; 29]. In general, optimization algorithms applied to optical systems can be divided into classical gradient-based optimization algorithms based on the least-squares (LS) method [30; 31; 32; 33; 34; 35] and modern optimization algorithms based on the analogy with natural evolution. The application of the classical LS method in the optimization of optical systems was first proposed by [PERSON] and [PERSON] [30]; since then, a considerable number of researchershave applied or modified this method in different fields. The appealing reason for the application of the LS method in the merit function is the preservation of the information relating to the distribution of the various aberrations. [PERSON] [36] defined a value, referred to as a step length, with the target of controlling and limiting the changes of the constructional parameters in the optical system, and formed the damped least-squares (DLS) method. After that, numerous methods, including altering the additive damping in the DLS into multiplicative damping [31], were proposed to improve the convergence of the DLS. Except for the LS methods, [PERSON] [37] has specified that computers could only be regarded as a tool capable of offering optical designers temporary solutions because qualitative judgments and compromises were required in the optimization of optical systems. A novel concept of aberrations brought up by [PERSON] [38; 39] is prominent, and this is principally due to the practical realization of his computer program, where a novel orthonormal theory of aberrations was applied in the optimization of optical systems. Moreover, the orthonormalization in this theory was improved through the Gram-Schmidt transformation proposed by [PERSON] et al. [40]. The fundamental ideas forming the concept of simulated annealing originated from [PERSON] et al. [41] and were suggested by [PERSON] et al. [42] to be used as an optimization method in various systems, such as optical.