## 1 Introduction  
Structural systems operate in variable and uncertain environments that impact building performance and integrity. While maintaining a satisfactory level of safety and reliability, structural systems are expected to satisfy their design criteria, such as performance and serviceability. Uncertainties and tolerances are commonly present and even anticipated during the design process and construction. The prediction of building response to these events has a crucial role in the design and optimization of structures. In the optimal design of structural systems, variations in the material, loads, and parameters of a structure are random variables with a certain probability distribution rather than fixed coefficients. With the existence of such uncertainties, the problem of reliability analysis and its reflection must be addressed during the design process of structures. The inclusion of reliability analysis within the design process could drastically reduce the total cost of development and the risk of failure. Structural design optimization can be viewed as a process that methodically searches for improved engineering solutions and designs while considering requirements and constraints that exist in the field. In deterministic design optimization (DDO) [1], the uncertainties of a structural system, such as material properties, forces, models, and sizes, are taken into account subjectively and indirectly, by means of safety factors or load combinations with coefficients specified in the design codes. The procedures of deterministic design optimization consider the effects of uncertainties in a qualitative way, without incorporation of quantitative characterization of the probabilistic nature of system and design. Non-deterministic optimization has emerged as an alternative form to accurately model the safety-under-uncertainty aspect of the optimization problem. Methods for considering uncertainty in the optimal design process can be categorized into two groups. Reliability-based design optimization (RBDO) aims to meet a specific target failure probability for objectives and/or constraint functions that include uncertainty parameters. Robust design optimization (RDO), on the other hand, minimizes the sensitivity of the objectives and constraints to the uncertainty design parameters. The growing interest in the identification of feasible, reliable, and robust solutions in structural design optimization has led to significant advances in theoretical and numerical developments, such as simulation and analysis approaches and optimization algorithms. [PERSON] and [PERSON] [2] proposed a robust design optimization method for structures with considerations of parameter variation and sensitivity to variation by using genetic and non-linear programming algorithms. [PERSON] et al. [3] developed a robust optimal design criterion for a tuned mass damper device under random vibrations. The robustness of optimal solutions was obtained by solving a multi-objective optimization problem in which the mean and standard deviations of the deterministic objection were minimized. A robust design of a vibration absorber under uncertainties in mass and stiffness of the dynamic system was utilized to demonstrate the robust design approach by [PERSON] et al. [4]. Robust design optimization was performed by a direct first-order perturbation method based on a Taylor expansion. The development of nonlinear programming-based RDO [5; 6; 7] in various fields with diverse applications and a thorough review of RDO methods are illustrated in the literature, e.g., by [PERSON] and [PERSON] [8], as well as [PERSON] et al. [4]. In RBDO, a design at each iteration step is checked to determine whether it satisfies probabilistic constraints by reliability analysis, which requires repetitive computational efforts [9]. To assess the probabilistic constraints in RBDO, various methods have been developed [10; 11].