# Truss Structure Optimization with Subset Simulation and Augmented Lagrangian Multiplier Method
## 1 Introduction  
In modern design practice, structural engineers are often faced with structural optimization problems, which aim to find an optimal structure with minimum weight (or cost) under multiple general constraints, e.g., displacement, stress, or bulking limits, etc. It is theoretically straightforward to formulate the structural optimization problem as a constrained optimization problem in terms of equations under the framework of mathematical programming. This, however, is very challenging to accomplish in practice, at least, due to three reasons: (a) the number of design variables is large; (b) the feasible region is highly irregular; and (c) the number of design constraints is large. Flexible optimization methods, which are able to deal with multiple general constraints that may have non-linear, multimodal, or even discontinuous behaviors, are desirable to explore complex design spaces and find the global optimal design. Optimization methods can be roughly classified into two groups: gradient-based methods and gradient-free methods. Gradient-based methods use the gradient information to search the optimal design starting from an initial point [1]. Although these methods have been often employed to solve structural optimization problems, solutions may not be good if the optimization problem is complex, particularly when the abovementioned major difficulties are involved. An alternative could be the deterministic global optimization methods, e.g., the widely used DIRECT method [2; 3], which is a branch of gradient-free methods. Recently, [PERSON] and his colleagues provided a good guide on the deterministic global optimization methods [4] and carried out a comprehensive comparison study between the deterministic and stochastic global optimization methods for one-dimensional problems [5]. One of the main disadvantages of the deterministic global optimization methods is the high-dimensionality issue caused by the larger number of design variables [3] or constraints. In contrast, another branch of gradient-free methods (or stochastic optimization methods), such as Genetic Algorithms (GA) [6; 7; 8; 9; 10; 11; 12], Simulated Annealing (SA) [13; 14; 15],Ant Colony Optimization (ACO) [16; 17; 18; 19], Particle Swarm Optimizer (PSO) [17; 20; 21; 22; 23; 24; 25; 26; 27], Harmony Search (HS) [28; 29; 30; 31], Charged System Search (CSS) [32], Big Bang-Big Crunch (BB-BC) [33], Teaching-Learning based optimization (TLBO) [34; 35], Artificial Bee Colony optimization (ABC-AP) [36], Cultural Algorithm (CA) [37], Flower Pollination Algorithm (FPA) [38], Water Evaporation Optimization (WEO) [39], and hybrid methods combining two or more stochastic methods [40; 41] have been developed to find the global optimal designs for both continuous and discrete structural systems. They have been attracting increasing attention for structural optimization because of their ability to overcome the drawbacks of gradient-based optimization methods and the high-dimensionality issue. A comprehensive review of stochastic optimization of skeletal structures was provided by [PERSON] and [PERSON] [42]. All of the stochastic optimization methods share a common feature, that is, they are inspired by the observations of random phenomena in nature. For example, GA mimics natural genetics and the survival-of-the-fittest code. To implement a stochastic optimization method, random manipulation plays the key role to \"jump\" out local optima, such as the crossover and mutation in GA, the random velocity and position in PSO, etc. Although these stochastic optimization methods have achieved many applications in structural design optimization, structural engineers are still concerned with seeking more efficient and robust methods because no single universal method is capable of handling all types of structural optimization problems. This paper aims to propose an efficient and robust structural optimization method that combines subset simulation optimization (SSO) and the augmented Lagrangian multiplier method [43].