# Truss Structure Optimization with Subset Simulation and Augmented Lagrangian Multiplier Method
## 3 The Augmented Lagrangian Subset Simulation Optimization
### The Augmented Lagrangian Multiplier Method  
Dealing with multiple constraints is pivotal to applying SSO in a nonlinear constrained optimization. Although the basic idea of SSO is highly different from other stochastic optimization algorithms, one can still make use of the constraint-handling strategies used in them. Substantial research studies have been devoted to GA, PSO, etc. In our previous study [45; 46], a modified feasibility-based rule motivated by [PERSON] et al. [24] in PSO was proposed to handle multiple constraints for SSO, which incorporates the effect of constraints during the generation process of random samples of design variables. By this method, the feasible domain is properly depicted by the population of random samples. However, this rule fails to detect the active constraints directly.  
The purpose of this paper is to present an alternative strategy to deal with multiple constraints in SSO for constrained optimization problems, which makes use of the augmented Lagrangian multiplier method. Consider, for example, a nonlinear constrained optimization problem with equality and inequality constraints. It can be converted into an unconstrained optimization problem by introducing Lagrange multiplier vector \(\mathbf{\lambda}\) and penalty parameter vector \(\mathbf{\sigma}\)[43] through the Lagrangian multiplier method. Then, the unconstrained optimization problem is given by  
\[\begin{array}{l}\min\,L(\mathbf{x},\mathbf{\lambda},\mathbf{\sigma})\\ \text{s.t.}\,\,\mathbf{x}^{\text{L}}\leq\mathbf{x}\leq\mathbf{x}^{\text{U}}\end{array} \tag{5}\]  
where \(L(\mathbf{x},\mathbf{\lambda},\mathbf{\sigma})\) is the following augmented Lagrangian function [25; 26; 50]:  
\[L(\mathbf{x},\mathbf{\lambda},\mathbf{\sigma})=W(\mathbf{x})+\sum_{i=1}^{n_{e}+n_{i}}\lambda_ {i}\theta_{i}(\mathbf{x})+\sum_{i=1}^{n_{e}+n_{i}}\lambda_{i}\theta_{i}^{2}(\mathbf{x}) \tag{6}\]  
with  
\[\theta_{i}=\left\{\begin{array}{ll}g_{i}(\mathbf{x}),&i=1,\ldots,n_{i}\\ \max\Bigl{(}h_{i}(\mathbf{x}),\frac{-\lambda_{i}}{2\sigma_{i}}\Bigr{)},&i=1,\ldots,n_{e}\end{array}\right.. \tag{7}\]  
The advantages of this augmented Lagrangian function in Equation (6) lie in bypassing the ill-conditioning caused by the need for infinite penalty parameters and transforming a stationary point in an ordinary Lagrangian function into a minimum point. It can be easily proved that the Karush-Kuhn-Tucker solution \((\mathbf{x}^{*},\mathbf{\lambda}^{*})\) of the problem in Equation (6) is a solution to the problem in Equation (1) [26; 43]. As a result, SSO for unconstrained optimization [44] can be applied to the problem in Equation (1) after it has been converted into the problem in Equation (6). It is also well-known that if the magnitude of penalty parameters is larger than a positive real value, the solution to the unconstrained problem is identical to that of the original constrained problem [43].  
Figure 1 shows a flowchart of the proposed method that contains an outer loop and inner loop. The outer loop is performed to formulate the augmented Lagrangian function, update the Lagrange multipliers and penalty parameters, and check the convergence. For the \(k\)th outer loop, the Lagrange multipliers \(\mathbf{\lambda}^{k}\) and penalty parameters \(\mathbf{\sigma}^{k}\) are considered as constant in the inner loop. Then, SSO is applied to solve the global minimum design of the augmented Lagrangian function \(L\Bigl{(}\mathbf{x},\mathbf{\lambda}^{k},\mathbf{\sigma}^{k}\Bigr{)}\) with the given set of \(\mathbf{\lambda}^{k}\) and \(\mathbf{\sigma}^{k}\) in the \(k\)th outer loop. This also means that the Lagrange multipliers and penalty parameters are held at a fixed value in the inner loop.