## 3 The Proposed C-MDSDO
### Reliability Assessment with the Formulated Sub-Optimization Problem
#### 3.2.2 Formulated Sub-Optimization Problem and Problem-Solving Process
Then, the deduction trajectory of each accompanying sample is obtained with the control strategy \(\mathbf{u}_{k}(\mathbf{t})\) and the given discrete-time \(\mathbf{t}(\cdot)\) that are obtained in deterministic MDSDO. Once the deduction trajectories of the accompanying samples are obtained, the calculation of the crossrate \(\mathrm{C}_{rate}\) can be performed as follows. \[C_{rate}(\mathbf{R}_{\mathbf{k}})=1+\frac{Num_{R}{}^{\prime}}{Num_{R}} \tag{13}\]  
where \(num_{R}\) represents the number of accompanying samples and \(Num_{R}{}^{\prime}\) represents the constraint violation number of the accompanying samples. Thus, the composition of the sub-optimization problem, the deduction of the state trajectory, the selection of the control trajectory, and the calculation of the crossrate are presented. The complete algorithmic flow of the sub-optimization problem is described in the following, and the flow chart of the proposed reliability assessment stage that contains the sub-optimization problem-solving loop is shown in Figure 5.  
1. In each step \(k\), the solution \(\{\mathbf{d}_{\mathbf{k}}\mathbf{u}_{\mathbf{k}}(\mathbf{t})\}\) from the deterministic MDSDO is employed as the initial condition of the sub-optimization problem;
2. According to the uncertainty information, the accompanying samples \(\mathbf{R}_{k}\) that satisfy the objective reliability index \(\beta_{obj}\). are generated;
3. The random optimization algorithm is employed to solve the sub-optimization problem shown in Formula (10). Particle swarm optimization is employed in this paper;
4. In each iteration, according to the control trajectory \(\mathbf{u}_{k}(\mathbf{t})\), the deduction trajectory of the \(\mathbf{R}_{k}-\mathbf{v}_{k}\) is obtained. The \(C_{rate}\) and the performance of the objective function in Formula (10) of each particle are calculated. The particle optimum and global optimum are compared and updated, the particle position is updated, and the next iteration is performed;
5. The convergence condition of the sub-optimization problem is satisfied. The shifting vector \(\mathbf{v}_{k}\) is output. Otherwise, the step-d iteration is continued;
6. The sub-optimization loop is terminated, and the design \(d_{k}\) in step \(k\) is shifted with \(\mathbf{d}_{k+1}=\mathbf{d}_{k}+\mathbf{v}_{k}\), which is treated as the initial condition of the next deterministic MDSDO. This section attempts to define the reliability-based MDSDO problem studied in this paper. For definition, the main structure of the C-MDSDO approach proposed in this paper is first introduced, which consists of a deterministic MDSDO stage and a sub-optimization-based reliability assessment stage. Then, the construction of the suggested sub-optimization stage and the algorithm flow are presented. Moreover, the generation of the accompanying samples, the deduction of the state trajectories, and the selection of the control strategies are described. In the next section, several case studies are employed to verify the effectiveness of the proposed C-MDSDO, and the result from a similar study is introduced for comparison.