# A Multi-Constrained Optimization Method for THz Backward Wave Oscillators
## 3 Optimization Method
### Successive Iteration for Local Optimum
#### 3.2.3 Iterations toward Local Optimum by Greedy Algorithm  
Being different from typical local search algorithms, the proposed optimization method does not solve the best local solution for the next move at each iteration step. A greedy algorithm is applied after finding the best local solution at the start. The best local solution is not only considered as the next move, but also considered as a 'gradient direction' along which successive moves are made without evaluating local solutions at each step until the last move itself does not satisfy the constraints any more.  
The successive moves, which reduce a lot of solving and evaluating calculations, can be defined using a geometric series such as  
\[\left\{\begin{array}{l}\overset{\rightarrow}{S^{\prime}}_{i+1}=\overset{ \rightarrow}{S^{\prime}}_{i}\cdot\left(1+\delta\left((\alpha_{1},\ldots,\alpha _{k})-\frac{v+1}{2}\right)\right)\\ i=1,2,\ldots,i_{\max}-1\\ \overset{\rightarrow}{S^{\prime}}_{1}=\overset{\rightarrow}{S}_{1}\end{array}\right. \tag{8}\]  
where \(S^{\prime}_{ex}\) is an example of series of solutions, \(S^{\prime}_{ex,j}\) is the \(i\)th solution in the series, \(\overset{\rightarrow}{S}_{1}\) has been defined in Equation (4), \(\alpha_{1}\),..., \(\alpha_{k}\) are the levels in the orthogonal array corresponding to the best local solution, \(i_{\max}\) is the maximum number of iterations.  
The geometrical parameters of the successively variant solutions increase or decrease exponentially with a slow rate. Figure 5 shows the corresponding working parameters as a function of the serial number that are calculated by [PERSON]'s theory [PERSON] (1975). The monotonicity of each working parameter remains within several moves, but it could be broken as the ratio for the geometrical series may not influence the working parameters monotonically. Thus, a greedy strategy is let \(\alpha_{1}\),..., \(\alpha_{k}\) be constant during the successive iteration following Equation (7), which means that the local solutions will not be sampled at each step for reducing the total calculation cost.  
Figure 5: Normalized output power, beam current density and maximum beam voltage in operating frequency range versus the serial number.  
\(i_{\max}\) controls the number of iterations for each 'direction', i.e., combination of \(\alpha_{1}\),..., \(\alpha_{k}\). It is expected that the output performances and the working parameters can be improved monotonically during the \(i_{\max}\) times of iterations. Thus, this successive iteration process should stop once the reached solution provides the output performances and the working parameters that broke their previous monotonic variation. The last solution must be a bad choice for further iteration, and the second to last solution in the series will be suitable as a new initial solution for next successive moves. The search for local optimum will end if no better solution can pass the evaluation.  
Figure 6 shows the flow chart of the optimization for local optimum. A local search will terminate if all the local solutions, having not been arrived at, have the same evaluated value of zero. The result should satisfy the constraints mentioned in Section 2.