## 7 Solving Engineering Design Optimization Problems Using FVIMDE
### Constrained Optimization  
A COP generally involves an objective function in conjunction with various equality and inequality constraints. It also often includes specified lower and upper bounds for the design variables. For a situation involving \(n\) design variables, the COP can be expressed in the following manner:  
The objective function \(f(\mathbf{X})\) is the target for minimization. Here, \(g_{i}(\mathbf{X})\) and \(h_{k}(\mathbf{X})\) represent the inequality and equality constraint functions, respectively, with \(m\) inequality constraints and \(p\) equality constraints. The problem becomes a nonlinear optimization problem if at least one of the functions \(f(\mathbf{X})\), \(g_{i}(\mathbf{X})\), or \(h_{k}(\mathbf{X})\) is nonlinear. \[ \begin{split}\text{Minimize}:\quad f(\mathbf{X})\\ \text{Subject to:}\\ g_{i}(\mathbf{X})\leq 0,\quad i=1,2,\ldots,m\\ h_{k}(\mathbf{X})=0,\quad k=1,2,\ldots,p\\ a_{j}\leq x_{j}\leq b_{j},\quad j=1,\ldots,n\\ \mathbf{X}=\left(x_{1},x_{2},\ldots,x_{n}\right)\end{split}  \tag{19}\]  
Metaheuristic algorithms are generally devised for both constrained and unconstrained search spaces. However, to apply these algorithms to COPs, additional mechanisms are required to integrate the constraints into their objective function. In addressing COPs, it becomes essential to manage both feasible and infeasible solutions, with greater emphasis on the latter. Although completely ignoring infeasible solutions could simplify the process, such an approach might lead to the loss of potentially valuable information about certain promising areas within the function landscape. To circumvent this issue, a traditional method involves imposing a penalty for infeasible solutions. This method includes a constraint violation in the penalized candidate solutions, effectively transforming them into an unconstrained objective function, which is then optimized using techniques suited for unconstrained optimization. The measure of how much a candidate solution \(\mathbf{X}\) deviates from the given constraints is defined as the constraint violation \(V(\mathbf{X})\), as shown in Equations (20) and (21). The evaluation of this violation in the COP typically utilizes the (22) equations. \[V(\mathbf{X})= \begin{cases}0,&\text{if }\mathbf{X}\in F\\ >0,&\text{if }\mathbf{X}\
otin F\end{cases} \text{, where }F\text{ is the feasible region} \tag{20}\]  
\[V(\mathbf{X})=\max\biggl{(}\max_{i}\{0,g_{i}(\mathbf{X})\},\max_{k}\{|h_{k}( \mathbf{X})|\}\biggr{)} \tag{21}\]  
\[V(\mathbf{X})=\sum_{i}\max\{0,g_{i}(\mathbf{X})\}^{m}+\sum_{k}|h_{k}(\mathbf{X} )|^{m} \tag{22}\]  
In COP, the technique of constraint handling is a critical criterion to achieve the optimal solution within the feasible region, if such a region exists. This approach primarily aims to exploit infeasible candidate solutions and extract useful information for the stochastic search process. For managing constraints, [PERSON]'s rules [58] are employed, considering both the constraint violation and the value of the objective function. Handling active constraints in COPs presents significant challenges. All equality constraints are inherently active, and for inequality constraints, those which satisfy \(g_{i}(\mathbf{X})=0\) at the global optimum are deemed active constraints. Hence, issues involving equality constraints require cautious handling to ensure high-quality solutions. A common practice is to transform equality constraints into inequality constraints. This transformation often involves a variety of techniques, one of which is the use of a tolerance parameter (\(tp\)). This parameter assists in converting equality constraints into their inequality counterparts. Thus, the constraints of Equation (19) can be expressed as follows.