# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 2 Related Work
### Theoretical Basis of Multi-Objective Optimization  
(1) Definition of multi-objective optimization problem  
In the optimization process of practical engineering problems, multi-objective optimization problems with complex and high-dimensional nonlinear constraints are often encountered [27]. There may be conflicts between different targets, which makes it very difficult to find a global optimal solution. For single-objective optimization problems, the optimal solution is usually a specific point in the solution space. In contrast, the computational complexity of multi-objective optimization problems is much higher, because it needs to satisfy multiple objective functions at the same time. In this case, there may not be a globally unique optimal solution, but rather a series of solutions that strike a balance between different objectives [28]. In a standard multi-objective optimization problem, there are usually n objective functions, m decision variables, and related constraints, which are usually expressed as shown in (1). \[ \begin{array}{c}\underset{x\in X}{\text{min}}=\underset{x\in X}{\text{min}} \,F(x)=(f_{1}(x),f_{2}(x),\cdots,f_{n}(x))\\ s.t.\begin{cases}g_{i}(x)\leq 0\;i=1,2,\cdots,q\\ h_{j}(x)=0\;j=1,2,\cdots,p\end{cases}.\end{array}  \tag{1}\]  
Among the variables, \(x=(x_{1},x_{2},\cdots,x_{m})\in X\subset R^{m}\) represents the m-dimensional decision variable vector, which constitutes the feasible region of the variable, where \(X\) is the decision space; \(y=(y_{1},y_{2},\cdots,y_{n})\in Y\subset R^{n}\) represents the objective function vector, where \(Y\) represents the target space; \(g_{i}\leq 0\;(i=1,2,\cdots,q)\) denotes \(q\) inequality constraints; and \(h_{j}(x)=0\) represents \(p\) equality constraints. (2) Multi-objective optimization evaluation index  
If for any goal i, there exists \(f_{i}(x_{1})\leq f_{i}(x_{2})\), and there exists at least one goal j such that \(f_{j}(x_{1})\leq f_{j}(x_{2})\), then the solution \(x_{1}\) dominates the solution \(x_{2}\) on all goals. If there is no \(x\) solution dominating \(x^{*}\), then the solution \(x^{*}\in X_{f}\) is the Pareto optimal solution. The set P composed of all Pareto optimal solutions is called the Pareto optimal solution set. There is usually no obvious linear or regular functional relationship between the elements in the P set, except that they all belong to the Pareto set. The Pareto front is composed of the target vectors corresponding to all the solutions in the Pareto optimal solution set [27]. The double objective function \(f_{1}\), \(f_{2}\) and the minimum Pareto solution are shown in Figure 1. It can be seen from the figure that both the green point and the blue point are solutions in the feasibility region, and there is a Pareto dominance relationship between the two solutions. The performance of solution A on the two objective functions of \(f_{1}\) and \(f_{2}\) is better than that of solution D. At this time, solution A is called Pareto dominant solution D. Similarly, solution B and solution C are also Pareto dominant solution D. The optimal solution of the bi-objective function is on the Pareto front, and there may be multiple solutions on the front. It can be seen in the graph that the performance of solution A on the \(f_{1}\) objective is better than that of solution B, but the performance on the \(f_{2}\) objective is slightly worse than that of solution B. At this time, solution A is not different from solution B, and there is no Pareto dominance relationship. If there is a solution, it can Pareto dominate all other solutions, that is, it performs better than other solutions on all objective functions, which is called the absolute optimal solution. When the optimal solution does not exist, non-inferior solutions on the Pareto front are sought out, that is, Pareto optimal solutions, in order to achieve a comprehensive and effective solution to the problem [29; 30].