# Topology Optimization Based Parametric Design of Balloon Borne Telescope's Primary Mirror
## 3 Parametric Design of Primary Mirror
### Theory of Compromise Programming Method  
Using the compromise programming method to set the optimization function, the maximum value of any function can be transformed into the minimum value of the negative function, so all multi-objective optimization problems can be expressed as  
\[\left\{ \begin{array}{ll}find&x=(x_{1},x_{2},\ldots,x_{m})\\ \min&f_{i}(x),i=1,2,\ldots,n\\ s.t.&x\in X\end{array} \right. \tag{13}\]  
where \(x\) is the optimization variable, \(m\) is the number of optimization variables, \(f_{i}(x)\) is the ith sub-objective function, \(n(n\geq 2,n\in N)\) is the total number of objective functions, and \(X\) is the constraint set of the optimization variables. Assuming that the ideal solution of the objective function of multi-objective optimization (the optimal solution in single-objective optimization) is \(a^{\min}=\left\{a_{1}^{\min},a_{2}^{\min},\ldots,a_{n}^{\min}\right\}\), then the compromise solution of the multi-objective optimization problem is the vector with minimum distance \(X\) from the ideal solution [25]. In this case, the multi-objective optimization problem can be transformed into a single-objective optimization problem, and the objective function can be expressed as:  
\[\min F(t)=\left\{\sum_{k=1}^{n}w_{k}^{p}\Big{[}f_{k}(t)-a_{k}^{\min}\Big{]}^{p }\right\}^{\frac{1}{p}} \tag{14}\]  
where \(w_{k}\) is the weight ratio of the kth objective, \(p\) is the distance index, and the meaning of the objective function is different when \(p\) takes different values:  
(1) When \(p=1\), the meaning of the objective function is the sum of the ideal solution. (2) When \(1<p<\infty\), the meaning of the objective function is the weighted geometric distance from the ideal solution. Specifically, when \(p=2\), the objective function is the Euclidean distance. (3) When \(p=\infty\), the meaning of the objective function is the Chebyshev distance, which is the maximum weighted distance. When multi-objective optimization problems with different measurement methods exist, the absolute distance between the compromise solution of the objective function and its ideal solution can be transformed into a relative distance, and the objectives of different measurement methods can be compared to achieve the goal of a single-objective optimization problem [26,27]. \[\min F(t)=\left\{\sum_{k=1}^{n}w_{k}^{p}\Bigg{[}\frac{f_{k}(t)-a_{k}^{\min}}{a _{k}^{\max}-a_{k}^{\min}}\Bigg{]}^{p}\right\}^{\frac{1}{p}} \tag{15}\]  
where \(a_{k}^{\max}\) and \(a_{k}^{\min}\) are the maximum and minimum values of each objective under its boundary conditions. Using the theory of the compromise programming method, the parametric mathematical model of the primary mirror in near-space is expressed as formula (16). \[\min F(t)=\left\{w_{x}^{2}\Bigg{[}\frac{R_{x}(t)-R_{x}^{\min}}{R_{x}^{\max}- R_{x}^{\min}}\Bigg{]}^{2}+w_{z}^{2}\Bigg{[}\frac{R_{z}(t)-R_{z}^{\min}}{R_{x}^{ \max}-R_{z}^{\min}}\Bigg{]}^{2}+w_{m}^{2}\Bigg{[}\frac{m(t)-m^{\min}}{m^{\max} -m^{\min}}\Bigg{]}^{2}+w_{f}^{2}\Bigg{[}\frac{f(t)-f^{\min}}{f^{\max}-f^{\min} }\Bigg{]}^{2}\right\}^{\frac{1}{2}} \tag{16}\]where \(R_{z}(t)\) and \(R_{x}(t)\) are the mirror surface root mean squares caused by gravity in the \(Z\) and \(X\) directions, respectively; \(R_{\rm min}\) is the minimum value of the RMS under each working condition; \(R_{\rm max}\) is the maximum value of the RMS under each working condition; m(t) is the mass of the primary mirror; \(m_{\rm min}\) and \(m_{\rm max}\) are the minimum and maximum values of the mass respectively; \(f(t)\) is the fundamental frequency of the primary mirror;\(f_{\rm min}\) and \(f_{\rm max}\) are the minimum and maximum fundamental frequencies, respectively; and \(w\) is the weight of each optimization sub-objective. The weight factor is defined according to the importance of each parameter to the primary mirror. In order to reduce the cost of flight and the difficulty of balloon manufacturing, the mass of the primary mirror is as low as possible.