# An SQP Algorithm for Structural Topology Optimization Based on Majorization-Minimization Method
## 2 Quadratic Programming Subproblem
### Quadratic Programming Subproblem  
The method [22] of combining the MM algorithm with the SQP algorithm is to find an approximate Hessian matrix \(\mathbf{H}\), where \(\mathbf{H}\succ\mathbf{B}-\mathbf{A}\). Since matrices \(\mathbf{A}\) and \(\mathbf{B}\) are positive definite, the most straightforward approach is to take \(\mathbf{H}=\mathbf{B}\) like the TopSQP method, which can ensure that the matrix \(\mathbf{H}-(\mathbf{B}-\mathbf{A})\) is positive definite. However, matrix \(\mathbf{B}\) needs to calculate the inverse of the global stiffness matrix, which is very expensive for large-scale topology optimization problems and does not meet the requirements of low computational cost. Since the matrix \(\mathbf{A}\) is diagonally positive definite, there must exist a curvature parameter \(q(q>0)\), such that \(q\mathbf{A}\succ\mathbf{B}-\mathbf{A}\). Then, the surrogate function of Equation (2) can be expressed as  
\[\Phi(\mathbf{x})=\frac{q}{2}(\mathbf{x}-\mathbf{x}^{*})^{T}\mathbf{A}( \mathbf{x}-\mathbf{x}^{*})+C^{\prime}(\mathbf{x}^{*})^{T}(\mathbf{x}-\mathbf{ x}^{*})+C(\mathbf{x}^{*}). \tag{10}\]  
The solution of Equation (10) is given by  
\[\mathbf{x}=\mathbf{x}^{*}-\frac{\mathbf{A}^{-1}C^{\prime}(\mathbf{x}^{*})}{q}. \tag{11}\]  
Figure 1: MM algorithm procedure. In this paper, we do not directly calculate the value of \(q\), but use numerical experiments to verify the effect of different \(q\) on the convergence result. The QP subproblem of Equation (2) is as follows:  
\[ \begin{cases}min&\Phi(\mathbf{x})=\frac{q}{2}(\mathbf{x}-\mathbf{x}^{*})^{T} \mathbf{A}(\mathbf{x}-\mathbf{x}^{*})+C^{\prime}(\mathbf{x}^{*})^{T}(\mathbf{ x}-\mathbf{x}^{*})+C(\mathbf{x}^{*})\\ st.&V(\mathbf{x})=fV_{0}=\sum\limits_{i=1}^{N}x_{i}v_{i}\quad,\\ &x_{i}^{lb}\leq x_{i}\leq x_{i}^{ub}\end{cases}  \tag{12}\]  
where  
\[ \begin{array}{l}x_{i}^{lb}=\max(x_{\min},x_{i}-move)\\ x_{i}^{ub}=\min(x_{\max},x_{i}+move)\end{array}  \tag{13}\]  
where _move_ is the moving limit, which is usually set in topology optimization to ensure the stability of the solutions. The surrogate function has several properties desired by the literature [22]: convex separable and the existence of a closed-form minimizer. The design of efficient and neat solution algorithms to verify the feasibility of the surrogate function is presented in the next section. **3. Solution of Subproblem**  
3.1. _QPOC Method_  
Since the subproblem in Equation (12) is convex and separable, the algorithm for solving it should take full advantage of this property. First, before solving Equation (12), we construct an easy to solve quadratic programming problem, which is  
\[ \begin{cases}min&\Phi(\mathbf{x};\lambda)=\frac{q}{2}(\mathbf{x}-\mathbf{x}^{* })^{T}A(\mathbf{x}-\mathbf{x}^{*})+C^{\prime}(\mathbf{x}^{*})^{T}(\mathbf{x}- \mathbf{x}^{*})+C(\mathbf{x}^{*})+\lambda(\sum\limits_{i=1}^{N}x_{i}v_{i}-V^{ *})\;,\\ st.&x_{i}^{lb}\leq x_{i}\leq x_{i}^{ub}\end{cases}  \tag{14}\]  
where \(\lambda\) is a scalar parameter. Because \(\mathbf{A}\) is a diagonal positive definite matrix, Equation (14) can be decomposed into \(N\) problems; the optimal solution of \(\Phi(\mathbf{x};\lambda)\) is  
\[ \begin{array}{l}\mathbf{x}(\lambda)=\frac{\mathbf{A}^{-1}}{q}(-C^{\prime}( \mathbf{x}^{*})-\lambda v\mathbf{I})+\mathbf{x}^{*}\\ x_{i}(\lambda)=\frac{(q(p-1)+1)C^{\prime}(\mathbf{x}_{i}^{*})+\lambda v_{i}}{ q(p-1)C^{\prime}(\mathbf{x}_{i}^{*})}x_{i}^{*}\end{array}  \tag{15}\]  
Constrained by upper and lower bounds, the solution of Equation (14) is given by  
\[x_{i}(\lambda)=\text{middle}(x_{i}^{lb},x_{i}(\lambda),x_{i}^{ub}). \tag{16}\]  
Then, we solve the univariate nonlinear equation,  
\[r(\lambda)=\sum\limits_{i=1}^{N}x_{i}(\lambda)v_{i}-fV_{0}=0. \tag{17}\]  
According to Equations (15) and (16), \(x_{i}(\lambda)\) and \(r(\lambda)\) are monotonically decreasing continuous functions. Note that \(r(0)>0,r(\infty)<0\); hence, Equation (17) must have a unique solution.