# An SQP Algorithm for Structural Topology Optimization Based on Majorization-Minimization Method
## 1 Introduction  
Structural topology optimization aims to optimize the material layout in a given design space under a given set of loads, boundary conditions, and constraints to achieve the optimal performance of the structure. At present, structural topology optimization has been widely used in related technical fields, such as aerospace [1], 3D print [2], and reliability study of structures [3]. Since [PERSON] and [PERSON] conducted seminal work [4] on numerical topology optimization in 1988, more and more structural topology optimization methods have been proposed, such as density-based method [5], level set method (LSM) [6], moving morphable components (MMC) [7], and bidirectional evolutionary structural optimization (BESO) [8]. Among them, the density-based method has received wide attention due to its strong adaptability and simplicity of implementation. Accordingly, a series of mature solution algorithms have been developed. These algorithms can be divided into two main categories: one is the optimality criteria (OC) [9] algorithm and its extensions, such as the PDOC [10] method based on the idea of proportional differential control. The advantages of this type of method are the high computational efficiency, and the speed of the solving is less sensitive to the number of design variables. However, this type of method can be used for single-constraint optimization problems, while it is difficult to be extended to multi-constraint optimization problems. Another class of algorithms to solve the density-based method is represented by the approximation algorithms. The main idea of this class of methods is to transform the original problem into a simpler subproblem, and then use the solution of the subproblem as the initial point for the next iteration until convergence. The main approximation algorithms are sequential linear programming (SLP) [11] SQP [12], the method of moving asymptote (MMA) [13], and its globally convergent counterpart(GCMMA) [14]. This type of method can solve multi-constraint problems, but its solving speed is slower than that of the previous methods. The SQP algorithm has been one of the most successful general methods for solving nonlinear constrained optimization problems. The solving of nonlinear problems with SQP is divided into two main steps. The first step is calculating the Hessian matrix to obtain the subproblem, and the second step is solving this subproblem. Compared with other nonlinear programming problems, QP problems are more attractive. [PERSON] et al. [15] transformed the sequential convex programming subproblem into a Newton Lagrange QP subproblem, and [PERSON] et al. [16] directly used the Hessian matrix in the MMA subproblem to construct the quadratic programming subproblem. In practice, the Hessian matrix of most optimization problems is difficult to calculate. The SQP algorithms generally use quasi-Newton [17] methods to approximate the Hessian matrix to obtain the QP subproblem. In topology optimization, TopSQP [18] uses information from part of the second-order derivatives to ensure the positive definiteness of the Hessian matrix. Albeit successful in solving nonlinear optimization problems, the SQP method is not as widely used as the MMA and OC methods in structural topology optimization. The main reason is that the calculation of the Hessian matrix may become a heavy burden for large-scale topology optimization problems. On the one hand, the existence of the equilibrium equation makes it difficult to calculate the exact Hessian matrix, which may also not be positive definite; on the other hand, using the quasi-Newton method to approximate the Hessian matrix will lead to a massive amount of computation and storage. Facing these obstacles, using the problem-driven majorization-minimization (MM) [19] algorithm to compute the approximate Hessian matrix would be a good breakthrough point. This paper proposes a method to calculate the approximate Hessian matrix according to the MM algorithm, which takes advantage of the problem structure.