Sequential conservative integer programming method for multi-constrained discrete variable structure topology optimization

The sequential approximate integer programming (SAIP) method successfully solves multiple types of large-scale topology optimization problems by solving a sequence of separable approximate integer programming subproblems. However, these subproblems must be with quadratic/linear objective function with linear constraints that can be analytically solved by the canonical relaxation algorithm (CRA). Besides, SAIP relies on the decreasing volume fraction strategy so that it owns difficulties to confront topology optimization problems without active volume constraints. The success of MMA (method of moving asymptotic) inspires us to introduce the classical sequential conservative approximate programming that can generate a sequence of steadily improving feasible designs and present a sequential conservative integer programming (SCIP) method for the development of SAIP. This new method generates a sequence of nonlinear approximate integer programming subproblems containing the reciprocal variables, whose conservation is controlled by moving asymptotes. Since CRA is invalid for nonlinear subproblems, a simple design variable update rule derived by the KKT (Karush-Kuhn-Tucker) conditions is given. Based on the above idea, SCIP obtains stable optimization processes without relying on linear constraints, such as the volume constraint, and thus complex discrete variable topology optimization problems can be efficiently solved. Various requirements of topology optimization problems are handled, including equality or inequality constraints, and inactive or active volume constraints. Numerical results show that since the conservative property has been inherited, the convergence of the optimization process can be regulated without additional structural analysis.