Extending the SIMP Method to General Integer Optimization Problems

The SIMP (\textit{Solid Isotropic Material with Penalization}) method is one of the best-known approaches for solving Topology Optimization  problems, which consist of determining the optimal material distribution in a domain, usually aiming at maximizing structural stiffness under volume constraints. The classical formulation of these problems leads to large-scale mixed-integer programming problems. In the SIMP method, continuous relaxations of binary variables are combined with penalization terms in the stiffness function, reducing the occurrence of intermediate-density solutions. In this work, we investigate the possibility of extending the SIMP philosophy to more general classes of mixed integer optimization problems. In particular, we consider problems in which activating a procedure implies a fixed cost while also providing additional capacity or resources. In such cases, the classical modeling approach involves binary variables simultaneously associated with cost terms in the objective function and capacity constraints. We show how to adapt this formulation in order to make the SIMP theory applicable, penalizing fractional solutions and encouraging decisions close to zero or one. Computational experiments illustrate the applicability and limitations of the proposed approach in logistical applications related to military problems.