Abstract
Many design, planning and decision problems arising in engineering, sciences, finance, and statistics can be modeled as mixed-integer nonlinear optimization problems. A challenging class of mixedinteger problems are topology design problem, arising in additive manufacturing or the design of cloaking devices. In topology optimization, the physical response of the design is modeled as partial-differential equations (PDEs) and the design is modeled with binary variables defined on each element of the discretization of the PDE. This approach results in mixed integer PDE-constrained optimization (MIPDECO) problem that combine the computational challenges of PDEs with the combinatorial challenges of a massive number of discrete variables. We a number of efficient and scalable optimization algorithms based on rounding and randomized search techniques and discuss their optimality properties. We illustrate these solution techniques with examples from topology optimization.