**Title: **Topological Design Problems and Massive Integer Optimization

**Speaker:** Sven Leyffer, Argonne National Laboratory

**Day, Date and Time: **Thursday, 25th January, 2024, 10:30AM

**Venue:** VMCC Room 5, Ground Floor

**Abstract: **Topological design problems arise in many important manufacturing and scientific applications, such as additive manufacturing and the optimization of structures built from trusses. Topology design optimizes the material layout within a design space to satisfy given global load constraints and boundary conditions, represented by partial differential equations (PDEs). Traditionally, the PDEs have been discretized using the finite-element method with additional continuous variables that model the density of the material within each finite element, giving a (discretized) layout of the material.

In contrast, we model the material layout using binary variables to represent the presence or absence of material in each finite element. This approach results in a massive mixed-integer nonlinear optimization (MINLO) problem, where the nonlinearities arise from the discretization of the governing PDEs. At first sight, such an approach seems impractical, because it requires a massive number of binary variables (one per finite element): the number of binary variables is driven by the accuracy requirements of the finite-element discretization. We show empirically that traditional MINLO algorithms that search a branch-and-bound tree cannot solve these topology optimization problems with even a coarse finite-element discretization.

Recently, a new class of methods have been proposed for solving MINLOs that arise from the discretization of differential equations. These methods solve a single continuous relaxation of the MINLO, which relaxes the integrality requirement on the binary variables. Given the optimal solution of this relaxation, we apply a rounding technique that relies on space-filling curves. These rounding techniques have a remarkable convergence property, and can be shown to asymptotically converge to the global optimum as we refine the finite-element mesh. We illustrate these solution techniques with examples from topology optimization arising in the design of electro-magnetic cloaking devices.

**Speaker's Bio:** Sven Leyffer joined the Mathematics and Computer Science Division at Argonne in 2002, where he is now a senior computational mathematician. Sven is a SIAM Fellow, and a senior fellow of the University of Chicago/Argonne Computation Institute. He is the current SIAM President and serves on the editorial boards of Computational Optimization and Applications, and Mathematics of Computation. In 2006, Leyffer (along with two colleagues) received the Lagrange Prize in Continuous Optimization, which is awarded only once every 3 years. In 2016, he received the Farkas Prize from the INFORMS Optimization Society. Leyffer obtained his Ph.D. in 1994 from the University of Dundee, Scotland, and held postdoctoral research positions at Dundee, Argonne, and Northwestern University. Check here for more info about speaker.