Multi-Material Optimization - Global Optimization and Relaxations

Large scale structural optimization problems with many design variables are challenging from a computational point of view in terms of memory consumption and CPU time. In particular it is a challenge to obtain guaranteed globally optimal solutions to (mixed-)integer nonlinear problems such as those arising in e.g. topology optimization. An example of such a large scale problem is topology design of (laminated) multi-material composite structures which may be regarded as a matter of distributing different materials such that a given objective function is minimized subject to physical and resource constraints. One way to model multi-material distribution problems is the so-called discrete material optimization (DMO) approach proposed by [1]. The original problem is discrete, but continuous relaxations of the discrete problem exist and enable the use of gradient based optimizers as part of the solution procedure. In the continuous relaxations, effective material propert ies are obtained as weighted averages of the constituent properties of the individual candidate materials . These averages are non-physical in the sense that they may not correspond to realizable materials, but if the solution to the continuous relaxation eventually attains discrete values this issue poses no further p roblems. It is also possible to directly attack the original discrete problem by use of methods from global optimization such as Branch-and-Cut algorithms [2] or Decomposition Techniques as demonstrated for topology optimization by [3, 4]. However, global optimization techniques are quite limited in the siz e of the problems that can be solved within reasonable resources (CPU time and memory). Direct use of e.g. Generalized Benders Decomposition limits the problem size to about a few thousand design variables and some hundred design domains. On the other hand, with continuous relaxations examples with as many as 750,000 design variables and approximately 1000 domains have been solved, see [5]. The solution of the continuous relaxation may be obtained using any modern nonlinear programming solver such as [6]. The goal of this paper is to extend the tractable range of problems that can be solved to global optimality. This is obtained using the following approach. With the multi-material design problem defined in terms of design domains, finite element discretiza tion and candidate materials to distribute, the idea is to solve a convex continuous relaxation and use the solution to obtain as much information as possible to be used in the global optimization technique which in this case is Generalized Benders Decomposition. With the continuous solution at hand, a lower bound on the attainable performance is obtained since the continuous relaxation is a solution obta ined in a larger set than the integer-valued set. Also, some of the variables in the continuous solution may attain integer values which is the basis of a rounding heuristic used to obtain an integer and feasible solution to be used as a good incumbent solution for the global optimization technique. The idea of the rounding heuristic is to fix all integralvalued variables at their current value and consider a sub-MINLP in the remaining fractional variables. The sub-MINLP is solved to optimality using GBD and the integer solution obtained is passed back as an incumbent solution for the global problem. This idea has been proposed as a heuristic for linear integer problems and is called Relaxation Enforced Neighborhood Search (RENS), and it is used as a starting heuristic within the open source MIP-framework SCIP, see [7, 8]. The improvement obtained using RENS naturally depends on the degree of integrality of the continuous solution. If the continuous