A gradient-type algorithm for constrained optimization with application to microstructure optimization

Abstract

We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.

Null space gradient flows for constrained optimization with applications to shape optimization

“…Therefore, the set I(x n ) used in the latter strategy and that I(x n ) featured in ours (given by (3.16)) do not coincide in general; one could think of configurations where the procedure of [16] would fail to find the optimal set I(x n ) (for example if i 0 ∈ I(x n )) and would project the gradient on a less optimal subset of constraints. We note that no convergence result is given by the authors of [15,16] about their procedure.…”

“…[15,16] reads 25) where the set I(x n ) is obtained by removing indices from I(x n ) one by one, starting from the index i 0 associated with the most negative multiplier ν n,i0 < 0, until all of them become non negative. Therefore, the set I(x n ) used in the latter strategy and that I(x n ) featured in ours (given by (3.16)) do not coincide in general; one could think of configurations where the procedure of [16] would fail to find the optimal set I(x n ) (for example if i 0 ∈ I(x n )) and would project the gradient on a less optimal subset of constraints. We note that no convergence result is given by the authors of [15,16] about their procedure.…”

“…al. [16], where an iterative algorithm for equality constrained optimization is devised, which turns out to be a discretization of (1.2) with a variable scaling for the parameter α C . When dealing with inequality constraints, the authors propose an active set strategy which is based-like ours-on the extraction of an appropriate subset of the active constraints (without convergence guarantee, however).…”