  An algorithm is devised for solving minimization problems with equality
constraints. The algorithm uses first-order derivatives of both the objective
function and the constraints. The step is computed as a sum between a
steepest-descent step (which minimizes the objective functional) and a
correction step related to the Newton method(which aims 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 satisfied only in the limit
(after convergence). This algorithm was proposed by one of the authors in a
previous paper. In the present paper, a local convergence result is proven for
a general non-linear setting, where both the objective functional and the
constraints are not necessarily convex functions. The algorithm is extended, by
means of an active set strategy, to account also for inequality constraints and
to address minimax problems. The method is then applied to the optimization of
periodic microstructures for obtaining homogenized elastic tensors having
negative Poisson ratio (so-called auxetic materials) using shape and/or
topology variations in the model hole. In previous works of the same authors,
anisotropic homogenized tensors have been obtained which exhibit negative
Poisson ratio in a prescribed direction of the plane. In the present work, a
new approach is proposed, that employs multi-objective optimization in order to
minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic
tensor in several prescribed directions of the plane. Numerical examples are
presented.
