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An algorithm is devised for solving minimization problems with equalityconstraints. The algorithm uses first-order derivatives of both the objectivefunction and the constraints. The step is computed as a sum between asteepest-descent step (which minimizes the objective functional) and acorrection step related to the Newton method(which aims to solve the equalityconstraints). The linear combination between these two steps involvescoefficients similar to Lagrange multipliers which are computed in a naturalway based on the Newton method. The algorithm uses no projection and thus theiterates are not feasible; the constraints are satisfied only in the limit(after convergence). This algorithm was proposed by one of the authors in aprevious paper. In the present paper, a local convergence result is proven fora general non-linear setting, where both the objective functional and theconstraints are not necessarily convex functions. The algorithm is extended, bymeans of an active set strategy, to account also for inequality constraints andto address minimax problems. The method is then applied to the optimization ofperiodic microstructures for obtaining homogenized elastic tensors havingnegative Poisson ratio (so-called auxetic materials) using shape and/ortopology variations in the model hole. In previous works of the same authors,anisotropic homogenized tensors have been obtained which exhibit negativePoisson ratio in a prescribed direction of the plane. In the present work, anew approach is proposed, that employs multi-objective optimization in order tominimize the Poisson ratio of the (possibly anisotropic) homogenized elastictensor in several prescribed directions of the plane. Numerical examples arepresented.

A gradient-type algorithm for constrained optimization with applications to multi-objective optimization of auxetic materials