A multiobjective approach for optimization of passive damping for vibration reduction in sandwich structures is presented in this paper. A layerwise finite element model for sandwich plates with viscoelastic core and anisotropic laminated face layers is used along with the complex modulus approach and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for maximisation of modal loss factors and minimisation of weight of sandwich beams and plates with elastic laminated constraining layers and a viscoelastic core, with layer thickness and laminate layer ply orientation angles as design variables. The problem is solved using the Direct MultiSearch (DMS) solver for derivative-free multiobjective optimization and solutions are compared with alternative ones obtained using genetic algorithms. DMS is a solver for multiobjective optimization problems which does not use any derivatives of the objective functions. It is based on a novel technique called direct multisearch, developed by extending direct search from single to multiobjective optimisation.