---

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications. It is proved that nonsmooth complementary approximate Karush-Kuhn-Tucker conditions are stronger than nonsmooth approximate gradient projection conditions. Sufficiency for differentiable generalized convex programming is established.

---

In this paper we describe the formal Lagrange-technique to optimize the production process of solid state crystals from a mixture crystal melt. After the construction of the adjoint equation system of the Boussinesq equation of the crystal melt the forward and backward problems (KKT-system) are discretized by a conservative finite volume method.