Método de direções interiores ao epígrafo para a solução de problemas de otimização não-convexos e não-diferenciáveis via dualidade lagrangeana

Abstract

This work presents a method for solving constrained nonsmooth and nonconvex optimization problems. Themethod, called IED (Interior Epigraph Directions) can be applied to optimization problems with continuos objective functions defined over compact subsets of Rn and subjected to equalities and/or inequalities constraints. The IED method considers the dual problem induced by a generalized augmented Lagrangian function and obtains the primal solution by generating a sequence of iterates in the interior of the dual function. First, a subgradient is used to build a linear approximation to the dual problem. Then, this linear approximation is used to define a search direction in the interior of the dual function. From an interior point of the epigraph, a new point is obtained and an interior sequence to the epigraph is built, This sequence of interior points generates a dual sequence which in its turn generates a primal sequence by solving a problem originated by duality. The convergence analysis is also presented as well as numerical result of several problems obtained from de literature.