Abstract
The Interior Epigraph Directions (IED) method for solving constrained nonsmooth and nonconvex optimization problem via Generalized Augmented Lagrangian Duality considers the dual problem induced by a Generalized Augmented Lagrangian Duality scheme and obtains the primal solution by generating a sequence of iterates in the interior of the epigraph of the dual function. In this approach, the value of the dual function at some point in the dual space is given by minimizing the Lagrangian. The first version of the IED method uses the Matlab routine fminsearch for this minimization. The second version uses NFDNA, a tailored algorithm for unconstrained, nonsmooth and nonconvex problems. However, the results obtained with fminsearch and NFDNA were not satisfactory. The current version of the IED method, presented in this work, employs a Genetic Algorithm, which is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as GA, is applied alone to solve constrained optimization problems. Two sets of constrained optimization problems from mathematics and mechanical engineering were solved and compared with literature. It is shown that the proposed hybrid algorithm is able to solve problems where fminsearch and NFDNA fail.
Highlights
We present a new version of the Interior Epigraph Directions Method published by Burachik et al [1] for solving constrained nonsmooth and nonconvex optimization problems of the following kind: minimize f (x) over all x ∈ X satisf ying g (x) = 0, (P)
The second version of the Interior Epigraph Directions (IED) method [15] uses an algorithm called Nonsmooth Feasible Directions Nonconvex Algorithm (NFDNA) [16, 17] for solving nonsmooth and nonconvex unconstrained optimization problems which correspond to our problem (2)
We have proposed a version of the IED method for nonsmooth and nonconvex optimization problems that employs a Genetic Algorithm for minimization the Lagrangian function
Summary
The IED method works as follows: starting at a point in the interior of the epigraph of the dual function, a deflected subgradient direction is used to define a linear approximation to the epigraph. The second version of the IED method [15] uses an algorithm called Nonsmooth Feasible Directions Nonconvex Algorithm (NFDNA) [16, 17] for solving nonsmooth and nonconvex unconstrained optimization problems which correspond to our problem (2). We have used a Genetic Algorithm (GA) for minimizing the Lagrangian function With this modification, the method gained robustness and was able to solve all the test problems analyzed here.
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