Further Application of H

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP(f,g) where f and g are H-differentiable. We describe H-differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on the H-differentials of f and g, we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP(f,g) to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved for C1, semismooth, and locally Lipschitzian

On Characterizations of Relatively P– and P0– Properties in Nonsmooth Functions

For H-differentiable function f from a closed rectangle Q in Rn into Rn, a result of Song, Gowda and Ravindran [On Characterizations of P- and P0-Properties in Nonsmooth Functions. Mathematics of Operations Research. 25: 400-408 (2000)] asserts that f is a P(P0)− function on Q if the HQ-differential TQ(x) at each x ∈ Q consisting of P(P0)− matrices. In this paper, we introduce the concepts of relatively P(P0)− properties in order to extend these results to nonsmooth functions when the underlying functions are H-differentiable.We give characterizations of relatively P(P0)− of vector nonsmooth functions. Also, our results give characterizations of relatively P(P0)− when the underlying functions are C1-functions, semismooth-functions, and for locally Lipschitzian functions. Moreover, we show useful applications of our results by giving illustrations to generalized complementarity problems

Solving Linear Bilevel Programming via Particle Swarm Algorithm with Heuristic Pattern Searc

A metaheuristic approach is proposed for solving linear bilevel programming problem using the Memetic Particle Swarm Algorithm which uses a Heuristic Pattern Search as the local search. The proposed algorithm has proven to be stable and capable of generating the optimal solution to the linear bilevel programming problem. The numerical results show that the metaheuristic approach is both feasible and efficient

Honey Bee Mating Optimization with Nelder-Mead for Constrained Optimization, Integer Programming and Minimax Problems

In this article, we propose a new hybrid Honey Bee Mating Optimization (HBMO) algorithm with simplex Nelder-Mead method in order to solve constrained optimization, integer programming and minimax problems. We call the proposed algorithm a hybrid Honey Bee Mating Optimization(HBMONM) algorithm. In the the proposed algorithm, we combine HBMO algorithm with Nelder-Mead method in order to refine the best obtained solution from the standard HBMO algorithm.We perform several experiments on a wide variety of well known test functions, 6 constrained optimization problems, 7 integer programming and 7 minimax benchmark problems.We compare the performance of HBMONMagainst standard HBMO algorithm and Genetic Algorithm (GA). In the majority of tests, HBMONM is shown to converge faster, and reach a better solution than both HBMO and GA in reasonable time

Direct Search Firefly Algorithm for Solving Global Optimization Problems

In this paper, we propose a new hybrid algorithm for solving global optimization problems, namely, integer programming and minimax problems. The main idea of the proposed algorithm, Direct Search Firefly Algorithm (DSFFA), is to combine the firefly algorithm with direct search methods such as pattern search and Nelder-Mead methods. In the proposed algorithm, we try to balance between the global exploration process and the local exploitation process. The firefly algorithm has a good ability to make a wide exploration process while the pattern search can increase the exploitation capability of the proposed algorithm. In the final stage of the proposed algorithm, we apply a final intensification process by applying the Nelder-Mead method on the best solution found so far, in order to accelerate the search instead of letting the algorithm running with more iterations without any improvement of the results. Moreover, we investigate the general performance of the DSFFA algorithm on 7 integer programming problems and 10 minimax problems, and compare it against 5 benchmark algorithms for solving integer programming problems and 4 benchmark algorithms for solving minimax problems. Furthermore, the experimental results indicate that DSFFA is a promising algorithm and outperforms the other algorithms in most cases

Conjugate Direction DE Algorithm for Solving Systems of Nonlinear Equations

In this paper, Differential Evolution with Powell conjugate direction method (DE-Powell)is proposed in order solve a system of nonlinear equations. A given system of nonlinear equations is formulated as an unconstrained optimization problem. Integrating Powell conjugate direction method into DE improves the performance of DE and enables DE to optimize effectively the system of nonlinear equations. For example, applying DE to solve our formulation of the system of nonlinear equations, in some iterations DE may get trapped in local minima, then Powell conjugate direction method is applied to help DE to overcome local minima by changing the initial solution for Powell with best obtained one by DE. Our proposed algorithm, DE-Powell, has superiority over Powell Conjugate Direction (CD) and Differential Evolution (DE), separately, it it overcomes the inaccuracy of Powell conjugate direction method and DE for solving systems of nonlinear equations. The DE-Powell is tested on nine well known problems and our numerical results show that the proposed algorithm is solving the highly nonlinear problems effectively and outperforms over many algorithms in literature