Nonmonotone Derivative-free Optimization Algorithm

Nonmonotone derivative-free methods are important ingredient for optimization problems.Generally,the methods use sufficient reduction of objective function in the iteration.But in practical application,the satisfaction of sufficient reduction is difficult. This is the motivation for this paper to research the nonmonotone derivative-free.The algorithm does not computer any derivatives and some iterative points need not to be generated strictly decreasing,but we can also obtain the convergence.Based on the theory of the new algebra structure,two optimization models(unconstrained optimization and linear equality constrained optimization) are investigated in this paper.In what follows, we will introduce the main results of the paper briefly.1.The second chapter presents a nonmonoteone derivative-free optimization algorithm for unconstrained optimization problem.The algorithm make use of the sets of search directions which have the property that the local behavior of the objective function along them provides sufficient information to overcome the lack the gra-dient.The paper introduce "nonmonotone" technique,in a way that some points need not to be generated strictly decreasing,but ultimately convergence towards stationary point of unconstrained optimization problem which is not a maximum point.2.The third chapter constructs a derivative-free optimization method for linear equality constrained optimization problem and gives the proofs of the global convergence. The method mainly adopts the "nonmonotone" technique and projected gradient technique.Through calculating the fundus of the null space of matrix A in the constraint,there exists at least one feasible descent direction at every nonstationary point and every iterate generated by the methods is feasible so long as the initial iterate is feasible.Furthermore,the technique decreases the space dimension of the selected direction set from n to n-m,thus simplifying the implementation procedure and reducing the calculational cost.