Search

In this paper, we present a QP-free algorithm without a penalty function or a filter for nonlinear general-constrained optimization. At each iteration, three systems of linear equations with the same coefficient matrix are solved to yield search direction; the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. There is no feasibility restoration phase in our algorithm, which is necessary for filter methods. The algorithm possesses global convergence as well as superlinear convergence under some mild conditions including a weaker assumption of positive definiteness. Finally, some preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new penalty-free-type method for nonlinear equality constrained problems. The new algorithm uses trust region framework and feasibility safeguarding technique. Moreover, it has no choice of penalty parameter and penalty function as a merit function, and it does not use the filter technique to avoid the penalty function either. We analyze the global convergence of the main algorithm under the standard assumptions. The preliminary numerical tests are reported. [Copyright &y& Elsevier]

Abstract: It is well-known that the HS method and the PRP method may not converge for nonconvex optimization even with exact line search. Some globalization techniques have been proposed, for instance, the PRP+ globalization technique and the Grippo–Lucidi globalization technique for the PRP method. In this paper, we propose a new efficient globalization technique for general nonlinear conjugate gradient methods for nonconvex minimization. This new technique utilizes the information of the previous search direction sufficiently. Under suitable conditions, we prove that the nonlinear conjugate gradient methods with this new technique are globally convergent for nonconvex minimization if the line search satisfies Wolfe conditions or Armijo condition. Extensive numerical experiments are reported to show the efficiency of the proposed technique. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the smoothing self-adaptive Levenberg–Marquardt algorithm for the system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations H(x)=0. A smoothing self-adaptive Levenberg–Marquardt algorithm is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The Levenberg–Marquardt parameter μ k is chosen as the product of μ k =∥H k ∥ δ with δ ∈(0,2] being a positive constant. We will show that if ∥H k ∥ δ provides a local error bound, which is weaker than the non-singularity, the proposed method converges superlinearly to the solution for δ ∈(0,1), while quadratically for δ ∈[1,2]. Numerical results show that the new method performs very well for system of inequalities. [Copyright &y& Elsevier]

Abstract: In this paper, we take a little modification to the Wei–Yao–Liu nonlinear conjugate gradient method proposed by Wei et al. [Z. Wei, S. Yao, L. Liu, The convergence properties of some new conjugate gradient methods, Appl. Math. Comput. 183 (2006) 1341–1350] such that the modified method possesses better convergence properties. In fact, we prove that the modified method satisfies sufficient descent condition with greater parameter in the strong Wolfe line search and converges globally for nonconvex minimization. We also extend these results to the Hestenes–Stiefel method and prove that the modified HS method is globally convergent for nonconvex functions with the standard Wolfe conditions. Numerical results are reported by using some test problems in the CUTE library. [Copyright &y& Elsevier]