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In this paper, a modified Liu–Storey conjugate gradient method is proposed. The method can generate sufficient descent directions for non-linear unconstrained optimization problems. A global convergence result is established when the line search fulfils the strong Wolfe conditions. Moreover, the-linear convergence rate of the methods is proved. The extensive numerical results show that the proposed method is efficient for the unconstrained optimization problems in the CUTEr library. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we try to solve the semidefinite program with box constraints. Since the traditional projection method for constrained optimization with box constraints is not suitable to the semidefinite constraints, we present a new algorithm based on the feasible direction method. In the paper, we discuss two cases: the objective function in semidefinite programming is linear and nonlinear, respectively. We establish the convergence of our algorithm, and report the numerical experiments which show the effectiveness of the algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with conjugate gradient methods for solving nonlinear least squares problems. Several Newton-like methods have been studied for solving nonlinear least squares problems, which include the Gauss–Newton method, the Levenberg–Marquardt method and the structured quasi-Newton methods. On the other hand, conjugate gradient methods are appealing for general large-scale nonlinear optimization problems. By combining the structured secant condition and the idea of Dai and Liao (2001) , the present paper proposes conjugate gradient methods that make use of the structure of the Hessian of the objective function of nonlinear least squares problems. The proposed methods are shown to be globally convergent under some assumptions. Finally, some numerical results are given. [Copyright &y& Elsevier]

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]

We propose a smoothing Newton method for solving nonlinear complementarity problems based on a new smoothing function. We adopt a variant merit function. Under suitable conditions, we obtain the global convergence and local quadratic convergence of the proposed algorithm. Some numerical results indicate that our algorithm is effective for the given problems. [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]

We study the convergence properties of several conjugate gradient methods for nonlinear optimization under the assumptions that the objective function is bounded below and its gradient is Lipschitz continuous. Specifically, we strengthen the existing convergence result of the Polak–Ribière–Polyak method with constant stepsizes. For the method of shortest residuals, we establish global convergence of both the Fletcher–Reeves version and the Polak–Ribière–Polyak version using constant stepsizes. A numerical example is also presented. [ABSTRACT FROM AUTHOR]

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]

This paper addresses the problem of solving a system of nonlinear equations using complex numbers, rib this end, second order Taylor expansions of the functions of the equations are used to minimize the sum of the modula of the errors in the equations. This gives a single multivariate equation which is not differentiable in complex space, but is differentiable when it is transposed to real space. Using complex number's properties we demonstrate the existence of negative eigenvalues for the Hessian, in the neighborhood of a null Gradient. A sufficient condition for these negative eigenvalues to exist is: when the search vector is not a solution and the Gradient is null, then the bilinear form associated with second order derivatives is full rank. This result leads to an algorithm which avoids first order local minima of the total error in the equations. Convergence to a solution at an accumulation point is then demonstrated. Indeed, with complex numbers, negative curvatures are available when needed. [ABSTRACT FROM AUTHOR]

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, a new feasible primal–dual interior point algorithm for solving inequality constrained optimization problems is presented. At each iteration, the algorithm solves only two or three reduced systems of linear equations with the same coefficient matrix. The searching direction is feasible and the object function is monotone decreasing. The proposed algorithm is proved to possess global and superlinear convergence under mild conditions. Finally, some numerical experiments with the algorithm are reported. [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]

A new trust-region method is proposed for symmetric nonlinear equations. In this given algorithm, if the trial step is unsuccessful, one line search will be used instead of repeatedly solving the subproblem of the normal trust-region method. Moreover, the global convergence is established under mild conditions by a new way. The quadratic convergence of the presented method is also proved. Numerical results show that the method is interesting for the given problems. [ABSTRACT FROM AUTHOR]