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This paper deals with generalized pattern search (GPS) algorithms for linearly constrained optimization. At each iteration, the GPS algorithm generates a set of directions that conforms to the geometry of any nearby linear constraints. This set is then used to construct trial points to be evaluated during the iteration. In a previous work, Lewis and Torczon developed a scheme for computing the conforming directions, however, the issue of degeneracy merits further investigation. The contribution of this paper is to provide a detailed algorithm for constructing the set of directions whether or not the constraints are degenerate. One difficulty in the degenerate case is the classification of constraints as redundant or nonredundant. We give a short survey of the main definitions and methods for treating redundancy and propose an approach to identify nonredundant ε-active constraints. We also introduce a new approach for handling nonredundant linearly dependent constraints, which maintains GPS convergence properties without significantly increasing computational cost. Some simple numerical tests illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported. [ABSTRACT FROM PUBLISHER]

In this paper, based on a new class of asymptotic l 1 exact penalty functions, we propose a smooth penalty function for solving nonlinear programming problems. One of the main features of our algorithm is that at each iteration, we do not need to solve the global minimum of penalty functions. Furthermore, global convergence property is established without requiring any constraint qualifications. By addressing perturbation functions, we obtain that the lower semi-continuity of the perturbation function at zero is a necessary and sufficient condition to ensure the convergence of the objective function values generated by the algorithm to the optimal value of the primal problem. Since the perturbation function is only dependent on the data of the primal problem, these results enable us to check the convergence property of the algorithm in advance. Finally, we discuss the finite termination of the algorithm when the solution set is non-degenerate or weakly sharp. [ABSTRACT FROM AUTHOR]

We propose a quasi-Newton line-search method that uses negative curvature directions for solving unconstrained optimization problems. In this method, the symmetric rank-one (SR1) rule is used to update the Hessian approximation. The SR1 update rule is known to have a good numerical performance; however, it does not guarantee positive definiteness of the updated matrix. We first discuss the details of the proposed algorithm and then concentrate on its practical behaviour. Our extensive computational study shows the potential of the proposed method from different angles, such as its performance compared with some other existing packages, the profile of its computations, and its large-scale adaptation. We then conclude the paper with the convergence analysis of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider correcting an inconsistent set of linear inequalities by minimal changes in problem data using the l2-norm. A new formulation of the problem is introduced, which involves minimizing a nonconvex fractional function. Then by utilizing the generalized Newton (GN) method, two heuristic algorithms are developed to solve the fractional minimization problem. We further consider correcting inconsistent linear inequalities with extra non-negativity constraints on the variables. For this specific case, we also propose two heuristic algorithms. Finally, we report encouraging numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a trust region algorithm with memory for equality constrained optimization problems. Different from the traditional trust region algorithms, our trust region model includes memory of the past iterations, which makes the algorithm more farsighted in the sense that its behavior is not completely dominated by the local nature of the objective function, but rather by a more global view. The global convergence is established by using a nonmonotone technique. We report numerical tests to examine the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]