Search

In this paper, we study the optimization problem (PWE) of minimizing a convex function over the set of weakly efficient solutions of a convex multiobjective problem. This is done by using the fact that each lower semicontinuous convex function is an upper envelope of its affine minorants together with a generalized cutting plane method. We give necessary conditions for optimal solutions of the problem (PWE). Moreover, a novel algorithm for solving the problem (PWE) together with numerical results are presented. We also prove that the proposed algorithm terminates after a finite numbers of iterations, and the algorithm is coded in MATLAB language and evaluated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the convergence of the DFP algorithm with revised search direction. We prove that the algorithm is globally convergent for continuously differentiable functions and the rate of convergence of the algorithm is one-step superlinear and n-step second-order for uniformly convex objective functions. [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]

In this paper, we consider the least l2-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg-Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l2-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions. [ABSTRACT FROM AUTHOR]

In this paper, we establish a set of necessary optimality conditions and discuss a fairly large number of sets of global nonparametric sufficient optimality criteria under various generalized (η, ρ)-invexity assumptions for a minmax fractional programming problem with infinitely many nonlinear inequality and equality constraints. [ABSTRACT FROM AUTHOR]

In this article, unconstrained minimax problems are discussed, and a sequential quadratic programming (SQP) algorithm with a new nonmonotone linesearch is presented. At each iteration, a search direction of descent is obtained by solving a quadratic programming (QP). To circumvent the Maratos effect, a high-order correction direction is achieved by solving another QP and a new nonmonotone linesearch is performed. Under reasonable conditions, the global convergence and the rate of superlinear convergence are established. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this article, we study convergence of the extragradient method for constrained convex minimization problems in a Hilbert space. Our goal is to obtain an ε-approximate solution of the problem in the presence of computational errors, where ε is a given positive number. Most results known in the literature establish convergence of optimization algorithms, when computational errors are summable. In this article, the convergence of the extragradient method for solving convex minimization problems is established for nonsummable computational errors. We show that the the extragradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [ABSTRACT FROM AUTHOR]

In this article, an ODE-based trust region filter algorithm for unconstrained optimization is proposed. It can be regarded as a combination of trust region and filter techniques with ODE-based methods. Unlike the existing trust-region-filter methods and ODE-based methods, a distinct feature of this method is that at each iteration, a reduced linear system is solved to obtain a trial step, thus avoiding solving a trust region subproblem. Under some standard assumptions, it is proven that the algorithm is globally convergent. Preliminary numerical results show that the new algorithm is efficient for large scale problems. [ABSTRACT FROM AUTHOR]

A class of affine-scaling interior-point trust region methods with line search for bound constrained optimization problems is presented in this article. The method is globally convergent under the standard assumptions and is locally q-superlinear or q-quadratic convergent in the case where the strong second order sufficient optimality conditions at the solution are satisfied, but the strict complementarity condition is violated. Numerical experiments show that the method is efficient. [ABSTRACT FROM AUTHOR]

This article is concerned with a class of nonsmooth constrained convex optimization in a real Hilbert space. Coupling with the penalty method, we propose an automatic system (AS) and a nonautomatic system (NS) modeled by differential inclusions. Under a suitable assumption on the feasible region and a proper condition on the objective and constrained functions, some valuable convergence properties of (AS) are obtained. In order to obtain strong convergence result in general cases, based on evolution differential inclusion, we propose a nonautomatic system (NS). When the control item ε(t) of (NS) satisfies some basal conditions, global and unique existence of the solution, finite time convergence to the feasible region and slow solution choice are obtained. Moreover, under different conditions of ε(t), we give some strong convergence results of (NS). Furthermore, we end the article by numerical experiments to illustrate the efficiency and good performance of the proposed systems in this article. [ABSTRACT FROM AUTHOR]

We introduce a two-step direct method, like Korpelevich's, for solving monotone variational inequalities. The advantage of our method over that one is that ours converges strongly in Hilbert spaces, whereas only weak convergence has been proved for Korpelevich's algorithm. Our method also has the following desirable property: the sequence converges to the solution of the problem that lies closest to the initial iterate. [ABSTRACT FROM AUTHOR]

Combining the ideas of generalized projection and the strongly subfeasible sequential quadratic programming (SQP) method, we present a new strongly subfeasible SQP algorithm for nonlinearly inequality-constrained optimization problems. The algorithm, in which a new unified step-length search of Armijo type is introduced, starting from an arbitrary initial point, produces a feasible point after a finite number of iterations and from then on becomes a feasible descent SQP algorithm. At each iteration, only one quadratic program needs to be solved, and two correctional directions are obtained simply by explicit formulas that contain the same inverse matrix. Furthermore, the global and superlinear convergence results are proved under mild assumptions without strict complementarity conditions. Finally, some preliminary numerical results show that the proposed algorithm is stable and promising. [ABSTRACT FROM AUTHOR]