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In this paper, we study two types of approximate solutions for a vector optimization problem in Banach space setting. Our main concern is to define two new concepts of approximate saddle points and relate them to the above solution concepts. As a result, a dual is formulated, and duality results are established. [ABSTRACT FROM AUTHOR]

In the analysis of steel structures, several modern codes such as LRFD and Eurocode 3 provide for second-order approximate analysis. This paper presents a method of optimization for use in design that is directed toward improving the overall stability and strength of moment-resistant building frames. The method makes use of codified expressions for approximate second-order analysis. The objective function used in the optimization is the dominant eigenvalue of the linearized buckling problem. Only a first-order analysis is required, along with the calculation of the first eigenvalue of the linearized buckling problem of the structure. This new method provides a story by story procedure that is easily visualized. Several examples are provided to illustrate applications of the procedure. [ABSTRACT FROM AUTHOR]

In this article, we investigate the optimal control problem governed by parabolic inclusion. We describe the Galerkin approximation and we demonstrate the existence of the strong condensation points of the set of solutions of approximate optimization problems. Each of these points is a solution of the initial optimization problem. [ABSTRACT FROM AUTHOR]

We give a lower semicontinuous type regularity condition and a closedness type one which turn out to be necessary and sufficient for the fulfilment of two different formulae involving the ε-subdifferential of a perturbation function, respectively. These regularity conditions prove to be sufficient also for having formulae for the classical subdifferential of a perturbation function. Some recently published results concerning ε-subdifferentials are rediscovered as special cases. [ABSTRACT FROM AUTHOR]

In support set expansion sensitivity analysis, one concerns to find the range of parameter variation where the perturbed problem has an optimal solution with the support set that includes the support set of the given optimal solution of the unperturbed problem. In this article, we consider the perturbed convex quadratic optimization problem and present a method to identify the support set expansion sets for this problem. [ABSTRACT FROM AUTHOR]

A scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving unconstrained optimization problems is presented. The basic idea is to combine the scaled memoryless BFGS method and the preconditioning technique in the frame of the conjugate gradient method. The preconditioner, which is also a scaled memoryless BFGS matrix, is reset when the Beale-Powell restart criterion holds. The parameter scaling the gradient is selected as the spectral gradient. In very mild conditions, it is shown that, for strongly convex functions, the algorithm is globally convergent. Computational results for a set consisting of 750 unconstrained optimization test problems show that this new scaled conjugate gradient algorithm substantially outperforms the known conjugate gradient methods including the spectral conjugate gradient by Birgin and Martínez [Birgin, E. and Martínez, J.M., 2001, A spectral conjugate gradient method for unconstrained optimization. Applied Mathematics and Optimization, 43, 117-128], the conjugate gradient by Polak and Ribière [Polak, E. and Ribière, G., 1969, Note sur la convergence de méthodes de directions conjuguées. Revue Francaise Informat. Reserche Opérationnelle, 16, 35-43], as well as the most recent conjugate gradient method with guaranteed descent by Hager and Zhang [Hager, W.W. and Zhang, H., 2005, A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM Journal on Optimization, 16, 170-192; Hager, W.W. and Zhang, H., 2004, CG-DESCENT, A conjugate gradient method with guaranteed descent ACM Transactions on Mathematical Software, 32, 113-137]. [ABSTRACT FROM AUTHOR]

Generally, global convergence of the conjugate gradient methods for unconstrained optimization problems needs Wolfe line search or strong Wolfe line search. In this article, we propose a cautious DY conjugate gradient method and prove that this method with Armijo line search converges globally if the objective function has Lipschitz continuous gradients. We also present some preliminary numerical results to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]