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Abstract: In this paper, we suggest and analyze a new inexact proximal point method for solving general variational inequalities, which can be considered as an implicit predictor–corrector method. An easily measurable error term is proposed with further relaxed error bound and an optimal step length is obtained by maximizing the profit-function and is dependent on the previous points. Our results include several known and new techniques for solving variational inequalities and related optimization problems. Results obtained in this paper can be viewed as an important improvement and refinement of the previously known results. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov–Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, , which is as good as the linear analogue. [Copyright &y& Elsevier]

Abstract: In this paper, two conjugate dual problems based on weak efficiency to a constrained vector optimization problem are introduced. Some inclusion relations between the dual objective mappings and the properties of the Lagrangian maps and their saddle points for primal problem are discussed. Gap functions for a vector equilibrium problem are established by using the weak and strong duality. [Copyright &y& Elsevier]

Abstract: In this paper, we establish some relationships between vector variational-like inequality and vector optimization problems under the assumptions of α-invex functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problems, under pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizan, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113–119] to a wider class of functions, namely the pseudo-α-invex functions studied in a recent work of Noor [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. 5 (2004) 1–9]. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the classes of generalized type I vector-valued functions introduced by Aghezzaf and Hachimi in [J. Global Optim. 18 (2000) 91–101] to generalized univex type I vector-valued functions and consider a multiple-objective optimization problem involving generalized type I univex functions. A number of Kuhn–Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond–Weir and general Mond–Weir type duality results are also presented. [Copyright &y& Elsevier]