JIMO
Nonlinear augmented Lagrangian for nonconvex multiobjective optimization
Chunrong Chen T. C. Edwin Cheng Shengji Li Xiaoqi Yang
Journal of Industrial & Management Optimization 2011, 7(1): 157-174 doi: 10.3934/jimo.2011.7.157
In this paper, based on the ordering relations induced by a pointed, closed and convex cone with a nonempty interior, we propose a nonlinear augmented Lagrangian dual scheme for a nonconvex multiobjective optimization problem by applying a class of vector-valued nonlinear augmented Lagrangian penalty functions. We establish the weak and strong duality results, necessary and sufficient conditions for uniformly exact penalization and exact penalization in the framework of nonlinear augmented Lagrangian. Our results include several ones in the literature as special cases.
keywords: ordering cone nonlinear augmented Lagrangian strong duality set-valued maps. Multiobjective optimization exact penalization
JIMO
Generalized weak sharp minima of variational inequality problems with functional constraints
Wenyan Zhang Shu Xu Shengji Li Xuexiang Huang
Journal of Industrial & Management Optimization 2013, 9(3): 621-630 doi: 10.3934/jimo.2013.9.621
In this paper, the notion of generalized weak sharp minima is introduced for variational inequality problems with functional constraints in finite-dimensional spaces by virtue of a dual gap function. Some equivalent and necessary conditions for the solution set of the variational inequality problems to be a set of generalized weak sharp minima are obtained.
keywords: Robinson's constraint qualification. dual gap function Variational inequality problems generalized weak sharp minima
JIMO
Calculus rules of generalized $\epsilon-$subdifferential for vector valued mappings and applications
Shengji Li Xiaole Guo
Journal of Industrial & Management Optimization 2012, 8(2): 411-427 doi: 10.3934/jimo.2012.8.411
In this paper, a generalized $\epsilon-$subdifferential, which was defined by a norm, is first introduced for a vector valued mapping. Some existence theorems and the properties of the generalized $\epsilon-$subdifferential are discussed. A relationship between the generalized $\epsilon-$subdifferential and a directional derivative is investigated for a vector valued mapping. Then, the calculus rules of the generalized $\epsilon-$subdifferential for the sum and the difference of two vector valued mappings were given. The positive homogeneity of the generalized $\epsilon-$subdifferential is also provided. Finally, as applications, necessary and sufficient optimality conditions are established for vector optimization problems.
keywords: vector optimization problem. Generalized $\epsilon-$subdifferential calculus rule optimality condition vector valued mapping
JIMO
Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem
Qilin Wang Shengji Li
Journal of Industrial & Management Optimization 2014, 10(4): 1225-1234 doi: 10.3934/jimo.2014.10.1225
This paper deals with the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Under new assumptions, which do not contain any information about solution mappings, we establish the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem by using a scalarization method. These results improve the corresponding ones in recent literature. Some examples are given to illustrate our results.
keywords: Lower semicontinuity parametric generalized vector equilibrium problems solution mappings scalarization.
JIMO
Semicontinuity of approximate solution mappings to generalized vector equilibrium problems
Qilin Wang Shengji Li
Journal of Industrial & Management Optimization 2016, 12(4): 1303-1309 doi: 10.3934/jimo.2016.12.1303
In this paper, the lower semicontinuity of the approximate solution mapping to generalized strong vector equilibrium problems is established by using a new proof method which is different from the ones used in the literature. Simultaneously, we also obtain the upper semicontinuity of the approximate solution mapping without the assumptions about monotonicity and approximate solution mappings. Some examples are given to illustrate our results.
keywords: approximate solution mappings. upper semicontinuity Generalized strong vector equilibrium problems lower semicontinuity
JIMO
Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria
Chunrong Chen Shengji Li
Journal of Industrial & Management Optimization 2012, 8(3): 691-703 doi: 10.3934/jimo.2012.8.691
In this paper, on one hand, we discuss upper Hölder type estimates of solutions to parametric vector quasi-equilibria with general settings, which generalize and extend the results of Chen et al. (Optim. Lett. 5: 85-98, 2011). On the other hand, combining the technique used for primal problems with suitable modifications, we also study upper Hölder type estimates of solutions to Minty-type parametric dual vector quasi-equilibria. The consequences obtained for dual problems are new in the literature.
keywords: upper Hölder estimates Hölder continuity Minty-type dual problems. primal and dual problems Parametric vector quasi-equilibria
NACO
Preface
Shengji Li Nan-Jing Huang Xinmin Yang
Numerical Algebra, Control & Optimization 2011, 1(3): i-ii doi: 10.3934/naco.2011.1.3i
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Franco Giannessi on the occasion of his 75th birthday and in recognition of his many fundamental contributions in Optimization and Nonlinear Analysis. It is a great honor and pleasure for the Guest Editors to have this opportunity to edit this Special Issue.

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NACO
Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization
Qilin Wang Shengji Li Kok Lay Teo
Numerical Algebra, Control & Optimization 2011, 1(3): 417-433 doi: 10.3934/naco.2011.1.417
In this paper, some properties are established for second-order adjacent derivatives of set-valued maps. Upper and lower semicontinuity and closedness are obtained for second-order adjacent derivatives of weak perturbation maps in vector optimization problems. Several examples are given for illustrating our results.
keywords: Vector optimization second-order adjacent derivatives weak perturbation maps upper and lower semicontinuity.
NACO
Stability analysis of parametric variational systems
Shengji Li Chunmei Liao Minghua Li
Numerical Algebra, Control & Optimization 2011, 1(2): 317-331 doi: 10.3934/naco.2011.1.317
In this paper, Robinson's metric regularity of a positive order around/at some point of parametric variational systems is discussed. Under some suitable conditions, the relationships among H$\ddot{o}$lder-likeness, H$\ddot{o}$lder calmness, metric regularity of a positive order and Robinson's metric regularity of a positive order are discussed for the parametric variational systems. Then, some applications to the stabilities of the optimal value map and the solution map are studied for a parametric vector optimization problem, respectively.
keywords: H$\ddot{o}$lder-likeness Parametric variational systems metric regularity of a positive order. Robinson's metric regularity of a positive order

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