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Numerical Algebra, Control & Optimization

2013 , Volume 3 , Issue 4

Special issue in the honor of Bernard Bonnard. Part II.

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A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function
Behrouz Kheirfam
2013, 3(4): 601-614 doi: 10.3934/naco.2013.3.601 +[Abstract](137) +[PDF](367.6KB)
We present a full Nesterov-Todd step infeasible interior-point algorithm based on a kernel function. Each main iteration of the algorithm consists of a feasibility step and some centering steps. We introduce a kernel function in the algorithm to induce the feasibility step. The iteration bound coincides with the best iteration bound for infeasible interior-point methods, that is, $O(r\log\frac{r}{\epsilon})$, where $r$ is the rank of Euclidean Jordan algebra.
Some properties of a class of $(F,E)$-$G$ generalized convex functions
Lijia Yan
2013, 3(4): 615-625 doi: 10.3934/naco.2013.3.615 +[Abstract](36) +[PDF](306.4KB)
In this paper, firstly, a new class of $(F,E)$-$G$ generalized convex functions, which is the generalization of $F$-$G$ generalized convex functions, is introduced. Secondly, some properties of $(F,E)$-$G$ generalized convex functions are obtained. Finally, some relations between $(F,E)$-$G$ generalized convex functions and other generalizations of convex functions are established.
Error bounds for symmetric cone complementarity problems
Xin-He Miao and  Jein-Shan Chen
2013, 3(4): 627-641 doi: 10.3934/naco.2013.3.627 +[Abstract](31) +[PDF](369.5KB)
In this paper, we investigate the issue of error bounds for symmetric cone complementarity problems (SCCPs). In particular, we show that the distance between an arbitrary point in Euclidean Jordan algebra and the solution set of the symmetric cone complementarity problem can be bounded above by some merit functions such as Fischer-Burmeister merit function, the natural residual function and the implicit Lagrangian function. The so-called $R_0$-type conditions, which are new and weaker than existing ones in the literature, are assumed to guarantee that such merit functions can provide local and global error bounds for SCCPs. Moreover, when SCCPs reduce to linear cases, we demonstrate such merit functions cannot serve as global error bounds under general monotone condition, which implicitly indicates that the proposed $R_0$-type conditions cannot be replaced by $P$-type conditions which include monotone condition as special cases.
Characterizations of the $E$-Benson proper efficiency in vector optimization problems
Kequan Zhao and  Xinmin Yang
2013, 3(4): 643-653 doi: 10.3934/naco.2013.3.643 +[Abstract](37) +[PDF](357.5KB)
In this paper, under the nearly $E$-subconvexlikeness, some characterizations of the $E$-Benson proper efficiency are established in terms of scalarization, Lagrange multipliers, saddle point criteria and duality for a vector optimization problem with set-valued maps. Our main results generalize and unify some previously known results.
Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method
Honglei Xu , Peng Sui , Guanglu Zhou and  Louis Caccetta
2013, 3(4): 655-664 doi: 10.3934/naco.2013.3.655 +[Abstract](26) +[PDF](305.4KB)
In this paper, we consider the bullwhip effect problem of an Order-Up-To (OUT) inventory strategy for a supply chain system. We firstly establish a new discrete-time dynamical model which is suitable to describe the OUT inventory strategy. Then, we analyze the bullwhip effect for the dynamical model of the supply chain system. We thus transform the bullwhip effect's dampening problem to a discrete-time optimal control problem. By using the Pontryagin's maximum principle, we compute the corresponding optimal control and obtain the optimal manufacturer productivity of goods. Finally, we carry out numerical simulation experiments to show that the devised optimal control strategy is useful to dampen the bullwhip effect which always happens in the supply chain system.
Semi-definite programming based approaches for real-time tractor localization in port container terminals
Wei Huang , Ka-Fai Cedric Yiu and  Henry Y. K. Lau
2013, 3(4): 665-680 doi: 10.3934/naco.2013.3.665 +[Abstract](24) +[PDF](780.0KB)
In order to effectively manage and deploy internal tractors in a port container marine terminal, real-time information concerning the location of the tractors is required so that timely scheduling and planning of tractors control and dispatching can be derived. This paper propose a wireless sensor network-based Truck Flow Management System (TFMS) to help tracking the real-time location of internal tractors in a container terminal so as to streamline the management of the terminal operation. Focusing on the real-time localization, the semi-definite programming (SDP) based approaches are employed by introducing the terminal context information, including prior known road constraints and available time-serial data recorded in the network, into the traditional SDP formulation. Experimental results are presented to show that the proposed formulation and treatments to the problem can greatly decrease the estimated errors compared to the traditional formulation.




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