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Numerical Algebra, Control & Optimization

2014 , Volume 4 , Issue 1

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On the Hermite--Hadamard inequality for convex functions of two variables
Shu-Lin Lyu
2014, 4(1): 1-8 doi: 10.3934/naco.2014.4.1 +[Abstract](498) +[PDF](311.1KB)
Abstract:
Inspired by the results in [S. S. Dragomir and I. Gomm, Num. Alg. Cont. $\&$ Opt., 2 (2012), 271--278], we give some new bounds for two mappings related to the Hermite--Hadamard inequality for convex functions of two variables, and apply them to special functions to get some results for the $p$-logarithmic mean. We also apply the Hermite--Hadamard inequality to matrix functions in this paper.
The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions
Yuying Zhou and Gang Li
2014, 4(1): 9-23 doi: 10.3934/naco.2014.4.9 +[Abstract](472) +[PDF](393.4KB)
Abstract:
In this paper, by virtue of the epigraph technique, we construct a new kind of closedness-type constraint qualification, which is the sufficient and necessary condition to guarantee the strong duality between a cone constraint composite optimization problem and its dual problem holds. Under this closedness-type constraint qualification condition, we obtain a formula of subdifferential for composite functions and study a cone constraint composite DC optimization problem in locally convex Hausdorff topological vector spaces.
Partial $S$-goodness for partially sparse signal recovery
Lingchen Kong, Naihua Xiu and Guokai Liu
2014, 4(1): 25-38 doi: 10.3934/naco.2014.4.25 +[Abstract](338) +[PDF](380.0KB)
Abstract:
In this paper, we will consider the problem of partially sparse signal recovery (PSSR), which is the signal recovery from a certain number of linear measurements when its part is known to be sparse. We establish and characterize partial $s$-goodness for a sensing matrix in PSSR. We show that the partial $s$-goodness condition is equivalent to the partial null space property (NSP), and is weaker than partial restricted isometry property. Moreover, this provides a verifiable approach for partial NSP via partial $s$-goodness constants. We also give exact and stable partially $s$-sparse recovery via the partial $l_1$-norm minimization under mild assumptions.
Some useful inequalities via trace function method in Euclidean Jordan algebras
Yu-Lin Chang and Chin-Yu Yang
2014, 4(1): 39-48 doi: 10.3934/naco.2014.4.39 +[Abstract](417) +[PDF](348.4KB)
Abstract:
In this paper, we establish convexity of some functions associated with symmetric cones, called SC trace functions. As illustrated in the paper, these functions play a key role in the development of penalty and barrier function methods for symmetric cone programs. With trace function method we offer much simpler proofs to these useful inequalities.
Adjacent vertex distinguishing edge-colorings and total-colorings of the Cartesian product of graphs
Shuangliang Tian, Ping Chen, Yabin Shao and Qian Wang
2014, 4(1): 49-58 doi: 10.3934/naco.2014.4.49 +[Abstract](466) +[PDF](347.3KB)
Abstract:
Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing edge-coloring of $G$ if $F_{\sigma}(u)\not= F_{\sigma}(v)$ for any $uv\in E(G)$, where $F_{\sigma}(u)$ denotes the set of colors of edges incident with $u$. A total-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing total-coloring of $G$ if $S_{\sigma}(u)\not= S_{\sigma}(v)$ for any $uv\in E(G)$, where $S_{\sigma}(u)$ denotes the set of colors of edges incident with $u$ together with the color assigned to $u$. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of $G$ is denoted by $\chi_a^{'}(G)$ (resp. $\chi^{''}_{a}(G)$). In this paper, we provide upper bounds for these parameters of the Cartesian product $G$ □ $H$ of two graphs $G$ and $H$. We also determine exact value of these parameters for the Cartesian product of a bipartite graph and a complete graph or a cycle, the Cartesian product of a complete graph and a cycle, the Cartesian product of two trees and the Cartesian product of regular graphs.
Grasping force based manipulation for multifingered hand-arm Robot using neural networks
Chun-Hsu Ko and Jing-Kun Chen
2014, 4(1): 59-74 doi: 10.3934/naco.2014.4.59 +[Abstract](433) +[PDF](459.8KB)
Abstract:
Multifingered hand-arm robots play an important role in dynamic manipulation tasks. They can grasp and move various shaped objects. It is important to plan the motion of the arm and appropriately control the grasping forces for the multifingered hand-arm robots. In this paper, we perform the grasping force based manipulation of the multifingered hand-arm robot by using neural networks. The motion parameters are analyzed and planned with the constraint of the multi-arms kinematics. The optimal grasping force problem is recast as the second-order cone program. The semismooth Newton method with the Fischer-Burmeister function is then used to efficiently solve the second-order cone program. The neural network manipulation system is obtained via the fitting of the data that are generated from the optimal manipulation simulations. The simulations of optimal grasping manipulation are performed to demonstrate the effectiveness of the proposed approach.
An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials
Nur Fadhilah Ibrahim
2014, 4(1): 75-91 doi: 10.3934/naco.2014.4.75 +[Abstract](447) +[PDF](481.0KB)
Abstract:
In this paper, we propose an iterative method for calculating the largest eigenvalue of nonhomogeneous nonnegative polynomials. This method is a generalization of the method in [19]. We also prove this method is convergent for irreducible nonhomogeneous nonnegative polynomials.

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