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Numerical Algebra, Control & Optimization

2013 , Volume 3 , Issue 2

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Preface
Jiu Ding, Bingsheng He, Qin Ni and Wenyu Sun
2013, 3(2): i-ii doi: 10.3934/naco.2013.3.2i +[Abstract](475) +[PDF](94.6KB)
Abstract:
This special issue is dedicated to the memory of the late Professor Xuchu He of Nanjing University, China. Professor He passed away on April 30, 1990 at the age of 69. An international workshop was held at Nanjing Normal University in May 2011 on the occasion of his 90th birthday.

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A two-phase method for multidimensional number partitioning problem
Feng Ma and Mingfang Ni
2013, 3(2): 203-206 doi: 10.3934/naco.2013.3.203 +[Abstract](608) +[PDF](229.5KB)
Abstract:
The multidimensional number partitioning problem is to split a given set of integer vectors into two disjoint subsets, such that the sums of the elements in the two subsets are equal or nearly equal on every coordinate. This partitioning problem is in fact NP-hard. In this paper, we propose an optimization method for this problem. The proposed method involves recasting the original problem as a two-phase optimization problem. First, candidate partitions are obtained by using the method proposed in [4]. Then the optimal one is selected from the obtained candidate partitions. An example is given to illustrate the method.
Partial eigenvalue assignment with time delay robustness
Xiaobin Mao and Hua Dai
2013, 3(2): 207-221 doi: 10.3934/naco.2013.3.207 +[Abstract](631) +[PDF](371.8KB)
Abstract:
The partial eigenvalue assignment problem concerns reassigning a few of undesired eigenvalues of a linear system to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spill-over). This paper considers the partial eigenvalue assignment problem with time delay robustness. A time delay robustness measure is presented by analyzing the sensitivity of the assigned eigenvalues with respect to time delay. The problem is formulated as an unconstrained minimization problem with the cost function involving the time delay robustness measure. A numerical algorithm with analytical formulation of the gradient for the cost function is provided. A numerical example is included to show the effectiveness of the proposed method.
Subspace trust-region algorithm with conic model for unconstrained optimization
Xin Zhang, Jie Wen and Qin Ni
2013, 3(2): 223-234 doi: 10.3934/naco.2013.3.223 +[Abstract](689) +[PDF](378.0KB)
Abstract:
In this paper, a new subspace algorithm is proposed for unconstrained optimization. In this new algorithm, the subspace technique is used in the trust region subproblem with conic model, and the dogleg method is modified to solve this subproblem. The global convergence of this algorithm under some reasonable conditions is established. Numerical experiment shows that this algorithm may be superior to the corresponding algorithm without using subspace technique especially for large scale problems.
A unified maximum entropy method via spline functions for Frobenius-Perron operators
Jiu Ding and Noah H. Rhee
2013, 3(2): 235-245 doi: 10.3934/naco.2013.3.235 +[Abstract](737) +[PDF](355.4KB)
Abstract:
We present a general frame of finite element maximum entropy methods for the computation of a stationary density of Frobenius-Perron operators associated with one dimensional transformations, based on spline function approximations. This gives a unified numerical approach to the density recovery for this class of positive operators by combining the principle of maximum entropy with the idea of finite elements. The norm convergence of the method is proved and the numerical results with the piecewise cubic method show its fast convergence.
Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming
Bingsheng He and Xiaoming Yuan
2013, 3(2): 247-260 doi: 10.3934/naco.2013.3.247 +[Abstract](1080) +[PDF](461.8KB)
Abstract:
Recently, we have proposed combining the alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show that the decomposed subproblems in the ADMM procedure can be substantially alleviated by linearizing the involved quadratic terms arising from the augmented Lagrangian penalty. When the resolvent operators of the separable functions in the objective have closed-form representations, embedding the linearization into the ADMM subproblems becomes necessary to yield easy subproblems with closed-form solutions. We thus show theoretically that the blend of ADMM, Gaussian back substitution and linearization works effectively for the separable convex minimization model under consideration.
The stationary iterations revisited
Xuzhou Chen, Xinghua Shi and Yimin Wei
2013, 3(2): 261-270 doi: 10.3934/naco.2013.3.261 +[Abstract](573) +[PDF](351.7KB)
Abstract:
In this paper, we first present a necessary and sufficient conditions for the weakly and strongly convergence of the general stationary iterations $x^{(k+1)} = T x^{(k)} +c$ with initial iteration matrix $T$ and vectors $c$ and $x^{(0)}$. Then we apply these general results and present convergence conditions for the stationary iterations for solving singular linear system $A x = b$. We show that our convergence conditions are weaker and more general than the known results.
Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $
Daniel Morales-Silva and David Yang Gao
2013, 3(2): 271-282 doi: 10.3934/naco.2013.3.271 +[Abstract](500) +[PDF](367.1KB)
Abstract:
The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double-min duality is solved for a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple solutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.
A structured trust region method for nonconvex programming with separable structure
Dan Xue, Wenyu Sun and Hongjin He
2013, 3(2): 283-293 doi: 10.3934/naco.2013.3.283 +[Abstract](576) +[PDF](184.5KB)
Abstract:
In this paper, we present a structured trust region algorithm for nonconvex programming with separable structure. We obtain the trial step by decomposing the step into its normal and tangential components. The structure of the problem is dealt with in the framework of the trust region. The global convergence is proved for the proposed algorithm. The preliminary numerical results show that the proposed algorithm is potentially efficient.
A modification of the forward-backward splitting method for maximal monotone mappings
Xiao Ding and Deren Han
2013, 3(2): 295-307 doi: 10.3934/naco.2013.3.295 +[Abstract](1362) +[PDF](192.3KB)
Abstract:
In this paper, we propose a modification of the forward-backward splitting method for maximal monotone mappings, where we adopt a new step-size scheme in generating the next iterate. This modification is motivated by the ingenious rule proposed by He and Liao in modified Korpelevich's extra-gradient method [13]. Under suitable conditions, we prove the global convergence of the new algorithm. We apply our method to solve some monotone variational inequalities and report its numerical results. Comparisons with modified Khobotov-Korpelevich's extragradient method [13,14] and Tseng's method [30] show the significance of our work.
Nonmonotone retrospective conic trust region method for unconstrained optimization
Lijuan Zhao and Wenyu Sun
2013, 3(2): 309-325 doi: 10.3934/naco.2013.3.309 +[Abstract](668) +[PDF](245.6KB)
Abstract:
We propose a retrospective conic trust region method for unconstrained optimization. It can be regarded as an extension of the retrospective trust region method based on a quadratic model which was first proposed by Bastin et al (2010). Nonmonotone technique is added to accelerate the speed of the algorithm. Under some mild conditions, the sequence generated by our algorithm converges to a stationary point. Numerical tests on a set of standard testing problems confirm the efficiency of our new method.
An adaptive wavelet method and its analysis for parabolic equations
Qiang Guo and Dong Liang
2013, 3(2): 327-345 doi: 10.3934/naco.2013.3.327 +[Abstract](546) +[PDF](261.2KB)
Abstract:
In this paper, we analyze an adaptive wavelet method with variable time step sizes and space refinement for parabolic equations. The advantages of multi-resolution wavelet processes combined with certain equivalences involving weighted sequence norms of wavelet coefficients allow us to set up an efficient adaptive algorithm producing locally refined spaces for each time step. Reliable and efficient a posteriori error estimate is derived, which assesses the discretization error with respect to a given quantity of physical interest. The influence of the time and space discretization errors is separated into different error indicators. We prove that the proposed adaptive wavelet algorithm terminates in a finite number of iterations for any given accuracy.
Solutions of the Yang-Baxter matrix equation for an idempotent
A. Cibotarica, Jiu Ding, J. Kolibal and Noah H. Rhee
2013, 3(2): 347-352 doi: 10.3934/naco.2013.3.347 +[Abstract](600) +[PDF](253.5KB)
Abstract:
Let $A$ be a square matrix which is an idempotent. We find all solutions of the matrix equation of $AXA=XAX$ by using the diagonalization technique for $A$.
Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems
Zaki Chbani and Hassan Riahi
2013, 3(2): 353-366 doi: 10.3934/naco.2013.3.353 +[Abstract](511) +[PDF](406.5KB)
Abstract:
The aim of this paper is to obtain, in a Hilbert space $H$, the weak and strong convergence of a penalty proximal algorithm and a splitting one for a bilevel equilibrium problem: find $ x\in S_F $ such that $\ G(x,y)\geq 0\ $ for all $ \ y\in S_F$, where $S_F :=\lbrace y\in K\; :\; F(y,u)\geq 0\;\; \forall u\in K \rbrace$, and $F,G:K\times K\longrightarrow \mathbb{R}$ are two bifunctions with $K$ a nonempty closed convex subset of $H$. In our framework, results of convergence generalize those recently obtained by Attouch et al. (SIAM Journal on Optimization 21, 149-173 (2011)). We show in particular that for the strong convergence of the penalty algorithm, the geometrical condition they impose is not required. We also give applications of the iterative schemes to fixed point problems and variational inequalities.
Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming
Cristian Dobre
2013, 3(2): 367-378 doi: 10.3934/naco.2013.3.367 +[Abstract](584) +[PDF](394.6KB)
Abstract:
In this paper we give a proof for the special structure of the Wedderburn decomposition of the regular *-representation of a given matrix $*$-algebra. This result was stated without proof in: de Klerk, E., Dobre, C. and Pasechnik, D.V.: Numerical block diagonalization of matrix $*$-algebras with application to semidefinite programming, Mathematical Programming-B, 129 (2011), 91--111; and is used in applications of semidefinite programming (SDP) for structured combinatorial optimization problems. In order to provide the proof for this special structure we derive several other mathematical properties of the regular *-representation.
Index-range monotonicity and index-proper splittings of matrices
Litismita Jena and Sabyasachi Pani
2013, 3(2): 379-388 doi: 10.3934/naco.2013.3.379 +[Abstract](584) +[PDF](176.6KB)
Abstract:
Index-range monotonicity is proposed, and some characterizations of this notion are obtained. Then different convergence and comparison theorems are presented using several new subclasses of index-proper splittings.

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