Vector and tensor optimization

Vector and tensor optimization

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Please wait a minute... For Selected: Download Citations EndNote Ris BibTeX Toggle Thumbnails Select A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor Yi-Yong Li· Qing-Zhi Yang · Xi He Journal of the Operations Research Society of China    2017, 5 (1): 3-.   DOI: 10.1007/s40305-016-0132-4 Abstract （469）      PDF       Knowledge map    Save In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported. Related Articles | Metrics | Comments（0） Select A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors Jiao-Jiao Hou · Chen Ling · Hong-Jin He Journal of the Operations Research Society of China    2017, 5 (1): 45-.   DOI: 10.1007/s40305-016-0137-z Abstract （538）      PDF       Knowledge map    Save In this paper, we consider the second-order cone tensor eigenvalue complementarity problem (SOCTEiCP) and present three different reformulations to the model under consideration. Specifically, for the general SOCTEiCP, we first show its equivalence to a particular variational inequality under reasonable conditions. A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem. Then, for the symmetric and sub-symmetric SOCTEiCPs, we reformulate them as appropriate nonlinear programming problems, which are extremely beneficial for designing reliable solvers to find solutions of the considered problem. Finally, we report some preliminary numerical results to verify our theoretical results. Related Articles | Metrics | Comments（0） Select An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application Mao-Lin Che · Yi-Min Wei Journal of the Operations Research Society of China    2017, 5 (1): 65-.   DOI: 10.1007/s40305-016-0138-y Abstract （403）      PDF       Knowledge map    Save The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingularM-tensor. We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors. We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples. Related Articles | Metrics | Comments（0） Select Upper Bounds for the Spectral Radii of Nonnegative Tensors Jing-Jing Jia · Qing-Zhi Yang Journal of the Operations Research Society of China    2017, 5 (1): 83-.   DOI: 10.1007/s40305-016-0150-2 Abstract （427）      PDF       Knowledge map    Save In this paper, we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors. Meanwhile, we show that these bounds could be tight for some special tensors. For a general nonnegative tensor which can be transformed into a matrix, we prove the maximal singular value of this matrix is an upper bound of its Z-eigenvalues. Some examples are provided to show these proposed bounds greatly improve some existing ones. Related Articles | Metrics | Comments（0） Select Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique Bing Hua · Gu-Yan Ni · Meng-Shi Zhang Journal of the Operations Research Society of China    2017, 5 (1): 111-.   DOI: 10.1007/s40305-016-0135-1 Abstract （480）      PDF       Knowledge map    Save The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue (US-eigenvalue) for symmetric complex tensors, which can be taken as a multilinear optimization problem in complex number field. In this paper, we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem. Then we use Jacobian semidefinite relaxation method to solve it. Some numerical examples are presented. Related Articles | Metrics | Comments（0） Cited: Baidu(4) Select On the Bound of the Eigenvalue in Module for a Positive Tensor Wen Li · Wei-Hui Liu · Seak-Weng Vong Journal of the Operations Research Society of China    2017, 5 (1): 123-.   DOI: 10.1007/s40305-016-0142-2 Abstract （2524）      PDF       Knowledge map    Save In this paper, we propose a bound for ratio of the largest eigenvalue and second largest eigenvalue in module for a higher-order tensor. From this bound, one may deduce the bound of the second largest eigenvalue in module for a positive tensor, and the bound can reduce to the matrix cases. Related Articles | Metrics | Comments（0） Cited: Baidu(1) Select A Kind of Unified Strict Efficiency via Improvement Sets in Vector Optimization Hui Guo, Yan-Qin Bai Journal of the Operations Research Society of China    2018, 6 (4): 557-570.   DOI: https://doi.org/10.1007/s40305-017-0185-z Abstract （96）      PDF       Knowledge map    Save In this paper, we propose a kind of unified strict efficiency named E-strict efficiency via improvement sets for vector optimization. This kind of efficiency is shown to be an extension of the classical strict efficiency and ε-strict efficiency and has many desirable properties. We also discuss some relationships with other properly efficiency based on improvement sets and establish the corresponding scalarization theorems by a base-functional and a nonlinear functional. Moreover, some examples are given to illustrate the main conclusions. Related Articles | Metrics | Comments（0） Select Successive Partial-Symmetric Rank-One Algorithms for Almost Unitarily Decomposable Conjugate Partial-Symmetric Tensors Tao-Ran Fu, Jin-Yan Fan Journal of the Operations Research Society of China    2019, 7 (1): 147-167.   DOI: 10.1007/s40305-018-0194-6 Abstract （422）      PDF       Knowledge map    Save In this paper, we introduce the almost unitarily decomposable conjugate partial-symmetric tensors, which are different from the commonly studied orthogonally decomposable tensors by involving the conjugate terms in the decomposition and the perturbation term. We not only show that successive rank-one approximation algorithm exactly recovers the unitary decomposition of the unitarily decomposable conjugate partial-symmetric tensors. The perturbation analysis of successive rank-one approximation algorithm for almost unitarily decomposable conjugate partial-symmetric tensors is also provided to demonstrate the robustness of the algorithm. Reference | Related Articles | Metrics | Comments（0） Select An Introduction to the Computational Complexity of Matrix Multiplication Yan Li, Sheng-Long Hu, Jie Wang, Zheng-Hai Huang Journal of the Operations Research Society of China    2020, 8 (1): 29-43.   DOI: 10.1007/s40305-019-00280-x Abstract （362）      PDF       Knowledge map    Save This article introduces the approach on studying the computational complexity of matrix multiplication by ranks of the matrix multiplication tensors. Basic results and recent developments in this area are reviewed. Reference | Related Articles | Metrics | Comments（0） Select Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems Xin Xu, Yang-Dong Xu, Yue-Ming Sun Journal of the Operations Research Society of China    2021, 9 (2): 441-454.   DOI: 10.1007/s40305-019-00275-8 Abstract （2184）      PDF       Knowledge map    Save With the help of a level mapping, this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems. First, we introduce a kind of level mapping which generalizes one given in Han and Gong (Optimization 65:1337–1347, 2016). Then, we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping. Finally, in terms of the semicontinuity of the level mapping, we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric setvalued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge. Reference | Related Articles | Metrics | Comments（0） Select Necessary Optimality Conditions for Semi-vectorial Bi-level Optimization with Convex Lower Level: Theoretical Results and Applications to the Quadratic Case Julien Collonge Journal of the Operations Research Society of China    2021, 9 (3): 691-712.   DOI: 10.1007/s40305-020-00305-w Abstract （1310）      PDF       Knowledge map    Save This paper explores related aspects to post-Pareto analysis arising from the multicriteria optimization problem. It consists of two main parts. In the first one, we give first-order necessary optimality conditions for a semi-vectorial bi-level optimization problem:the upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multi-objective optimization problem to be solved by several followers acting in a cooperative way (greatest coalition multi-players game). For the lower level, we deal with weakly or properly Pareto (efficient) solutions and we consider the so-called optimistic problem, i.e. when followers choose amongst Pareto solutions one which is the most favourable for the leader. In order to handle reallife applications, in the second part of the paper, we consider the case where each follower objective is expressed in a quadratic form. In this setting, we give explicit first-order necessary optimality conditions. Finally, some computational results are given to illustrate the paper. Reference | Related Articles | Metrics | Comments（0） Select A Levenberg–Marquardt Method for Solving the Tensor Split Feasibility Problem Yu-Xuan Jin, Jin-Ling Zhao Journal of the Operations Research Society of China    2021, 9 (4): 797-817.   DOI: 10.1007/s40305-020-00337-2 Abstract （2157）      PDF       Knowledge map    Save This paper considers the tensor split feasibility problem. Let C and Q be non-empty closed convex set and $\mathcal{A}$ be a semi-symmetric tensor. The tensor split feasibility problem is to find x ∈ C such that $\mathcal{A} x^{m-1} \in Q$. If we simply take this problem as a special case of the nonlinear split feasibility problem, then we can directly get a projection method to solve it. However, applying this kind of projection method to solve the tensor split feasibility problem is not so efficient. So we propose a Levenberg– Marquardt method to achieve higher efficiency. Theoretical analyses are conducted, and some preliminary numerical results show that the Levenberg–Marquardt method has advantage over the common projection method. Reference | Related Articles | Metrics | Comments（0）