In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at le...In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at leastα=0.87856….In fact,the estimated worst-case performance ratio is dependent on the data of the problem withαbeing a uniform lower bound.In light of this new bound,we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at leastα+δ_d if every weight is strictly positive,whereδ_d>0 is a constant depending on the problem dimension and data.展开更多
In this paper,we consider the following indefinite complex quadratic maximization problem: maximize zHQz,subject to zk ∈ C and zkm = 1,k = 1,...,n,where Q is a Hermitian matrix with trQ = 0,z ∈ Cn is the decision ve...In this paper,we consider the following indefinite complex quadratic maximization problem: maximize zHQz,subject to zk ∈ C and zkm = 1,k = 1,...,n,where Q is a Hermitian matrix with trQ = 0,z ∈ Cn is the decision vector,and m 3.An (1/log n) approximation algorithm is presented for such problem.Furthermore,we consider the above problem where the objective matrix Q is in bilinear form,in which case a 0.7118 cos mπ 2approximation algorithm can be constructed.In the context of quadratic optimization,various extensions and connections of the model are discussed.展开更多
A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the...A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the modified relaxation problem,the number of introduced constraints and the lowest relaxation order decreases significantly.At the same time,the finite convergence property is guaranteed.In addition,the proposed method can be applied to the quadratically constrained problem with two quadratic constraints.Moreover,the efficiency of the proposed method is verified by numerical experiments.展开更多
We consider the design of semidefinite programming （SDP） based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance （MHC-LU）： Find a partition of the vertices of a weighted hypergraph...We consider the design of semidefinite programming （SDP） based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance （MHC-LU）： Find a partition of the vertices of a weighted hypergraph H = （V, E） into two subsets V1, V2 with ｜｜V2｜ - ｜1/1 ｜｜ ≤ u for some given u and maximizing the total weight of the edges meeting both V1 and V2. The problem MHC-LU generalizes several other combinatorial optimization problems including Max Cut, Max Cut with Limited Unbalance （MC-LU）, Max Set Splitting, Max Ek-Set Splitting and Max Hypergraph Bisection. By generalizing several earlier ideas, we present an SDP randomized approximation algorithm for MHC-LU with guaranteed worst-case performance ratios for various unbalance parameters τ = u/｜V｜. We also give the worst-case performance ratio of the SDP-algorithm for approximating MHC-LU regardless of the value of τ. Our strengthened SDP relaxation and rounding method improve a result of Ageev and Sviridenko （2000） on Max Hypergraph Bisection （MHC-LU with u = 0）, and results of Andersson and Engebretsen （1999）, Gaur and Krishnamurti （2001） and Zhang et al. （2004） on Max Set Splitting （MHC-LU with u = ｜V｜）. Furthermore, our new formula for the performance ratio by a tighter analysis compared with that in Galbiati and Maffioli （2007） is responsible for the improvement of a result of Galbiati and Maffioli （2007） on MC-LU for some range of τ.展开更多
无线传感网中的多类应用均需要准确的定位算法。为了降低定位成本,减少能量消耗,常采用基于接收信号强度RSS(Received Signal Strength)测距,再利用最大似然ML(Maximum likelihood)估计法求解节点的位置。然而,ML估计为非线性、非凸性,...无线传感网中的多类应用均需要准确的定位算法。为了降低定位成本,减少能量消耗,常采用基于接收信号强度RSS(Received Signal Strength)测距,再利用最大似然ML(Maximum likelihood)估计法求解节点的位置。然而,ML估计为非线性、非凸性,难以获取全局最优解。为此,提出凸半定规划SDP(Semidefinite Programming)的合作式定位方案,利用凸半定规划策略将ML估计转换成凸优问题。同时,该方案考虑两类场景:源节点发射功率已知、未知。针对第一类场景,利用半凸松驰策略,并结合最小化最小二乘法,建立凸优表达式,最后利用CVX求解;针对第二类场景,先建立联合ML估计函数,再利用SDP估计,并结合起来简单的三步骤方案进行位置估计。仿真结果表明,提出的SDP算法的定位精度比SD/SOCP-1、SDPRSS平均提高了近15%至20%。此外,提出的SDP算法在所有场景的误差小于3m的出现概率占0.8,而SD/SOCP-1、SDPRSS算法小于0.5。展开更多
Recently, researchers have been interested in studying the semidefinite programming （SDP） relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained qu...Recently, researchers have been interested in studying the semidefinite programming （SDP） relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained quadratic programming （QCQP）. Comparing to the basic SDP relaxation, this doubly-positive SDP model possesses additional O（n2） constraints, which makes the SDP solution complexity substantially higher than that for the basic model with O（n） constraints. In this paper, we prove that the doubly-positive SDP model is equivalent to the basic one with a set of valid quadratic cuts. When QCQP is symmetric and homogeneous （which represents many classical combinatorial and non- convex optimization problems）, the doubly-positive SDP model is equivalent to the basic SDP even without any valid cut. On the other hand, the doubly-positive SDP model could help to tighten the bound up to 36%, but no more. Finally, we manage to extend some of the previous results to quartic models.展开更多
This paper considers the NP (Non-deterministic Polynomial)-hard problem of finding a minimum value of a quadratic program (QP), subject to m non-convex inhomogeneous quadratic constraints. One effective algorithm is p...This paper considers the NP (Non-deterministic Polynomial)-hard problem of finding a minimum value of a quadratic program (QP), subject to m non-convex inhomogeneous quadratic constraints. One effective algorithm is proposed to get a feasible solution based on the optimal solution of its semidefinite programming (SDP) relaxation problem.展开更多
无线传感网中的多类应用均需要准确的定位算法。为了降低定位成本,减少能量消耗,常采用基于接收信号强度RSS（received signal strength）测距;再利用最大似然ML（maximum likelihood）估计法求解节点的位置。然而,ML估计为非线性、非凸...无线传感网中的多类应用均需要准确的定位算法。为了降低定位成本,减少能量消耗,常采用基于接收信号强度RSS（received signal strength）测距;再利用最大似然ML（maximum likelihood）估计法求解节点的位置。然而,ML估计为非线性、非凸性,难以获取全局最优解;为此,提出凸半定规划SDP（semidefinite programming）的合作式定位方案,利用凸半定规划策略将ML估计转换成凸优问题;同时,该方案考虑两类场景：源节点发射功率已知、未知。针对第一类场景,利用半凸松弛策略,并结合最小化最小二乘法,建立凸优表达式,最后利用CVX求解。针对第二类场景,先建立联合ML估计函数,再利用SDP估计,并结合起来简单的三步骤方案进行位置估计。仿真结果表明,提出的SDP算法的定位精度比SD/SOCP-1、SDPRSS平均提高了近15%~20%。此外,提出的SDP算法在所有场景的误差小于3 m的出现概率占0.8,而SD/SOCP-1、SDPRSS算法小于0.5。展开更多
基金This work was supported by the National Natural Science Foundation of China (Grant No.10401038)Startup Grant for Doctoral Research of Beijing University of Technology and Hong Kong RGC Earmarked Grant CUHK4242/04E
文摘In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at leastα=0.87856….In fact,the estimated worst-case performance ratio is dependent on the data of the problem withαbeing a uniform lower bound.In light of this new bound,we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at leastα+δ_d if every weight is strictly positive,whereδ_d>0 is a constant depending on the problem dimension and data.
基金supported by Hong Kong RGC Earmarked Grant (Grant No.CUHK418406)National Natural Science Foundation of China (Grant No.10771133)
文摘In this paper,we consider the following indefinite complex quadratic maximization problem: maximize zHQz,subject to zk ∈ C and zkm = 1,k = 1,...,n,where Q is a Hermitian matrix with trQ = 0,z ∈ Cn is the decision vector,and m 3.An (1/log n) approximation algorithm is presented for such problem.Furthermore,we consider the above problem where the objective matrix Q is in bilinear form,in which case a 0.7118 cos mπ 2approximation algorithm can be constructed.In the context of quadratic optimization,various extensions and connections of the model are discussed.
基金Fundamental Research Funds for the Central Universities,China(No.2232019D3-38)Shanghai Sailing Program,China(No.22YF1400900)。
文摘A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the modified relaxation problem,the number of introduced constraints and the lowest relaxation order decreases significantly.At the same time,the finite convergence property is guaranteed.In addition,the proposed method can be applied to the quadratically constrained problem with two quadratic constraints.Moreover,the efficiency of the proposed method is verified by numerical experiments.
基金supported by National Natural Science Foundation of China(Grant Nos.11171160,11331003 and 11471003)the Priority Academic Program Development of Jiangsu Higher Education Institutions+2 种基金the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.13KJB1100188)Natural Science Foundation of Guangdong Province(Grant No.S2012040007521)Sienceand Technology Planning Project in Guangzhou(Grant No.2013J4100077)
文摘We consider the design of semidefinite programming （SDP） based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance （MHC-LU）： Find a partition of the vertices of a weighted hypergraph H = （V, E） into two subsets V1, V2 with ｜｜V2｜ - ｜1/1 ｜｜ ≤ u for some given u and maximizing the total weight of the edges meeting both V1 and V2. The problem MHC-LU generalizes several other combinatorial optimization problems including Max Cut, Max Cut with Limited Unbalance （MC-LU）, Max Set Splitting, Max Ek-Set Splitting and Max Hypergraph Bisection. By generalizing several earlier ideas, we present an SDP randomized approximation algorithm for MHC-LU with guaranteed worst-case performance ratios for various unbalance parameters τ = u/｜V｜. We also give the worst-case performance ratio of the SDP-algorithm for approximating MHC-LU regardless of the value of τ. Our strengthened SDP relaxation and rounding method improve a result of Ageev and Sviridenko （2000） on Max Hypergraph Bisection （MHC-LU with u = 0）, and results of Andersson and Engebretsen （1999）, Gaur and Krishnamurti （2001） and Zhang et al. （2004） on Max Set Splitting （MHC-LU with u = ｜V｜）. Furthermore, our new formula for the performance ratio by a tighter analysis compared with that in Galbiati and Maffioli （2007） is responsible for the improvement of a result of Galbiati and Maffioli （2007） on MC-LU for some range of τ.
文摘无线传感网中的多类应用均需要准确的定位算法。为了降低定位成本,减少能量消耗,常采用基于接收信号强度RSS(Received Signal Strength)测距,再利用最大似然ML(Maximum likelihood)估计法求解节点的位置。然而,ML估计为非线性、非凸性,难以获取全局最优解。为此,提出凸半定规划SDP(Semidefinite Programming)的合作式定位方案,利用凸半定规划策略将ML估计转换成凸优问题。同时,该方案考虑两类场景:源节点发射功率已知、未知。针对第一类场景,利用半凸松驰策略,并结合最小化最小二乘法,建立凸优表达式,最后利用CVX求解;针对第二类场景,先建立联合ML估计函数,再利用SDP估计,并结合起来简单的三步骤方案进行位置估计。仿真结果表明,提出的SDP算法的定位精度比SD/SOCP-1、SDPRSS平均提高了近15%至20%。此外,提出的SDP算法在所有场景的误差小于3m的出现概率占0.8,而SD/SOCP-1、SDPRSS算法小于0.5。
基金The second author＇s research was supported by Program for Innovative Research Team of Shanghai University of Finance and Economics （IRTSHUFE） and by National Natural Science Foundation of China （NSFC） Project 11471205 the third author＇s research was supported in part by NSF GOALI 0800151.
文摘Recently, researchers have been interested in studying the semidefinite programming （SDP） relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained quadratic programming （QCQP）. Comparing to the basic SDP relaxation, this doubly-positive SDP model possesses additional O（n2） constraints, which makes the SDP solution complexity substantially higher than that for the basic model with O（n） constraints. In this paper, we prove that the doubly-positive SDP model is equivalent to the basic one with a set of valid quadratic cuts. When QCQP is symmetric and homogeneous （which represents many classical combinatorial and non- convex optimization problems）, the doubly-positive SDP model is equivalent to the basic SDP even without any valid cut. On the other hand, the doubly-positive SDP model could help to tighten the bound up to 36%, but no more. Finally, we manage to extend some of the previous results to quartic models.
文摘This paper considers the NP (Non-deterministic Polynomial)-hard problem of finding a minimum value of a quadratic program (QP), subject to m non-convex inhomogeneous quadratic constraints. One effective algorithm is proposed to get a feasible solution based on the optimal solution of its semidefinite programming (SDP) relaxation problem.
