基于矩量理论的电力系统全局优化算法研究
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摘要
优化理论在电网规划、运行等各方面得到广泛应用并发挥着越来越重要的作用,形成了各式各样的优化问题,其最优解的优劣直接影响电力系统的运行。几十年来,各种优化方法都被用于电力系统优化问题的求解,取得了许多有意义的成果。然而,由于电力系统优化问题具有非凸性,而传统优化方法难于确保其解的全局最优性,这使得电力系统全局最优解的求取面临着巨大的挑战。因此,研究新的全局优化理论,探究电力系统优化问题的全局最优解,具有重要的理论和现实意义。
本文依据全局优化理论的最新突破性成果——矩量半定规划,开展电力系统全局优化算法的理论研究工作。借助概率领域的矩量理论将电力系统多项式优化问题转换为矩量表达,通过构造半正定的矩量矩阵推导出矩量空间的半定规划凸松弛模型,即矩量半定规划模型,该模型可通过增大矩量矩阵的阶次而逐渐逼近于原问题的全局最优解。并且,在求解时引入全局最优判定准则,以保证解的全局最优性。
在电力系统优化问题中,{0,1}-经济调度和最优潮流问题是典型的非凸规划问题。其中{0,1}-经济调度问题属于混合整数规划问题,求解过程复杂,难于确保求得全局最优解,甚至得不到可行解；而最优潮流问题的全局最优解是学者们长期以来努力追求的目标,曾尝试采用半定规划凸松弛方法进行求取,但还是困难重重。本文采用矩量半定规划方法求解这两个问题,一般通过二阶松弛模型就能获得精确的全局最优解。主要研究成果如下：
1)提出了{0,1}-经济调度的矩量半定规划模型,将{0,1}-经济调度问题中的整数约束表示为多项式互补约束形式,并将问题转换到矩量空间,通过引入半正定约束,建立相应的矩量半定规划松弛模型进行求解。计算结果表明,该模型不用对原问题分解,其最优解中能直接得到0/1变量的整数解,并满足全局最优判定准则。
2)提出了求解{0,1}-经济调度问题多个全局最优解的矩量半定规划算法。{0,1}-经济调度属于组合优化问题,可能存在多个全局最优解,通过矩量半定规划的全局最优判定准则,可判断{0,1}-经济调度问题具有多少个全局最优解。当存在多个全局最优解时,所得矩量解为原问题的多个解在某取值概率下对应的矩量值,通过奇异值分解的特征值法可从矩量解中提取出{0,1}-经济调度问题的多个全局最优解。算例结果表明,该方法成功找到了多个有意义的全局最优解,这为电力系统组合优化问题的求解提供了有益的启示。
3)提出了最优潮流的矩量半定规划模型,将最优潮流问题表示为不等式约束的多项式优化问题,同样采用矩量空间的半定松弛技术,建立相应的矩量半定规划松弛模型进行求解。对最优潮流的标准算例及常规半定规划方法求解时的反例均能求得秩1的矩量解,从而确保得到全局最优解。由此表明,该模型能够克服现有的半定规划方法求解最优潮流时不能得到秩1解的问题,具有更高的可靠性。
4)提出了求解最优潮流问题的矩量半定规划全局优化算法。通过最优潮流矩量半定规划模型的秩1矩量解,可确定原问题的全局最优解是唯—的。此时,最优解的取值概率为狄拉克函数,则所得最优解的矩量值与原问题的全局最优解相等,因此最优潮流问题的全局最优解可从矩量解中直接获取。
本文在国家自然科学基金(51167001)和国家重点基础研究发展规划项目(973项目)(2013CB228205)的资助下完成。
Optimization theory has been widely applied in power system planning and operation, and it is playing an increasing significant role. This applications develop various optimization problems which solutions will affect the power system based on its merit. Over the past decades, all kinds of optimization are used with many positive results for the problems. However, power system optimization being nonconvex, it is an enormous challenge to solve the global optimum since traditional optimization methods can not assure the global optimality. Therefore, it has important theoretic and realistic meanings to study new global optimization theories and explore the global optimum for power system optimization problems.
Based on moment semidefinite programming (MSDP), which is the brand-new achievement of global optimization, this thesis focuses on the theory study for the global optimization algorithm of power system. With moment theory in the field of probability, power system polynomial optimization problems can be transformed as moment expressions, and relaxed as semidefinite programming (SDP) models in the moment space through constructing semidefinite moment matrices. This kind of SDP model is called MSDP model, and its optimal value can gradually approach the original global optimum as increasing the order of moment matrix. Moreover, a judgment criterion of global optimum is introduced to guarantee the global optimality for the solution.
In the power system optimization problems,{0,1}-economic dispatch (ED) and optimal power flow (OPF) are two typical nonconvex programming problems. The {0,l}-ED problem is a mixed integer program problem, its solution process is complex and is hard to get the global optimum, and even can not obtain a feasible solution. The global optimum of OPF problem is the goal of the long-term effort for the scholars, and it was attempted to be sovled by SDP convex relaxation method, but that is still difficult. This thesis employs MSDP method to solve these two problems, and obtains exact global optimum with2-order relaxation model in general. The main research achievements of this study are as follows.
1) A MSDP model for {O,1}-ED problem is proposed. The integer constraints in the {O,1}-ED model are written as polynomial complementary form, and the problem is converted to moment space, then a corresponding MSDP model can be established by introducting the semidefinite constraints. The results show that, its optimal solutions can meet the criterion of global optimum, and the integer solutions of the0/1variables can be got without dividing the original problem.
2) The MSDP algorithm of {0,1}-ED problem is proposed for solving multiple global optimal solutions. There may be one or more global optimal solutions for the {0,1}-ED since it belongs to the combined optimization problem. And the number of solutions of {0,1}-ED can be judged by the criterion of global optimum. When there are multiple global optimal solutions, the optimized result of MSDP is the moment of original solutions about a probability measure. And the multiple global optimal solutions of {0,1}-ED problem can be extracted from the MSDP solution by an eigenvalue method with singular value decomposition. An illustrative example showed that the method succeeded in finding multiple meaningful global optimal solutions. This process provides a useful insight into solving combinatorial optimization problems of power system.
3) A MSDP model for OPF problem is proposed. The OPF model being written as polynomial optimization problem with inequality constraints, a corresponding MSDP model can be established by the semidefinite relaxation techniques in moment space. This MSDP model can be solved with rank-1moment solution for OPF standard cases and the counterexamples of existing SDP method, and the global optimum is guaranteed. It is showed that the MSDP model overcome the problem of no rank-1solution in solving OPF by existing SDP method.
4) The MSDP global optimization algorithm of OPF problem is proposed. As the moment solution of MSDP-OPF model is rank-1, it ensures that the global optimal solution of OPF is unique. The probability measure of the solution is Dirac function now, and then the moment solution is equal to the original global optimum. Consequently, the global optimal solution of OPF can be got from moment solution directly.
This thesis was supported in part by the National Natural Science Foundation of China (51167001) and by The National Basic Research Program of China (2013CB228205)
本文依据全局优化理论的最新突破性成果——矩量半定规划,开展电力系统全局优化算法的理论研究工作。借助概率领域的矩量理论将电力系统多项式优化问题转换为矩量表达,通过构造半正定的矩量矩阵推导出矩量空间的半定规划凸松弛模型,即矩量半定规划模型,该模型可通过增大矩量矩阵的阶次而逐渐逼近于原问题的全局最优解。并且,在求解时引入全局最优判定准则,以保证解的全局最优性。
在电力系统优化问题中,{0,1}-经济调度和最优潮流问题是典型的非凸规划问题。其中{0,1}-经济调度问题属于混合整数规划问题,求解过程复杂,难于确保求得全局最优解,甚至得不到可行解；而最优潮流问题的全局最优解是学者们长期以来努力追求的目标,曾尝试采用半定规划凸松弛方法进行求取,但还是困难重重。本文采用矩量半定规划方法求解这两个问题,一般通过二阶松弛模型就能获得精确的全局最优解。主要研究成果如下：
1)提出了{0,1}-经济调度的矩量半定规划模型,将{0,1}-经济调度问题中的整数约束表示为多项式互补约束形式,并将问题转换到矩量空间,通过引入半正定约束,建立相应的矩量半定规划松弛模型进行求解。计算结果表明,该模型不用对原问题分解,其最优解中能直接得到0/1变量的整数解,并满足全局最优判定准则。
2)提出了求解{0,1}-经济调度问题多个全局最优解的矩量半定规划算法。{0,1}-经济调度属于组合优化问题,可能存在多个全局最优解,通过矩量半定规划的全局最优判定准则,可判断{0,1}-经济调度问题具有多少个全局最优解。当存在多个全局最优解时,所得矩量解为原问题的多个解在某取值概率下对应的矩量值,通过奇异值分解的特征值法可从矩量解中提取出{0,1}-经济调度问题的多个全局最优解。算例结果表明,该方法成功找到了多个有意义的全局最优解,这为电力系统组合优化问题的求解提供了有益的启示。
3)提出了最优潮流的矩量半定规划模型,将最优潮流问题表示为不等式约束的多项式优化问题,同样采用矩量空间的半定松弛技术,建立相应的矩量半定规划松弛模型进行求解。对最优潮流的标准算例及常规半定规划方法求解时的反例均能求得秩1的矩量解,从而确保得到全局最优解。由此表明,该模型能够克服现有的半定规划方法求解最优潮流时不能得到秩1解的问题,具有更高的可靠性。
4)提出了求解最优潮流问题的矩量半定规划全局优化算法。通过最优潮流矩量半定规划模型的秩1矩量解,可确定原问题的全局最优解是唯—的。此时,最优解的取值概率为狄拉克函数,则所得最优解的矩量值与原问题的全局最优解相等,因此最优潮流问题的全局最优解可从矩量解中直接获取。
本文在国家自然科学基金(51167001)和国家重点基础研究发展规划项目(973项目)(2013CB228205)的资助下完成。
Optimization theory has been widely applied in power system planning and operation, and it is playing an increasing significant role. This applications develop various optimization problems which solutions will affect the power system based on its merit. Over the past decades, all kinds of optimization are used with many positive results for the problems. However, power system optimization being nonconvex, it is an enormous challenge to solve the global optimum since traditional optimization methods can not assure the global optimality. Therefore, it has important theoretic and realistic meanings to study new global optimization theories and explore the global optimum for power system optimization problems.
Based on moment semidefinite programming (MSDP), which is the brand-new achievement of global optimization, this thesis focuses on the theory study for the global optimization algorithm of power system. With moment theory in the field of probability, power system polynomial optimization problems can be transformed as moment expressions, and relaxed as semidefinite programming (SDP) models in the moment space through constructing semidefinite moment matrices. This kind of SDP model is called MSDP model, and its optimal value can gradually approach the original global optimum as increasing the order of moment matrix. Moreover, a judgment criterion of global optimum is introduced to guarantee the global optimality for the solution.
In the power system optimization problems,{0,1}-economic dispatch (ED) and optimal power flow (OPF) are two typical nonconvex programming problems. The {0,l}-ED problem is a mixed integer program problem, its solution process is complex and is hard to get the global optimum, and even can not obtain a feasible solution. The global optimum of OPF problem is the goal of the long-term effort for the scholars, and it was attempted to be sovled by SDP convex relaxation method, but that is still difficult. This thesis employs MSDP method to solve these two problems, and obtains exact global optimum with2-order relaxation model in general. The main research achievements of this study are as follows.
1) A MSDP model for {O,1}-ED problem is proposed. The integer constraints in the {O,1}-ED model are written as polynomial complementary form, and the problem is converted to moment space, then a corresponding MSDP model can be established by introducting the semidefinite constraints. The results show that, its optimal solutions can meet the criterion of global optimum, and the integer solutions of the0/1variables can be got without dividing the original problem.
2) The MSDP algorithm of {0,1}-ED problem is proposed for solving multiple global optimal solutions. There may be one or more global optimal solutions for the {0,1}-ED since it belongs to the combined optimization problem. And the number of solutions of {0,1}-ED can be judged by the criterion of global optimum. When there are multiple global optimal solutions, the optimized result of MSDP is the moment of original solutions about a probability measure. And the multiple global optimal solutions of {0,1}-ED problem can be extracted from the MSDP solution by an eigenvalue method with singular value decomposition. An illustrative example showed that the method succeeded in finding multiple meaningful global optimal solutions. This process provides a useful insight into solving combinatorial optimization problems of power system.
3) A MSDP model for OPF problem is proposed. The OPF model being written as polynomial optimization problem with inequality constraints, a corresponding MSDP model can be established by the semidefinite relaxation techniques in moment space. This MSDP model can be solved with rank-1moment solution for OPF standard cases and the counterexamples of existing SDP method, and the global optimum is guaranteed. It is showed that the MSDP model overcome the problem of no rank-1solution in solving OPF by existing SDP method.
4) The MSDP global optimization algorithm of OPF problem is proposed. As the moment solution of MSDP-OPF model is rank-1, it ensures that the global optimal solution of OPF is unique. The probability measure of the solution is Dirac function now, and then the moment solution is equal to the original global optimum. Consequently, the global optimal solution of OPF can be got from moment solution directly.
This thesis was supported in part by the National Natural Science Foundation of China (51167001) and by The National Basic Research Program of China (2013CB228205)
引文
[1]谢政,李建平,陈挚.非线性最优化理论与方法[M].北京：高等教育出版社,2010.
[2]袁亚湘,孙文瑜.最优化理论与方法[M].北京：科学出版社,1997.
[3]何坚勇.最优化方法[M].北京：清华大学出版社,2007.
[4]I. Newton. Principia[M]. London, England,1687.
[5]J. L. Largrange. Mecanique analytique[M]. Paris, France,1788.
[6]G. B. Dantzig. "Origins of the simplex method," in A History of Scientific Computing, (Nash S G Ed.) [M]. Reading, MA:ACM Press,1990.
[7]H. W. Kuhn, Tucker A W. "Nonlinear programming," in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (J. Neyman Ed.) [M]. Berkeley, CA: University of California Press,1951:481-492.
[8]柳焯.最优化原理及其在电力系统中的应用[M].哈尔滨：哈尔滨工业大学出版社,1988.
[9]丘文千.电力系统优化规划模型与方法[M].杭州：浙江大学出版社,2012.
[10]李朝安.发电厂及电力系统经济运行[M].乌鲁木齐：新疆人民出版社,1985.
[11]于尔铿,相年德,赵国虹,王世缨.电力系统经济调度中网损修正方法的试验研究[J].中国电机工程学报,1985,5(3)：21-26.
[12]刘自发,张建华.一种求解电力经济负荷分配问题的改进微分进化算法[J].中国电机工程学报,2008,28(10)：100-105.
[13]吴阿琴,韦化,白晓清.基于半定规划的{0,1}-经济调度[J].电力系统及其自动化学报,2008,20(2)：121-125.
[14]J. Carpentier. Contribution to the economic dispatch problem[J]. Bull. Soc. Francaise Elect.,1962, 3(8):431-447.
[15]S. Frank, I. Steponavice, S. Rebennack. Optimal power flow:a bibliographic survey Ⅰ: Formulations and deterministic methods[J]. Energy Systems,2012,3(3):221-258.
[16]S. Frank, I. Steponavice, S. Rebennack. Optimal power flow:a bibliographic survey Ⅱ: Non-deterministic and hybrid methods[J]. Energy Systems,2012,3(3):259-289.
[17]O. Alsac, B. Scott. Optimal load flow with steady-state securily[J]. IEEE Trans, on PAS,1974, 93(3):745-751.
[18]D. I. Sun, B. Ashley, B. Brewer, A. Hughes, W. F. Tinney. Optimal Power Flow By Newton Approach[J]. IEEE Transactions on Power Apparatus and Systems,1984, PAS-103(10): 2864-2880.
[19]B. Stott, E. Hobson. Power System Security Control Calculations Using Linear Programming, Part Ⅰ[J]. IEEE Transactions on Power Apparatus and Systems,1978, PAS-97(5):1713-1720.
[20]B. Stott, E. Hobson. Power System Security Control Calculations Using Linear Programming, Part Ⅱ[J]. IEEE Transactions on Power Apparatus and Systems,1978, PAS-97(5):1721-1731.
[21]程浩忠.电力系统有功优化潮流算法——等效二次规划法[J].电力系统自动化,1991,15(4)：52-55.
[22]S. Granville. Optimal reactive dispatch through interior point methods[J]. IEEE Transactions on Power Systems,1994,9(1):136-146.
[23]W. Hua, H. Sasaki, J. Kubokawa, R. Yokoyama. An interior point nonlinear programming for optimal power flow problems with a novel data structure[J]. IEEE Transactions on Power Systems,1998,13(3):870-877.
[24]M. J. Todd. Semidefinite optimization[J]. Acta Numerica,2001,10:515-560.
[25]L. Vandenbergue. Applications of semidefinite programming[J]. Applied Numerical Mathematics, 1999,29(3):283-299.
[26]C. Helmberg. Semidefinite programming[J]. European Journal of Operational Research,2002, 137:461-482.
[27]M. Madrigal, V. H. Quintana. Semidefinite programming relaxations for{0,1} power dispatch problems[C]. IEEE Power Eng. Soc. Summer Meeting Conf.,1999:697-702.
[28]R. Fuentes-Loyola, V. H. Quintana. Medium-Term hydrothermal coordination by semidefinite programming[J]. IEEE Transactions on Power Systems,2003,18(4):1515-1522.
[29]韦化,吴阿琴,白晓清.一种求解机组组合问题的内点半定规划方法[J].中国电机工程学报,2008,28(1)：35-40.
[30]R. A. Jabr. Solution to economic dispatching with disjoint feasible regions via semidefinite programming[J]. IEEE Trans. Power Syst.,2012,27(1):572-573.
[31]R. A. Jabr. Rank-constrained semidefinite program for unit commitment[J]. Int. J. Elect. Power Energy Syst.,2013,47:13-20.
[32]X. Bai, H. Wei, K. Fujisawa, Y. Wang. Semidefinite programming for optimal power flow problems[J]. Int. J. Elect. Power Energy Syst.,2008,30(6-7):383-392.
[33]白晓清,韦化,K. Fujisawa.求解最优潮流问题的内点半定规划法[J].中国电机工程学报,2008,28(19)：56-64.
[34]J. Lavaei, S. Member, S. H. Low. Zero Duality Gap in Optimal Power Flow Problem[J]. IEEE Trans. Power Syst.,2012,27(1):92-107.
[35]王雅婷,何光宇,刘铠诚,王彬,申洪.求解状态估计问题的内点半定规划法[J].电网技术,2012,36(10)：209-215.
[36]R. A. Jabr. Exploiting sparsity in SDP relaxations of the OPF problem[J]. IEEE Trans. Power Syst.,2012,27(2):1138-1139.
[37]D. K. Molzahn, J. T. Holzer, B. C. Lesieutre, C. L. DeMarco. Implementation of a large-scale optimal power flow solver based on semidefinite programming[J]. IEEE Trans. Power Syst., to be published.
[38]S. Bose, D. F. Gayme, S. Low, K. M. Chandy. Optimal power flow over tree networks[C]. The 49th Annual Allerton Conference on Communication, Control, and Computing,2011:1342-1348.
[39]B. C. Lesieutre, D. K. Molzahn, A. R. Borden, C. L. DeMarco. Examining the limits of the application of semidefinite programming to power flow problems[C]. The 49th Annual Allerton Conference on Communication, Control, and Computing,2011:1492-1499.
[40]W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, P. A. Trodden. Local solutions of the optimal power flow problem[J]. IEEE Trans. Power Syst., to be published.
[41]R. Horst, P. M. Pardalos, V. T. Nguyen,全局优化引论(黄红选译)[M]..北京：清华大学出版社,2003.
[42]周雪刚.非凸优化问题的全局优化算法[D].长沙：中南大学,2010.
[43]R. E. Gomory. Outline of an algorithm for integer solution to linear programs[J]. Bulletin of Amerieal Mathematical Society,1958,64(5):275-278.
[44]J. E. Kelley. The cutting-plane method for solving convex programs[J]. Journal SIAM,1960,8(4): 703-712.
[45]T. V. Thieu, B. T. Tam, V. T. Ban. An outer approximation method for globally minimizing a concave function over a compact convex set[J]. Acta Mathematica Vietnamica,1983,8(1): 21-40.
[46]H. Tuy. Global minimization of a difference of two convex functions[J]. Mathematical Programming Studies,1987,30:150-182.
[47]A. I. Cohenand, M. Yoshimura. A branch-and-bound algorithm for unit commitment[J]. IEEE Trans. Power App. Syst.,1983, PAS-102(2):444-451.
[48]M. Tawarmalani, N. Sahinidis. Convexification and global optimization in continuous and mixed-integer nonlinear programming:theory, algorithms, software, and applications[M]. Kluwer Academic Publishers,2002.
[49]杜廷松,张明望,王浚岭.弱有效集上凹函数极大问题的分枝定界算法[J].黑龙江大学自然科学学报,2002,19(2)：14-17.
[50]R. P. Ge. The theory of filled funetion methods for finding global minimizer of nonlinearly constrained minimization problems[J]. Journal of Computational Mathematies,1987,5(1):1-9.
[51]X. Liu. Finding global minima with a computable filled function[J]. Journal of Global Optimization,2001,19(2):151-161.
[52]L. S. Zhang, D. Li. Global searchin nonliear integer programming:filled function Approach[C]. International Conference on Optimization Techniques and Applications,1998:446-452.
[53]D. Hilbert. Uber die Darstellung definiter Formen als Summe von Formenquadraten[J]. Mathematische Annalen,1888,32:342-350.
[54]B. Reznick. Some concrete aspects of Hilbert's 17th problem[J]. Contemporary Mathematics, 2000,253:251-272.
[55]C. Berg. The multidimensional moment problem and semi-groups[J]. Proc. Symp. Appl. Math., 1987,37:110-124.
[56]J. B. Lasserre. A sum of squares approximation of nonnegative polynomials[J]. SIAM Journal on Optimization,2006,16:751-765.
[57]K. Schmudgen. The K-moment problem for compact semi-algebraic sets[J]. Mathematische Annalen,1991,289:203-206.
[58]M. Putinar. Positive polynomials on compact semi-algebraic sets[J]. Indiana University Mathematics Journal,1993,42:969-984.
[59]M. D. Choi, T. Y. Lam, B. Reznick. Sums of squares of real polynomials[J]. Proc. Symposia Pure Math.,1995,58(2):103-126.
[60]V. Powers, T. Wormann. An algorithm for sums of squares of real polynomials[J]. J. Pure Appl. Algebra,1998,127:99-104.
[61]P. A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D]. California Institute of Technology,2000.
[62]J. B. Lasserre. Global optimization with polynomials and the problem of moments[J]. SIAM Journal on Optimization,2001,11 (3):796-817.
[63]J. B. Lasserre. Polynomials nonnegative on a grid and discrete representations[J]. Transactions of the American Mathematical Society,2001,354:631-649.
[64]P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems[J]. Mathematical Programming B,2003,96:293-320.
[65]P. A. Parrilo, B. Sturmfels. Minimizing polynomial functions[J]. DIMACS Series in Discrete Mathematics and Theoretical Computer Science,2003,60:83-99.
[66]M. Laurent. Sums of squares, moment matrices and optimization over polynomials[J]. IMA Volumes in Mathematics and its Applications,2009,149:157-270.
[67]H. Waki, S. Kim, M. Kojima, M. Muramatsu. Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity[J]. SIAM Journal on Optimization,2006,17(1):218-242.
[68]2009 Lagrange Prize Citation [EB/OL]. Seattle:Mathematical Optimization Society. http://www.mathopt.org/?nav=lagrange_2009.
[69]G. Chesi. LMI techniques for optimization over polynomials in control:a survey[J]. IEEE Transactions on Automatic Control,2010,55(11):2500-2510.
[70]A. Garulli, A. Vicino. Convex relaxations in circuits, systems, and control[J]. IEEE Circuits and Systems Magazine,2009,9(2):46-56.
[71]J. B. Lasserre. "An explicit exact SDP relaxation for nonlinear 0-1 programs," in Lecture Notes in Computer Science 2081, (K. Aardal and A. M. H. Gerards, Eds.) [M].2001:293-303.
[72]M. Laurent. Semidefinite Programming in Combinatorial and Polynomial Optimization[R]. Nieuw Archief voor Wiskunde,2008.
[73]J. Dattorro. Convex Optimization & Euclidean Distance Geometry[M]. Palo Alto, California: Meboo Publishing,2005.
[74]R. Bellman, K. Fan. On systems of linear inequalities in Hermitian matrix variables[C]. Convexity, proceedings of symposia in Pure Mathematics,1963:1-11.
[75]W. E. Donath, A. J. Hoffman. Lower bounds for the partitioning of graphs[J]. IBM J. of Research and Development,1973,17:420-425.
[76]J. Cullum, W. E. Donath, P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrixes[J]. Math. Programming Study,1975,3:35-55.
[77]L. Lovasz. On the Shannon capacity of a graph[J]. IEEE Trans. On Information Theory,1979,25: 1-7.
[78]B. D. Craven, B. Mond. Linear programming with matrix variables[J]. Linear Algebra and its Applications,1981,38:73-80.
[79]N. Karmarkar. A new polynomial-time algorithm for linear programming[J]. Combinatorica, 1984,4:373-395.
[80]Y. Nesterov, A. Nemirovskii, A general approach to polynomial-time algorithms design for convex programming[R]. Moscow,USSR,1988.
[81]Y. Nesterov, A. Nemirovskii. Interior point polynomial algorithms in convex programming[M]. Philadelphia, USA:SIAM,1994.
[82]F. Alizzdeh, Combinatorial optimization with interior point methods and semi-definite matrices[D]. Minneapolis:University of Minnesota,1991.
[83]F. Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization[J]. SIAM Journal on Optimization,1995,5:13-51.
[84]M. X. Goemans, F. Rendl. Semidefinite programming in combinatorial optimization[J]. Math Program,1997,79:143-161.
[85]P. Parrilo. Semidefinite programming relaxations and algebraic optimization in control[J]. Mathematical Programming, Series A,2003,93(3):331-359.
[86]H. Peng, L. K. Rasmussen. The application of semidefinite programming for detection in CDMA[J]. IEEE Journal on Selected areas in Communication,2001,19(8):1442-1449.
[87]S.-P. Wu, S. Boyd, L. Vandenberghe. FIR filter design via semidefinite programming and spectral factorization[C]. Proc.35 Conf. Decision and Control,1996:271-276.
[88]S. J. Benson, Y. Ye, X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization[J]. SIAM Journal on Optimization,2000,10(2):443-461.
[89]K. Fujisawa, M. Kojima, S. Shindoh. Exploiting Sparsity in Primal-Dual Interior-Point Methods for Semidefinite Programming[J]. Mathematical Programming B,1997,79:235-253.
[90]I. Adler, F. Alizadeh. Primal-dual interior point algorithms for convex quadratically constrained and semidefinite optimization problems[R]. New Brunswick, NJ,1995.
[91]J. Peng, C. Roos, T. Terlaky. New complexity analysis of the primal-dual method for semidefinite optimization based on the nesterov-todd direction[J]. Journal of Optimization Theory and Applications,2001,109(2):327-343.
[92]K.-C. Toh, M. J. Todd, R. H. Tutuncu. SDPT3 version 4.0-a MATLAB software for semidefinite-quadratic-linear programming.2009. Online]. Available: http://www.math.nus.edu.sg/-mattohkc/sdpt3.html.
[93]J. F. Strum. SeDuMi 1.02, a MATLAB toolbox for optimizatin over symmetric cones[J]. Optimization Methods and Software,1999,11&12:625-653.
[94]SDPA Online[EB/OL]. Seattle:Chuo University, Department of Industrial and Systems Engineering. http://sdpa.indsys.chuo-u.ac.jp/sdpa/.
[95]L. Lovasz, A. Schrijver. Cones of matrices and set-functions and 0-1 optimization[J]. SIAM Journal on Optimization,1991,1(2):160-190.
[96]H. J. Landau. Moments in Mathematics[M]. San Antonio:AMS short course lecture notes,1987.
[97]H. L. Pedersen. Stieltjes Moment Problems and the Friedrichs Extension of a Positive Definite Operator[J]. Journal of Approximation Theory,1995,83(3):289-307.
[98]J. B. Lasserre. Moments, positive polynomials and their applications[M]. London:Imperial College Press,2009.
[99]D. Henrion, J. B. Lasserre. Detecting global optimality and extracting solutions in GloptiPoly[R]. LAAS-CNRS, Toulouse (France),2003.
[100]G. H. Golub, C. F. Van Loan. Matrix computations (3rd edition)[M]. Baltimore:The Johns Hopkins University Press,1996.
[101]B. Sturmfels. Solving Systems of Polynomial Equations[M]. Providence, RI:American Mathematical Society,2002.
[102]A. Dickenstein, I. Z. Emiris. Solving Polynomial Equations:Foundations, Algorithms, and Applications[M]. Springer-Verlag,2005.
[103]李滨,韦化,李佩杰.电力系统无功优化的内点非线性互补约束算法[J].电力自动化设备,2010,30(2)：53-58.
[104]C. P. Cheng, C. W. Liu, C. C. Liu. Unit commitment by lagrangian relaxation and genetic algorithms[J]. IEEE Trans. Power Syst.,2000,15(2):707-714.
[105]P. M. Anderson, A. A. Fouad. Power System Control and Stability[M]. Ames (IA, USA):The Iowa State University Press,1977.
[106]W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, P. A. Trodden. Test Case Archive of Optimal Power Flow (OPF) Problems with Local Optima[EB/OL]. Available: http://www.maths.ed.ac.uk/optenergy/LocalOpt/.
[107]Power Systems Test Case Archive[EB/OL]. Seattle:University of Washington, Department of Electrical Engineering, http://www.ee.washington.edu/research/pstca.
[108]K. Fujisawa, H. Sato, S. Matsuoka, T. Endo, M. Yamashita, N. Maho. High-performance general solver for extremely large-scale semidefinite programming problems[C].2012 International Conference for High Performance Computing, Networking, Storage and Analysis,2012:1-11.
[109]M. Yamashita, K. Fujisawa, M. Fukuda, M. Kojima. Parallel primal-dual interior-point methods for semidefinite programs[R]. Department of Mathematical and Computing Sciences, Tokyo, 2005.
[110]L. S. Blackford, et al. ScaLA-PACK Users'Guide[M]. Philadelphia:Society for Industrial and Applied Mathematics,1997.
[Ill]M. Yamashita, K. Fujisawa, M. Kojima. SDPARA:semidefinite programming algorithm parallel version[J]. Parallel Computing,2003,29(8):1053-1067.
[112]SDPA Online Solver[EB/OL]. Seattle:Chuo University, Department of Industrial and Systems Engineering, http://laqua.indsys.chuo-u.ac.jp/portal/.
[113]M. Fukuda, M. Kojima, K. Murota, K. Nakata. Exploiting sparsity in semidefinite programming via matrix completion I:General framework[J]. SIAM Journal on Optimization,2000,11(3): 647-674.
[114]S. Kim, M. Kojima, M. Mevissen, M. Yamashita. Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion[J]. Mathematical Programming, 2011,129(1):33-68.
[115]H. Waki, S. Kim, M. Kojima, M. Muramatsu, H. Sugimoto. SparsePOP:a Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems[J]. ACM Transactions on Mathematical Software,2008,35(2):15.
[2]袁亚湘,孙文瑜.最优化理论与方法[M].北京：科学出版社,1997.
[3]何坚勇.最优化方法[M].北京：清华大学出版社,2007.
[4]I. Newton. Principia[M]. London, England,1687.
[5]J. L. Largrange. Mecanique analytique[M]. Paris, France,1788.
[6]G. B. Dantzig. "Origins of the simplex method," in A History of Scientific Computing, (Nash S G Ed.) [M]. Reading, MA:ACM Press,1990.
[7]H. W. Kuhn, Tucker A W. "Nonlinear programming," in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (J. Neyman Ed.) [M]. Berkeley, CA: University of California Press,1951:481-492.
[8]柳焯.最优化原理及其在电力系统中的应用[M].哈尔滨：哈尔滨工业大学出版社,1988.
[9]丘文千.电力系统优化规划模型与方法[M].杭州：浙江大学出版社,2012.
[10]李朝安.发电厂及电力系统经济运行[M].乌鲁木齐：新疆人民出版社,1985.
[11]于尔铿,相年德,赵国虹,王世缨.电力系统经济调度中网损修正方法的试验研究[J].中国电机工程学报,1985,5(3)：21-26.
[12]刘自发,张建华.一种求解电力经济负荷分配问题的改进微分进化算法[J].中国电机工程学报,2008,28(10)：100-105.
[13]吴阿琴,韦化,白晓清.基于半定规划的{0,1}-经济调度[J].电力系统及其自动化学报,2008,20(2)：121-125.
[14]J. Carpentier. Contribution to the economic dispatch problem[J]. Bull. Soc. Francaise Elect.,1962, 3(8):431-447.
[15]S. Frank, I. Steponavice, S. Rebennack. Optimal power flow:a bibliographic survey Ⅰ: Formulations and deterministic methods[J]. Energy Systems,2012,3(3):221-258.
[16]S. Frank, I. Steponavice, S. Rebennack. Optimal power flow:a bibliographic survey Ⅱ: Non-deterministic and hybrid methods[J]. Energy Systems,2012,3(3):259-289.
[17]O. Alsac, B. Scott. Optimal load flow with steady-state securily[J]. IEEE Trans, on PAS,1974, 93(3):745-751.
[18]D. I. Sun, B. Ashley, B. Brewer, A. Hughes, W. F. Tinney. Optimal Power Flow By Newton Approach[J]. IEEE Transactions on Power Apparatus and Systems,1984, PAS-103(10): 2864-2880.
[19]B. Stott, E. Hobson. Power System Security Control Calculations Using Linear Programming, Part Ⅰ[J]. IEEE Transactions on Power Apparatus and Systems,1978, PAS-97(5):1713-1720.
[20]B. Stott, E. Hobson. Power System Security Control Calculations Using Linear Programming, Part Ⅱ[J]. IEEE Transactions on Power Apparatus and Systems,1978, PAS-97(5):1721-1731.
[21]程浩忠.电力系统有功优化潮流算法——等效二次规划法[J].电力系统自动化,1991,15(4)：52-55.
[22]S. Granville. Optimal reactive dispatch through interior point methods[J]. IEEE Transactions on Power Systems,1994,9(1):136-146.
[23]W. Hua, H. Sasaki, J. Kubokawa, R. Yokoyama. An interior point nonlinear programming for optimal power flow problems with a novel data structure[J]. IEEE Transactions on Power Systems,1998,13(3):870-877.
[24]M. J. Todd. Semidefinite optimization[J]. Acta Numerica,2001,10:515-560.
[25]L. Vandenbergue. Applications of semidefinite programming[J]. Applied Numerical Mathematics, 1999,29(3):283-299.
[26]C. Helmberg. Semidefinite programming[J]. European Journal of Operational Research,2002, 137:461-482.
[27]M. Madrigal, V. H. Quintana. Semidefinite programming relaxations for{0,1} power dispatch problems[C]. IEEE Power Eng. Soc. Summer Meeting Conf.,1999:697-702.
[28]R. Fuentes-Loyola, V. H. Quintana. Medium-Term hydrothermal coordination by semidefinite programming[J]. IEEE Transactions on Power Systems,2003,18(4):1515-1522.
[29]韦化,吴阿琴,白晓清.一种求解机组组合问题的内点半定规划方法[J].中国电机工程学报,2008,28(1)：35-40.
[30]R. A. Jabr. Solution to economic dispatching with disjoint feasible regions via semidefinite programming[J]. IEEE Trans. Power Syst.,2012,27(1):572-573.
[31]R. A. Jabr. Rank-constrained semidefinite program for unit commitment[J]. Int. J. Elect. Power Energy Syst.,2013,47:13-20.
[32]X. Bai, H. Wei, K. Fujisawa, Y. Wang. Semidefinite programming for optimal power flow problems[J]. Int. J. Elect. Power Energy Syst.,2008,30(6-7):383-392.
[33]白晓清,韦化,K. Fujisawa.求解最优潮流问题的内点半定规划法[J].中国电机工程学报,2008,28(19)：56-64.
[34]J. Lavaei, S. Member, S. H. Low. Zero Duality Gap in Optimal Power Flow Problem[J]. IEEE Trans. Power Syst.,2012,27(1):92-107.
[35]王雅婷,何光宇,刘铠诚,王彬,申洪.求解状态估计问题的内点半定规划法[J].电网技术,2012,36(10)：209-215.
[36]R. A. Jabr. Exploiting sparsity in SDP relaxations of the OPF problem[J]. IEEE Trans. Power Syst.,2012,27(2):1138-1139.
[37]D. K. Molzahn, J. T. Holzer, B. C. Lesieutre, C. L. DeMarco. Implementation of a large-scale optimal power flow solver based on semidefinite programming[J]. IEEE Trans. Power Syst., to be published.
[38]S. Bose, D. F. Gayme, S. Low, K. M. Chandy. Optimal power flow over tree networks[C]. The 49th Annual Allerton Conference on Communication, Control, and Computing,2011:1342-1348.
[39]B. C. Lesieutre, D. K. Molzahn, A. R. Borden, C. L. DeMarco. Examining the limits of the application of semidefinite programming to power flow problems[C]. The 49th Annual Allerton Conference on Communication, Control, and Computing,2011:1492-1499.
[40]W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, P. A. Trodden. Local solutions of the optimal power flow problem[J]. IEEE Trans. Power Syst., to be published.
[41]R. Horst, P. M. Pardalos, V. T. Nguyen,全局优化引论(黄红选译)[M]..北京：清华大学出版社,2003.
[42]周雪刚.非凸优化问题的全局优化算法[D].长沙：中南大学,2010.
[43]R. E. Gomory. Outline of an algorithm for integer solution to linear programs[J]. Bulletin of Amerieal Mathematical Society,1958,64(5):275-278.
[44]J. E. Kelley. The cutting-plane method for solving convex programs[J]. Journal SIAM,1960,8(4): 703-712.
[45]T. V. Thieu, B. T. Tam, V. T. Ban. An outer approximation method for globally minimizing a concave function over a compact convex set[J]. Acta Mathematica Vietnamica,1983,8(1): 21-40.
[46]H. Tuy. Global minimization of a difference of two convex functions[J]. Mathematical Programming Studies,1987,30:150-182.
[47]A. I. Cohenand, M. Yoshimura. A branch-and-bound algorithm for unit commitment[J]. IEEE Trans. Power App. Syst.,1983, PAS-102(2):444-451.
[48]M. Tawarmalani, N. Sahinidis. Convexification and global optimization in continuous and mixed-integer nonlinear programming:theory, algorithms, software, and applications[M]. Kluwer Academic Publishers,2002.
[49]杜廷松,张明望,王浚岭.弱有效集上凹函数极大问题的分枝定界算法[J].黑龙江大学自然科学学报,2002,19(2)：14-17.
[50]R. P. Ge. The theory of filled funetion methods for finding global minimizer of nonlinearly constrained minimization problems[J]. Journal of Computational Mathematies,1987,5(1):1-9.
[51]X. Liu. Finding global minima with a computable filled function[J]. Journal of Global Optimization,2001,19(2):151-161.
[52]L. S. Zhang, D. Li. Global searchin nonliear integer programming:filled function Approach[C]. International Conference on Optimization Techniques and Applications,1998:446-452.
[53]D. Hilbert. Uber die Darstellung definiter Formen als Summe von Formenquadraten[J]. Mathematische Annalen,1888,32:342-350.
[54]B. Reznick. Some concrete aspects of Hilbert's 17th problem[J]. Contemporary Mathematics, 2000,253:251-272.
[55]C. Berg. The multidimensional moment problem and semi-groups[J]. Proc. Symp. Appl. Math., 1987,37:110-124.
[56]J. B. Lasserre. A sum of squares approximation of nonnegative polynomials[J]. SIAM Journal on Optimization,2006,16:751-765.
[57]K. Schmudgen. The K-moment problem for compact semi-algebraic sets[J]. Mathematische Annalen,1991,289:203-206.
[58]M. Putinar. Positive polynomials on compact semi-algebraic sets[J]. Indiana University Mathematics Journal,1993,42:969-984.
[59]M. D. Choi, T. Y. Lam, B. Reznick. Sums of squares of real polynomials[J]. Proc. Symposia Pure Math.,1995,58(2):103-126.
[60]V. Powers, T. Wormann. An algorithm for sums of squares of real polynomials[J]. J. Pure Appl. Algebra,1998,127:99-104.
[61]P. A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D]. California Institute of Technology,2000.
[62]J. B. Lasserre. Global optimization with polynomials and the problem of moments[J]. SIAM Journal on Optimization,2001,11 (3):796-817.
[63]J. B. Lasserre. Polynomials nonnegative on a grid and discrete representations[J]. Transactions of the American Mathematical Society,2001,354:631-649.
[64]P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems[J]. Mathematical Programming B,2003,96:293-320.
[65]P. A. Parrilo, B. Sturmfels. Minimizing polynomial functions[J]. DIMACS Series in Discrete Mathematics and Theoretical Computer Science,2003,60:83-99.
[66]M. Laurent. Sums of squares, moment matrices and optimization over polynomials[J]. IMA Volumes in Mathematics and its Applications,2009,149:157-270.
[67]H. Waki, S. Kim, M. Kojima, M. Muramatsu. Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity[J]. SIAM Journal on Optimization,2006,17(1):218-242.
[68]2009 Lagrange Prize Citation [EB/OL]. Seattle:Mathematical Optimization Society. http://www.mathopt.org/?nav=lagrange_2009.
[69]G. Chesi. LMI techniques for optimization over polynomials in control:a survey[J]. IEEE Transactions on Automatic Control,2010,55(11):2500-2510.
[70]A. Garulli, A. Vicino. Convex relaxations in circuits, systems, and control[J]. IEEE Circuits and Systems Magazine,2009,9(2):46-56.
[71]J. B. Lasserre. "An explicit exact SDP relaxation for nonlinear 0-1 programs," in Lecture Notes in Computer Science 2081, (K. Aardal and A. M. H. Gerards, Eds.) [M].2001:293-303.
[72]M. Laurent. Semidefinite Programming in Combinatorial and Polynomial Optimization[R]. Nieuw Archief voor Wiskunde,2008.
[73]J. Dattorro. Convex Optimization & Euclidean Distance Geometry[M]. Palo Alto, California: Meboo Publishing,2005.
[74]R. Bellman, K. Fan. On systems of linear inequalities in Hermitian matrix variables[C]. Convexity, proceedings of symposia in Pure Mathematics,1963:1-11.
[75]W. E. Donath, A. J. Hoffman. Lower bounds for the partitioning of graphs[J]. IBM J. of Research and Development,1973,17:420-425.
[76]J. Cullum, W. E. Donath, P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrixes[J]. Math. Programming Study,1975,3:35-55.
[77]L. Lovasz. On the Shannon capacity of a graph[J]. IEEE Trans. On Information Theory,1979,25: 1-7.
[78]B. D. Craven, B. Mond. Linear programming with matrix variables[J]. Linear Algebra and its Applications,1981,38:73-80.
[79]N. Karmarkar. A new polynomial-time algorithm for linear programming[J]. Combinatorica, 1984,4:373-395.
[80]Y. Nesterov, A. Nemirovskii, A general approach to polynomial-time algorithms design for convex programming[R]. Moscow,USSR,1988.
[81]Y. Nesterov, A. Nemirovskii. Interior point polynomial algorithms in convex programming[M]. Philadelphia, USA:SIAM,1994.
[82]F. Alizzdeh, Combinatorial optimization with interior point methods and semi-definite matrices[D]. Minneapolis:University of Minnesota,1991.
[83]F. Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization[J]. SIAM Journal on Optimization,1995,5:13-51.
[84]M. X. Goemans, F. Rendl. Semidefinite programming in combinatorial optimization[J]. Math Program,1997,79:143-161.
[85]P. Parrilo. Semidefinite programming relaxations and algebraic optimization in control[J]. Mathematical Programming, Series A,2003,93(3):331-359.
[86]H. Peng, L. K. Rasmussen. The application of semidefinite programming for detection in CDMA[J]. IEEE Journal on Selected areas in Communication,2001,19(8):1442-1449.
[87]S.-P. Wu, S. Boyd, L. Vandenberghe. FIR filter design via semidefinite programming and spectral factorization[C]. Proc.35 Conf. Decision and Control,1996:271-276.
[88]S. J. Benson, Y. Ye, X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization[J]. SIAM Journal on Optimization,2000,10(2):443-461.
[89]K. Fujisawa, M. Kojima, S. Shindoh. Exploiting Sparsity in Primal-Dual Interior-Point Methods for Semidefinite Programming[J]. Mathematical Programming B,1997,79:235-253.
[90]I. Adler, F. Alizadeh. Primal-dual interior point algorithms for convex quadratically constrained and semidefinite optimization problems[R]. New Brunswick, NJ,1995.
[91]J. Peng, C. Roos, T. Terlaky. New complexity analysis of the primal-dual method for semidefinite optimization based on the nesterov-todd direction[J]. Journal of Optimization Theory and Applications,2001,109(2):327-343.
[92]K.-C. Toh, M. J. Todd, R. H. Tutuncu. SDPT3 version 4.0-a MATLAB software for semidefinite-quadratic-linear programming.2009. Online]. Available: http://www.math.nus.edu.sg/-mattohkc/sdpt3.html.
[93]J. F. Strum. SeDuMi 1.02, a MATLAB toolbox for optimizatin over symmetric cones[J]. Optimization Methods and Software,1999,11&12:625-653.
[94]SDPA Online[EB/OL]. Seattle:Chuo University, Department of Industrial and Systems Engineering. http://sdpa.indsys.chuo-u.ac.jp/sdpa/.
[95]L. Lovasz, A. Schrijver. Cones of matrices and set-functions and 0-1 optimization[J]. SIAM Journal on Optimization,1991,1(2):160-190.
[96]H. J. Landau. Moments in Mathematics[M]. San Antonio:AMS short course lecture notes,1987.
[97]H. L. Pedersen. Stieltjes Moment Problems and the Friedrichs Extension of a Positive Definite Operator[J]. Journal of Approximation Theory,1995,83(3):289-307.
[98]J. B. Lasserre. Moments, positive polynomials and their applications[M]. London:Imperial College Press,2009.
[99]D. Henrion, J. B. Lasserre. Detecting global optimality and extracting solutions in GloptiPoly[R]. LAAS-CNRS, Toulouse (France),2003.
[100]G. H. Golub, C. F. Van Loan. Matrix computations (3rd edition)[M]. Baltimore:The Johns Hopkins University Press,1996.
[101]B. Sturmfels. Solving Systems of Polynomial Equations[M]. Providence, RI:American Mathematical Society,2002.
[102]A. Dickenstein, I. Z. Emiris. Solving Polynomial Equations:Foundations, Algorithms, and Applications[M]. Springer-Verlag,2005.
[103]李滨,韦化,李佩杰.电力系统无功优化的内点非线性互补约束算法[J].电力自动化设备,2010,30(2)：53-58.
[104]C. P. Cheng, C. W. Liu, C. C. Liu. Unit commitment by lagrangian relaxation and genetic algorithms[J]. IEEE Trans. Power Syst.,2000,15(2):707-714.
[105]P. M. Anderson, A. A. Fouad. Power System Control and Stability[M]. Ames (IA, USA):The Iowa State University Press,1977.
[106]W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, P. A. Trodden. Test Case Archive of Optimal Power Flow (OPF) Problems with Local Optima[EB/OL]. Available: http://www.maths.ed.ac.uk/optenergy/LocalOpt/.
[107]Power Systems Test Case Archive[EB/OL]. Seattle:University of Washington, Department of Electrical Engineering, http://www.ee.washington.edu/research/pstca.
[108]K. Fujisawa, H. Sato, S. Matsuoka, T. Endo, M. Yamashita, N. Maho. High-performance general solver for extremely large-scale semidefinite programming problems[C].2012 International Conference for High Performance Computing, Networking, Storage and Analysis,2012:1-11.
[109]M. Yamashita, K. Fujisawa, M. Fukuda, M. Kojima. Parallel primal-dual interior-point methods for semidefinite programs[R]. Department of Mathematical and Computing Sciences, Tokyo, 2005.
[110]L. S. Blackford, et al. ScaLA-PACK Users'Guide[M]. Philadelphia:Society for Industrial and Applied Mathematics,1997.
[Ill]M. Yamashita, K. Fujisawa, M. Kojima. SDPARA:semidefinite programming algorithm parallel version[J]. Parallel Computing,2003,29(8):1053-1067.
[112]SDPA Online Solver[EB/OL]. Seattle:Chuo University, Department of Industrial and Systems Engineering, http://laqua.indsys.chuo-u.ac.jp/portal/.
[113]M. Fukuda, M. Kojima, K. Murota, K. Nakata. Exploiting sparsity in semidefinite programming via matrix completion I:General framework[J]. SIAM Journal on Optimization,2000,11(3): 647-674.
[114]S. Kim, M. Kojima, M. Mevissen, M. Yamashita. Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion[J]. Mathematical Programming, 2011,129(1):33-68.
[115]H. Waki, S. Kim, M. Kojima, M. Muramatsu, H. Sugimoto. SparsePOP:a Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems[J]. ACM Transactions on Mathematical Software,2008,35(2):15.
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