메뉴 건너뛰기




Volumn 26, Issue 1, 2011, Pages 23-46

Projection methods for conic feasibility problems: Applications to polynomial sum-of-squares decompositions

Author keywords

convex cones; least squares problems; positive polynomials; projection algorithms; semidefinite programming

Indexed keywords

CONVEX CONE; LEAST SQUARE; POSITIVE POLYNOMIALS; PROJECTION ALGORITHMS; SEMI-DEFINITE PROGRAMMING;

EID: 77957300831     PISSN: 10556788     EISSN: 10294937     Source Type: Journal    
DOI: 10.1080/10556780903191165     Document Type: Article
Times cited : (33)

References (51)
  • 1
    • 34247385872 scopus 로고    scopus 로고
    • A finite step algorithm for solving convex feasibility problems
    • M. Ait Rami, U. Helmke, and J.B. Moore, A finite step algorithm for solving convex feasibility problems, J. Global Optim. 38 (2007), pp. 143-160.
    • (2007) J. Global Optim. , vol.38 , pp. 143-160
    • Ait Rami, M.1    Helmke, U.2    Moore, J.B.3
  • 2
    • 0003169116 scopus 로고
    • Interior point methods in semidefinite programming with applications to combinatorial optimization
    • F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim. 5(1) (1995), pp. 13-51.
    • (1995) SIAM J. Optim. , vol.5 , Issue.1 , pp. 13-51
    • Alizadeh, F.1
  • 3
    • 0001448913 scopus 로고
    • On the convergence of von Neumann's alternating projection algorithm for two sets
    • H.H. Bauschke and J.M. Borwein, On the convergence of von Neumann's alternating projection algorithm for two sets, Set Valued Anal. 1(2) (1993), pp. 185-212.
    • (1993) Set Valued Anal. , vol.1 , Issue.2 , pp. 185-212
    • Bauschke, H.H.1    Borwein, J.M.2
  • 6
    • 76549100532 scopus 로고    scopus 로고
    • A preconditioned Newton algorithm for the nearest correlation matrix
    • DOI: 10.1093/imanum/drn085
    • R. Borsdorf and N.J. Higham, A preconditioned Newton algorithm for the nearest correlation matrix, IMAJ. Numer. Anal., DOI: 10.1093/imanum/drn085.
    • IMAJ. Numer. Anal.
    • Borsdorf, R.1    Higham, N.J.2
  • 8
    • 33646249113 scopus 로고    scopus 로고
    • Least-squares covariance matrix adjustment
    • S. Boyd and L. Xiao, Least-squares covariance matrix adjustment, SIAM J. Matrix Anal. Appl. 27(2) (2005), pp. 532-546.
    • (2005) SIAM J. Matrix Anal. Appl. , vol.27 , Issue.2 , pp. 532-546
    • Boyd, S.1    Xiao, L.2
  • 11
    • 46449086587 scopus 로고    scopus 로고
    • HIFOO: A Matlab package for fixed-order controller design and H-infinity optimization
    • Toulouse, France, Available at
    • J.V. Burke, D. Henrion, A.S. Lewis, and M.L. Overton, HIFOO: A Matlab package for fixed-order controller design and H-infinity optimization, Proceedings of IFAC Symposium on Robust Control Design, Toulouse, France, 2006. Available at www.cs.nyu.edu/overton/software/hifoo.
    • (2006) Proceedings of IFAC Symposium on Robust Control Design
    • Burke, J.V.1    Henrion, D.2    Lewis, A.S.3    Overton, M.L.4
  • 13
    • 0002761429 scopus 로고
    • Generalization of Farkas' theorem
    • B.D. Craven and J.J. Koliha, Generalization of Farkas' theorem, SIAM J. Math. Anal. 8(6) (1977), pp. 983-997.
    • (1977) SIAM J. Math. Anal. , vol.8 , Issue.6 , pp. 983-997
    • Craven, B.D.1    Koliha, J.J.2
  • 16
    • 84948499836 scopus 로고
    • An algorithm for restricted least-square regression
    • R.L. Dykstra, An algorithm for restricted least-square regression, J. Amer. Statist. Assoc. 78(384) (1983), pp. 837-842.
    • (1983) J. Amer. Statist. Assoc. , vol.78 , Issue.384 , pp. 837-842
    • Dykstra, R.L.1
  • 17
    • 0000583199 scopus 로고
    • Rank minimization under LMI constraints: A framework for output feedback problems
    • Groningen, The Netherlands
    • L. El Ghaoui and P. Gahinet, Rank minimization under LMI constraints: A framework for output feedback problems, Proceedings of the European Control Conference, Groningen, The Netherlands, 1993.
    • (1993) Proceedings of the European Control Conference
    • El Ghaoui, L.1    Gahinet, P.2
  • 19
    • 0031144412 scopus 로고    scopus 로고
    • The projective method for solving linear matrix inequalities
    • P. Gahinet and A. Nemirovski, The projective method for solving linear matrix inequalities, Math. Program 77 (1997), pp. 163-190.
    • (1997) Math. Program , vol.77 , pp. 163-190
    • Gahinet, P.1    Nemirovski, A.2
  • 20
    • 56349136045 scopus 로고    scopus 로고
    • Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
    • M. Bakonyi, A. Gheondea, M. Putinar, and J. Rovnyak, eds., Theta Foundation, Bucharest
    • J.W. Helton and M. Putinar, Positive polynomials in scalar and matrix variables, the spectral theorem and optimization, in Operator Theory, Structured Matrices and Dilations: Tiberiu Constantinescu MemorialVolume,M. Bakonyi, A. Gheondea, M. Putinar, and J. Rovnyak, eds., Theta Foundation, Bucharest, 2007, pp. 229-306.
    • (2007) Operator Theory, Structured Matrices and Dilations: Tiberiu Constantinescu Memorial , pp. 229-306
    • Helton, J.W.1    Putinar, M.2
  • 22
    • 2542587541 scopus 로고    scopus 로고
    • GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi
    • Available at
    • D. Henrion and J.B. Lasserre, GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi, ACM Trans. Math. Softw. 29(2) (2003), pp. 165-194. Available at www.laas.fr/̃henrion/software/gloptipoly2.
    • (2003) ACM Trans. Math. Softw. , vol.29 , Issue.2 , pp. 165-194
    • Henrion, D.1    Lasserre, J.B.2
  • 24
    • 68949115889 scopus 로고    scopus 로고
    • GloptiPoly 3: Moments, optimization and semidefinite programming
    • D. Henrion, J.B. Lasserre, and J. Löfberg, GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods Softw. 24 (2009), pp. 761-779.
    • (2009) Optim. Methods Softw. , vol.24 , pp. 761-779
    • Henrion, D.1    Lasserre, J.B.2    Löfberg, J.3
  • 25
    • 0000379660 scopus 로고
    • Computing a nearest symmetric positive semidefinite matrix
    • N. Higham, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl. 103 (1988), pp. 103-118.
    • (1988) Linear Algebra Appl. , vol.103 , pp. 103-118
    • Higham, N.1
  • 26
    • 0344153904 scopus 로고    scopus 로고
    • Computing a nearest symmetric correlation matrix: A problem from finance
    • N. Higham, Computing a nearest symmetric correlation matrix: A problem from finance, IMA J. Numer. Anal. 22(3) (2002), pp. 329-343.
    • (2002) IMA J. Numer. Anal. , vol.22 , Issue.3 , pp. 329-343
    • Higham, N.1
  • 29
    • 0035238864 scopus 로고    scopus 로고
    • Global optimization with polynomials and the problem of moments
    • J.B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11(3) (2001), pp. 796-817.
    • (2001) SIAM J. Optim. , vol.11 , Issue.3 , pp. 796-817
    • Lasserre, J.B.1
  • 30
    • 67349148256 scopus 로고    scopus 로고
    • Sums of squares, moment matrices and optimization over polynomials
    • M. Putinar and S. Sullivant, eds., The IMA Volumes in Mathematics and its Applications, Springer-Verlag, Berlin
    • M. Laurent, Sums of squares, moment matrices and optimization over polynomials, in Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., The IMA Volumes in Mathematics and its Applications Vol. 149, Springer-Verlag, Berlin, 2009, pp. 157-270.
    • (2009) Emerging Applications of Algebraic Geometry , vol.149 , pp. 157-270
    • Laurent, M.1
  • 31
    • 48549094732 scopus 로고    scopus 로고
    • Alternating projections on manifolds
    • A.S. Lewis and J. Malick, Alternating projections on manifolds, Math. Oper. Res. 33(1) (2008), pp. 216-234.
    • (2008) Math. Oper. Res. , vol.33 , Issue.1 , pp. 216-234
    • Lewis, A.S.1    Malick, J.2
  • 32
    • 70349387487 scopus 로고    scopus 로고
    • Local linear convergence for alternating and averaged nonconvex projections
    • A.S. Lewis, D.R. Luke, and J. Malick, Local linear convergence for alternating and averaged nonconvex projections, Found. Comput. Math. 9(4) (2009), pp. 485-513.
    • (2009) Found. Comput. Math. , vol.9 , Issue.4 , pp. 485-513
    • Lewis, A.S.1    Luke, D.R.2    Malick, J.3
  • 33
    • 67349187539 scopus 로고    scopus 로고
    • Pre- and post-processing sum-of-squares programs in practice
    • J. Löfberg, Pre- and post-processing sum-of-squares programs in practice, IEEE Trans. Automat. Control 54(5) (2009), pp. 1007-1011.
    • (2009) IEEE Trans. Automat. Control , vol.54 , Issue.5 , pp. 1007-1011
    • Löfberg, J.1
  • 34
    • 14544268199 scopus 로고    scopus 로고
    • A dual approach to semidefinite least-squares problems
    • J. Malick, A dual approach to semidefinite least-squares problems, SIAM J. Matrix Anal. Appl. 26(1) (2004), pp. 272-284.
    • (2004) SIAM J. Matrix Anal. Appl. , vol.26 , Issue.1 , pp. 272-284
    • Malick, J.1
  • 35
    • 3142688043 scopus 로고    scopus 로고
    • An independent benchmarking of SDP and SOCP solvers
    • H.D. Mittelmann, An independent benchmarking of SDP and SOCP solvers, Math. Program. 95(2) (2003), pp. 407-430.
    • (2003) Math. Program. , vol.95 , Issue.2 , pp. 407-430
    • Mittelmann, H.D.1
  • 37
    • 0001776805 scopus 로고    scopus 로고
    • Squared functional systems and optimization problems
    • H. Frenk, C. Roos, T. Terlaky, and S. Zhang, eds., Kluwer Academic Publishers, Dordrecht
    • Y. Nesterov, Squared functional systems and optimization problems, in High Performance Optimization, H. Frenk, C. Roos, T. Terlaky, and S. Zhang, eds., Kluwer Academic Publishers, Dordrecht, 2000, pp. 405-440.
    • (2000) High Performance Optimization , pp. 405-440
    • Nesterov, Y.1
  • 38
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of nonsmooth functions
    • Y. Nesterov, Smooth minimization of nonsmooth functions, Math. Program. 103(1) (2005), pp. 127-152.
    • (2005) Math. Program. , vol.103 , Issue.1 , pp. 127-152
    • Nesterov, Y.1
  • 40
    • 33748923977 scopus 로고    scopus 로고
    • A Newton-like method for solving rank constrained linear matrix inequalities
    • R. Orsi, U. Helmke, and J.B. Moore, A Newton-like method for solving rank constrained linear matrix inequalities, Automatica 42(11) (2006), pp. 1875-1882.
    • (2006) Automatica , vol.42 , Issue.11 , pp. 1875-1882
    • Orsi, R.1    Helmke, U.2    Moore, J.B.3
  • 42
    • 55749106047 scopus 로고    scopus 로고
    • Computing sum of squares decompositions with rational coefficients
    • H. Peyrl and P.A. Parrilo, Computing sum of squares decompositions with rational coefficients, Theoret. Comput. Sci. 409(2) (2008), pp. 269-281.
    • (2008) Theoret. Comput. Sci. , vol.409 , Issue.2 , pp. 269-281
    • Peyrl, H.1    Parrilo, P.A.2
  • 43
    • 0032064831 scopus 로고    scopus 로고
    • An algorithm for sums of squares of real polynomials
    • V. Powers and T.Wörmann, An algorithm for sums of squares of real polynomials, J. Pure Appl. Algebra 127 (1998), pp. 99-104.
    • (1998) J. Pure Appl. Algebra , vol.127 , pp. 99-104
    • Powers, V.1    Wörmann, T.2
  • 44
    • 34247364678 scopus 로고    scopus 로고
    • Quadratic convergence and numerical experiments of Newton's method for computing the nearest correlation matrix
    • H. Qi and D. Sun, Quadratic convergence and numerical experiments of Newton's method for computing the nearest correlation matrix, SIAM J. Matrix Anal. Appl. 28 (2006), pp. 360-385.
    • (2006) SIAM J. Matrix Anal. Appl. , vol.28 , pp. 360-385
    • Qi, H.1    Sun, D.2
  • 45
    • 0004267646 scopus 로고
    • Princeton University Press, Princeton NJ
    • R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
    • (1970) Convex Analysis
    • Rockafellar, R.T.1
  • 47
    • 0033296299 scopus 로고    scopus 로고
    • Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones
    • J.F. Sturm, Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones, Optim. Methods Softw. 11-12 (1999), pp. 625-653.
    • (1999) Optim. Methods Softw. , vol.11-12 , pp. 625-653
    • Sturm, J.F.1
  • 48
    • 77957298033 scopus 로고    scopus 로고
    • Research Report CORR, Department of Combinatorics and Optimization, University of Waterloo, Canada, August
    • L. Tunçel and H. Wolkowicz, Strong duality and minimal representation for cone optimization, Research Report CORR 2008-2107, Department of Combinatorics and Optimization, University ofWaterloo, Canada, August 2008.
    • (2008) Strong Duality and Minimal Representation for Cone Optimization , pp. 2008-2107
    • Tunçel, L.1    Wolkowicz, H.2
  • 49
    • 55549112664 scopus 로고    scopus 로고
    • Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems
    • R.H. Tütüncü, K.C. Toh, and M.J. Todd, Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems, Pac. J. Optim. 3 (2007), pp. 135-164.
    • (2007) Pac. J. Optim. , vol.3 , pp. 135-164
    • Tütüncü, R.H.1    Toh, K.C.2    Todd, M.J.3
  • 50
    • 60549085200 scopus 로고    scopus 로고
    • Further relaxations of the SDP approach to sensor networks localization
    • Z. Wang, S. Zheng, Y. Ye, and S. Boyd, Further relaxations of the SDP approach to sensor networks localization, SIAM J. Optim. 19(2) (2008), pp. 655-673.
    • (2008) SIAM J. Optim. , vol.19 , Issue.2 , pp. 655-673
    • Wang, Z.1    Zheng, S.2    Ye, Y.3    Boyd, S.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.