메뉴 건너뛰기




Volumn , Issue , 2011, Pages 241-248

The minimum-rank gram matrix completion via modified fixed point continuation method

Author keywords

fixed point iterative method; gram matrix completion; nuclear norm minimization; schur decomposition; sum of squares

Indexed keywords

FIXED POINT ITERATIVE METHOD; GRAM MATRICES; NUCLEAR NORM MINIMIZATION; SCHUR DECOMPOSITION; SUM OF SQUARES;

EID: 79959639348     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1993886.1993924     Document Type: Conference Paper
Times cited : (8)

References (53)
  • 1
    • 0041629678 scopus 로고
    • The real positive definite completion problem for a simple cycle
    • BARRETT, W., JOHNSON, C. R., AND TARAZAGA, P. The real positive definite completion problem for a simple cycle. Linear Algebra Appl. 192 (1993), 3-31.
    • (1993) Linear Algebra Appl , vol.192 , pp. 3-31
    • Barrett, W.1    Johnson, C.R.2    Tarazaga, P.3
  • 2
    • 0001531895 scopus 로고
    • Two-point step size gradient methods
    • BARZILAI, J., AND BORWEIN, J. Two-point step size gradient methods. IMA J. Numer. Anal. 8 (1988), 141-148.
    • (1988) IMA J. Numer. Anal. , vol.8 , pp. 141-148
    • Barzilai, J.1    Borwein, J.2
  • 3
    • 85014561619 scopus 로고    scopus 로고
    • A fast iterative shrinkage-thresholding algorithm for linear inverse problems
    • BECK, A., AND TEBOULLE, M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 1 (2009), 183-202.
    • (2009) SIAM J. Imaging Sci. , vol.2 , Issue.1 , pp. 183-202
    • Beck, A.1    Teboulle, M.2
  • 4
    • 1542337156 scopus 로고    scopus 로고
    • A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
    • BURER, S., AND MONTEIRO, R. A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95, 2 (2003), 329-357.
    • (2003) Math. Program. , vol.95 , Issue.2 , pp. 329-357
    • Burer, S.1    Monteiro, R.2
  • 5
    • 21644465788 scopus 로고    scopus 로고
    • Local minima and convergence in low-rank semidefinite programming
    • BURER, S., AND MONTEIRO, R. Local minima and convergence in low-rank semidefinite programming. Math. Program. 103, 3 (2005), 427-444.
    • (2005) Math. Program. , vol.103 , Issue.3 , pp. 427-444
    • Burer, S.1    Monteiro, R.2
  • 6
    • 77951291046 scopus 로고    scopus 로고
    • A singular value thresholding algorithm for matrix completion
    • CAI, J.-F., CANDÈS, E. J., AND SHEN, Z. A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 4 (2010), 1956-1982.
    • (2010) SIAM J. Optim. , vol.20 , Issue.4 , pp. 1956-1982
    • Cai, J.-F.1    Candès, E.J.2    Shen, Z.3
  • 7
    • 71049116435 scopus 로고    scopus 로고
    • Exact matrix completion via convex optimization
    • CANDÈS, E. J., AND RECHT, B. Exact matrix completion via convex optimization. Found. Comput. Math. 9, 6 (2009), 717-772.
    • (2009) Found. Comput. Math. , vol.9 , Issue.6 , pp. 717-772
    • Candès, E.J.1    Recht, B.2
  • 8
    • 77951528523 scopus 로고    scopus 로고
    • The power of convex relaxation: Near-optimal matrix completion
    • CANDÈS, E. J., AND TAO, T. The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inform. Theory 56, 5 (2010), 2053-2080.
    • (2010) IEEE Trans. Inform. Theory , vol.56 , Issue.5 , pp. 2053-2080
    • Candès, E.J.1    Tao, T.2
  • 10
    • 0000082954 scopus 로고
    • Sums of squares of real polynomials
    • CHOI, M., LAM, T., AND REZNICK, B. Sums of squares of real polynomials. Symp. in Pure Math. 58, 2 (1995), 103-126.
    • (1995) Symp. in Pure Math. , vol.58 , Issue.2 , pp. 103-126
    • Choi, M.1    Lam, T.2    Reznick, B.3
  • 11
    • 0000921195 scopus 로고    scopus 로고
    • Initialization in semidefinite programming via a self-dual skew-symmetric embedding
    • DE KLERK, E., ROOS, C., AND TERLAKY, T. Initialization in semidefinite programming via a self-dual skew-symmetric embedding. Oper. Res. Lett. 20, 5 (1997), 213-221.
    • (1997) Oper. Res. Lett. , vol.20 , Issue.5 , pp. 213-221
    • De Klerk, E.1    Roos, C.2    Terlaky, T.3
  • 14
    • 79952483985 scopus 로고    scopus 로고
    • Convergence of fixed point continuation algorithms for matrix rank minimization
    • GOLDFARB, D., AND MA, S. Convergence of fixed point continuation algorithms for matrix rank minimization. Foundations of Computational Mathematics 11, 2 (2011), 183-210.
    • (2011) Foundations of Computational Mathematics , vol.11 , Issue.2 , pp. 183-210
    • Goldfarb, D.1    Ma, S.2
  • 15
    • 0004236492 scopus 로고    scopus 로고
    • third ed. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD
    • GOLUB, G. H., AND VAN LOAN, C. F. Matrix computations, third ed. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 1996.
    • (1996) Matrix Computations
    • Golub, G.H.1    Van Loan, C.F.2
  • 16
    • 47849109874 scopus 로고
    • Positive definite completions of partial Hermitian matrices
    • GRONE, R., JOHNSON, C. R., DE SÁ, E. M., AND WOLKOWICZ, H. Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58 (1984), 109-124.
    • (1984) Linear Algebra Appl , vol.58 , pp. 109-124
    • Grone, R.1    Johnson, C.R.2    De Sá, E.M.3    Wolkowicz, H.4
  • 17
    • 69649095451 scopus 로고    scopus 로고
    • Fixed-point continuation for l1-minimization: Methodology and convergence
    • HALE, E. T., YIN, W., AND ZHANG, Y. Fixed-point continuation for l1-minimization: methodology and convergence. SIAM J. Optim. 19, 3 (2008), 1107-1130.
    • (2008) SIAM J. Optim. , vol.19 , Issue.3 , pp. 1107-1130
    • Hale, E.T.1    Yin, W.2    Zhang, Y.3
  • 19
    • 77957300831 scopus 로고    scopus 로고
    • Projection methods for conic feasibility problems; application to sum-of-squares decompositions
    • HENRION, D., AND MALICK, J. Projection methods for conic feasibility problems; application to sum-of-squares decompositions. Optimization Methods and Software 26, 1 (2011), 23-46.
    • (2011) Optimization Methods and Software , vol.26 , Issue.1 , pp. 23-46
    • Henrion, D.1    Malick, J.2
  • 22
    • 77956241799 scopus 로고    scopus 로고
    • Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients, 2009
    • Accepted for publication in
    • KALTOFEN, E., LI, B., YANG, Z., AND ZHI, L. Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients, 2009. Accepted for publication in J. Symbolic Comput.
    • J. Symbolic Comput.
    • Kaltofen, E.1    Li, B.2    Yang, Z.3    Zhi, L.4
  • 23
    • 33846697416 scopus 로고    scopus 로고
    • On the solution of large-scale SDP problems by the modified barrier method using iterative solvers
    • Ser. B
    • KOČVARA, M., AND STINGL, M. On the solution of large-scale SDP problems by the modified barrier method using iterative solvers. Math. Program. 109, 2-3, Ser. B (2007), 413-444.
    • (2007) Math. Program. , vol.109 , Issue.2-3 , pp. 413-444
    • Kočvara, M.1    Stingl, M.2
  • 25
    • 0039506352 scopus 로고    scopus 로고
    • The real positive semidefinite completion problem for series-parallel graphs
    • LAURENT, M. The real positive semidefinite completion problem for series-parallel graphs. Linear Algebra Appl. 252 (1997), 347-366.
    • (1997) Linear Algebra Appl , vol.252 , pp. 347-366
    • Laurent, M.1
  • 26
    • 0035219189 scopus 로고    scopus 로고
    • Polynomial instances of the positive semidefinite and Euclidean distance matrix completion problems
    • LAURENT, M. Polynomial instances of the positive semidefinite and Euclidean distance matrix completion problems. SIAM J. Matrix Anal. Appl. 22, 3 (2000), 874-894.
    • (2000) SIAM J. Matrix Anal. Appl. , vol.22 , Issue.3 , pp. 874-894
    • Laurent, M.1
  • 28
    • 0030304775 scopus 로고    scopus 로고
    • Convex analysis on the Hermitian matrices
    • LEWIS, A. S. Convex analysis on the Hermitian matrices. SIAM J. Optim. 6, 1 (1996), 164-177.
    • (1996) SIAM J. Optim. , vol.6 , Issue.1 , pp. 164-177
    • Lewis, A.S.1
  • 29
    • 72549110327 scopus 로고    scopus 로고
    • Interior-point method for nuclear norm approximation with application to system identification
    • LIU, Z., AND VANDENBERGHE, L. Interior-point method for nuclear norm approximation with application to system identification. SIAM J. Matrix Anal. Appl. 31 (2009), 1235-1256.
    • (2009) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 1235-1256
    • Liu, Z.1    Vandenberghe, L.2
  • 31
    • 79957968988 scopus 로고    scopus 로고
    • Fixed point and bregman iterative methods for matrix rank minimization
    • MA, S., GOLDFARB, D., AND CHEN, L. Fixed point and bregman iterative methods for matrix rank minimization. Math. Program. (2009), 1-33.
    • (2009) Math. Program. , pp. 1-33
    • Ma, S.1    Goldfarb, D.2    Chen, L.3
  • 32
    • 79959664874 scopus 로고    scopus 로고
    • The minimum-rank Gram matrix completion via fixed point continuation method
    • in Chinese
    • MA, Y. The minimum-rank Gram matrix completion via fixed point continuation method (in Chinese). Journal of Systems Science and Mathematical Sciences 30, 11 (2010), 1501-1511.
    • (2010) Journal of Systems Science and Mathematical Sciences , vol.30 , Issue.11 , pp. 1501-1511
    • Ma, Y.1
  • 33
    • 70450212537 scopus 로고    scopus 로고
    • Regularization methods for semidefinite programming
    • MALICK, J., POVH, J., RENDL, F., AND WIEGELE, A. Regularization methods for semidefinite programming. SIAM J. Optim. 20, 1 (2009), 336-356.
    • (2009) SIAM J. Optim. , vol.20 , Issue.1 , pp. 336-356
    • Malick, J.1    Povh, J.2    Rendl, F.3    Wiegele, A.4
  • 35
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of non-smooth functions
    • NESTEROV, Y. Smooth minimization of non-smooth functions. Math. Program. 103, 1 (2005), 127-152.
    • (2005) Math. Program. , vol.103 , Issue.1 , pp. 127-152
    • Nesterov, Y.1
  • 38
    • 36448998081 scopus 로고    scopus 로고
    • A Macaulay 2 package for computing sum of squares decompositions of polynomials with rational coefficients
    • PEYRL, H., AND PARRILO, P. A Macaulay 2 package for computing sum of squares decompositions of polynomials with rational coefficients. In ProcSNC07 (2007), pp. 207-208.
    • ProcSNC07 (2007) , pp. 207-208
    • Peyrl, H.1    Parrilo, P.2
  • 39
    • 55749106047 scopus 로고    scopus 로고
    • Computing sum of squares decompositions with rational coefficients
    • PEYRL, H., AND PARRILO, P. Computing sum of squares decompositions with rational coefficients. Theoretical Computer Science 409 (2008), 269-281.
    • (2008) Theoretical Computer Science , vol.409 , pp. 269-281
    • Peyrl, H.1    Parrilo, P.2
  • 40
    • 0000668151 scopus 로고
    • Zur Darstellung definiter Funktionen als Summe von Quadraten
    • PFISTER, A. Zur Darstellung definiter Funktionen als Summe von Quadraten. Inventiones Math. 4, 4 (1967), 229-236.
    • (1967) Inventiones Math. , vol.4 , Issue.4 , pp. 229-236
    • Pfister, A.1
  • 41
    • 0002241304 scopus 로고
    • Sur la représentation en somme de carrés des polynômes à une indéterminée sur un corps de nombres algébriques
    • POURCHET, Y. Sur la représentation en somme de carrés des polynômes à une indéterminée sur un corps de nombres algébriques. Acta Arith. 19 (1971), 89-104.
    • (1971) Acta Arith , vol.19 , pp. 89-104
    • Pourchet, Y.1
  • 42
    • 0032064831 scopus 로고    scopus 로고
    • An algorithm for sums of squares of real polynomials
    • POWERS, V., AND WÖRMANN, T. An algorithm for sums of squares of real polynomials. J. Pure Appl. Algebra 127, 1 (1998), 99-104.
    • (1998) J. Pure Appl. Algebra , vol.127 , Issue.1 , pp. 99-104
    • Powers, V.1    Wörmann, T.2
  • 43
    • 78549288866 scopus 로고    scopus 로고
    • Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization
    • RECHT, B., FAZEL, M., AND PARRILO, P. A. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 3 (2010), 471-501.
    • (2010) SIAM Rev. , vol.52 , Issue.3 , pp. 471-501
    • Recht, B.1    Fazel, M.2    Parrilo, P.A.3
  • 44
    • 62949189702 scopus 로고    scopus 로고
    • Necessary and sufficient conditions for success of the nuclear norm heuristic for rank minimization
    • RECHT, B., XU, W., AND HASSIBI, B. Necessary and sufficient conditions for success of the nuclear norm heuristic for rank minimization. In CDC (2008), pp. 3065-3070.
    • CDC (2008) , pp. 3065-3070
    • Recht, B.1    Xu, W.2    Hassibi, B.3
  • 48
    • 0033296299 scopus 로고    scopus 로고
    • Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones
    • STURM, J. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11/12 (1999), 625-653.
    • (1999) Optimization Methods and Software , vol.11-12 , pp. 625-653
    • Sturm, J.1
  • 50
    • 70049111607 scopus 로고    scopus 로고
    • On accelerated proximal gradient methods for convex-concave optimization
    • TSENG, P. On accelerated proximal gradient methods for convex-concave optimization. Submitted to SIAM J. Optim (2008).
    • (2008) Submitted to SIAM J. Optim
    • Tseng, P.1
  • 51
    • 0011336092 scopus 로고
    • Characterization of the subdifferential of some matrix norms
    • WATSON, G. A. Characterization of the subdifferential of some matrix norms. Linear Algebra Appl. 170 (1992), 33-45.
    • (1992) Linear Algebra Appl. , vol.170 , pp. 33-45
    • Watson, G.A.1
  • 52
    • 77955666600 scopus 로고    scopus 로고
    • A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization, and continuation
    • WEN, Z., YIN, W., GOLDFARB, D., AND ZHANG, Y. A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization, and continuation. SIAM J. Sci. Comput. 32, 4 (2010), 1832-1857.
    • (2010) SIAM J. Sci. Comput. , vol.32 , Issue.4 , pp. 1832-1857
    • Wen, Z.1    Yin, W.2    Goldfarb, D.3    Zhang, Y.4
  • 53
    • 77951274410 scopus 로고    scopus 로고
    • A Newton-CG augmented Lagrangian method for semidefinite programming
    • ZHAO, X.-Y., SUN, D., AND TOH, K.-C. A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 4 (2010), 1737-1765.
    • (2010) SIAM J. Optim. , vol.20 , Issue.4 , pp. 1737-1765
    • Zhao, X.-Y.1    Sun, D.2    Toh, K.-C.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.