메뉴 건너뛰기




Volumn 45, Issue 1, 2009, Pages 39-61

Moments and sums of squares for polynomial optimization and related problems

Author keywords

Measures; Moments

Indexed keywords

CONVEX ENVELOPE; MEASURES; MOMENT PROBLEMS; MOMENTS; NON-NEGATIVE POLYNOMIALS; POLYNOMIAL OPTIMIZATION; POSITIVE POLYNOMIALS; RATIONAL FUNCTION; SEMIDEFINITE PROGRAMMING RELAXATIONS; SUMS OF SQUARES; SYSTEM OF POLYNOMIAL EQUATIONS;

EID: 68549094630     PISSN: 09255001     EISSN: 15732916     Source Type: Journal    
DOI: 10.1007/s10898-008-9394-7     Document Type: Article
Times cited : (34)

References (39)
  • 2
    • 0003292618 scopus 로고
    • Some questions in the theory of moments
    • American Mathematical Society, Providence, R.I.
    • Akhiezer, N.I., Krein, M.G.: Some questions in the theory of moments. Translation of Mathematical Monographs, vol. 2. American Mathematical Society, Providence, R.I. (1962)
    • (1962) Translation of Mathematical Monographs , vol.2
    • Akhiezer, N.I.1    Krein, M.G.2
  • 4
    • 0000913894 scopus 로고
    • The multidimensional moment problem and semi-groups
    • Landau, H.J. (ed.) Moments in Mathematics American Mathematical Society, Providence
    • Berg, C.: The multidimensional moment problem and semi-groups. In: Landau, H.J. (ed.) Moments in Mathematics. Proceedings of Symposia in Applied Mathematics, vol. 37, pp. 110-124. American Mathematical Society, Providence (1980)
    • (1980) Proceedings of Symposia in Applied Mathematics , vol.37 , pp. 110-124
    • Berg, C.1
  • 5
    • 33748567127 scopus 로고    scopus 로고
    • There are significantly more nonnegative polynomials than sums of squares
    • G. Blekherman 2006 There are significantly more nonnegative polynomials than sums of squares Israel J. Math. 153 355 380
    • (2006) Israel J. Math. , vol.153 , pp. 355-380
    • Blekherman, G.1
  • 6
    • 23044517911 scopus 로고    scopus 로고
    • The truncated complex K-moment problem
    • R.E. Curto L.A. Fialkow 2000 The truncated complex K-moment problem Trans. Am. Math. Soc. 352 2825 2855
    • (2000) Trans. Am. Math. Soc. , vol.352 , pp. 2825-2855
    • Curto, R.E.1    Fialkow, L.A.2
  • 7
    • 2542587541 scopus 로고    scopus 로고
    • GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi
    • DOI 10.1145/779359.779363
    • D. Henrion J.B. Lasserre 2003 GloptiPoly : global optimization over polynomials with Matlab and SeDuMi ACM Trans. Math. Softw. 29 165 194 (Pubitemid 40001449)
    • (2003) ACM Transactions on Mathematical Software , vol.29 , Issue.2 , pp. 165-194
    • Henrion, D.1    Lasserre, J.-B.2
  • 9
    • 33947537644 scopus 로고    scopus 로고
    • Detecting global optimality and extracting solutions in GloptiPoly
    • Henrion D., Garulli R. (eds.) Positive Polynomials in Control Springer Verlag
    • Henrion D., Lasserre J.B. (2005) Detecting global optimality and extracting solutions in GloptiPoly. In: Henrion D., Garulli R. (eds.) Positive Polynomials in Control. Lecture Notes on Control and Information Sciences, pp. 293-310. Springer Verlag
    • (2005) Lecture Notes on Control and Information Sciences , pp. 293-310
    • Henrion, D.1    Lasserre, J.B.2
  • 10
    • 68549115091 scopus 로고    scopus 로고
    • GloptiPoly 3: Moments, optimization and semidefinite programming
    • to appear
    • Henrion, D., Lasserre, J.B., Lofberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw. (to appear)
    • Optim. Methods Softw.
    • Henrion, D.1    Lasserre, J.B.2    Lofberg, J.3
  • 11
    • 0035531030 scopus 로고    scopus 로고
    • Distinguished representations of strictly positive polynomials
    • T. Jacobi A. Prestel 2001 Distinguished representations of strictly positive polynomials J. Reine Angew. Math. 532 223 235
    • (2001) J. Reine Angew. Math. , vol.532 , pp. 223-235
    • Jacobi, T.1    Prestel, A.2
  • 12
    • 29144520191 scopus 로고    scopus 로고
    • Global optimization of rational functions: A semidefinite programming approach
    • D. Jibetean E. de Klerk 2006 Global optimization of rational functions: a semidefinite programming approach Math. Prog. A 106 93 109
    • (2006) Math. Prog. A , vol.106 , pp. 93-109
    • Jibetean, D.1    De Klerk, E.2
  • 13
    • 0003372433 scopus 로고
    • Moments in mathematics
    • Landau, H.J. (ed.) American Mathematical Society, Providence
    • Landau, H.J. (ed.): Moments in Mathematics. In: Proceedings of Symposia in Applied Mathematics, vol. 37. American Mathematical Society, Providence (1980)
    • (1980) Proceedings of Symposia in Applied Mathematics , vol.37
  • 14
    • 48549086225 scopus 로고    scopus 로고
    • Computing uniform convex approximations for convex envelopes and convex hulls
    • R. Laraki J.B. Lasserre 2008 Computing uniform convex approximations for convex envelopes and convex hulls J. Conv. Anal. 15 635 654
    • (2008) J. Conv. Anal. , vol.15 , pp. 635-654
    • Laraki, R.1    Lasserre, J.B.2
  • 16
    • 0035238864 scopus 로고    scopus 로고
    • Global optimization with polynomials and the problem of moments
    • J.B. Lasserre 2001 Global optimization with polynomials and the problem of moments SIAM J. Optim. 11 796 817
    • (2001) SIAM J. Optim. , vol.11 , pp. 796-817
    • Lasserre, J.B.1
  • 17
    • 33646727433 scopus 로고    scopus 로고
    • S.o.s. approximations of a polynomial nonnegative on a real algebraic set
    • J.B. Lasserre 2005 S.o.s. approximations of a polynomial nonnegative on a real algebraic set SIAM J. Optim. 16 610 628
    • (2005) SIAM J. Optim. , vol.16 , pp. 610-628
    • Lasserre, J.B.1
  • 18
    • 34547492080 scopus 로고    scopus 로고
    • Convergent semidefinite relaxations in polynomial optimization with sparsity
    • J.B. Lasserre 2006 Convergent semidefinite relaxations in polynomial optimization with sparsity SIAM J. Optim. 17 822 843
    • (2006) SIAM J. Optim. , vol.17 , pp. 822-843
    • Lasserre, J.B.1
  • 19
    • 33747153560 scopus 로고    scopus 로고
    • A sum of squares approximation of nonnegative polynomials
    • Lasserre, J.B.: A sum of squares approximation of nonnegative polynomials. SIAM J. Optim. 16, 751-765 (2006).
    • (2006) SIAM J. Optim. , vol.16 , pp. 751-765
    • Lasserre, J.B.1
  • 20
    • 37349086611 scopus 로고    scopus 로고
    • Also in: SIAM Review 49, 651-669 (2007)
    • (2007) SIAM Review , vol.49 , pp. 651-669
  • 21
    • 34547470409 scopus 로고    scopus 로고
    • A Semidefinite programming approach to the generalized problem of moments
    • J.B. Lasserre 2008 A Semidefinite programming approach to the generalized problem of moments Math. Program. 112 65 92
    • (2008) Math. Program. , vol.112 , pp. 65-92
    • Lasserre, J.B.1
  • 22
    • 33847278560 scopus 로고    scopus 로고
    • SOS approximations of nonnegative polynomials via simple high degree perturbations
    • J.B. Lasserre T. Netzer 2007 SOS approximations of nonnegative polynomials via simple high degree perturbations Math. Zeitschrift. 256 99 112
    • (2007) Math. Zeitschrift. , vol.256 , pp. 99-112
    • Lasserre, J.B.1    Netzer, T.2
  • 23
    • 52949150070 scopus 로고    scopus 로고
    • Semidefinite characterization and computation of zero-dimensional real radical ideals
    • J.B. Lasserre M. Laurent P. Rostalski 2008 Semidefinite characterization and computation of zero-dimensional real radical ideals Found. Comp. Math. 8 607 647
    • (2008) Found. Comp. Math. , vol.8 , pp. 607-647
    • Lasserre, J.B.1    Laurent, M.2    Rostalski, P.3
  • 24
    • 26444549416 scopus 로고    scopus 로고
    • Revisiting two theorems of Curto and Fialkow on moment matrices
    • M. Laurent 2005 Revisiting two theorems of Curto and Fialkow on moment matrices Proc. Am. Math. Soc. 133 2965 2976
    • (2005) Proc. Am. Math. Soc. , vol.133 , pp. 2965-2976
    • Laurent, M.1
  • 25
    • 84972537501 scopus 로고
    • Exponentially bounded positive definite functions
    • P.H. Maserick C. Berg 1984 Exponentially bounded positive definite functions Ill. J. Math. 28 162 179
    • (1984) Ill. J. Math. , vol.28 , pp. 162-179
    • Maserick, P.H.1    Berg, C.2
  • 27
    • 3142688043 scopus 로고    scopus 로고
    • An independent benchmarking of SDP and SOCP solvers
    • H.D. Mittelmann 2003 An independent benchmarking of SDP and SOCP solvers Math. Program. 95 407 430
    • (2003) Math. Program. , vol.95 , pp. 407-430
    • Mittelmann, H.D.1
  • 28
    • 0000939331 scopus 로고
    • Quasi-analytic vectors
    • A.E. Nussbaum 1966 Quasi-analytic vectors Ark. Mat. 6 179 191
    • (1966) Ark. Mat. , vol.6 , pp. 179-191
    • Nussbaum, A.E.1
  • 30
    • 1542266056 scopus 로고    scopus 로고
    • Semidefinite programming relaxations for semialgebraic problems
    • P.A. Parrilo 2003 Semidefinite programming relaxations for semialgebraic problems Math. Program. 96 293 320
    • (2003) Math. Program. , vol.96 , pp. 293-320
    • Parrilo, P.A.1
  • 31
    • 0013043479 scopus 로고
    • On the relation between the multidimensional moment problem and the one dimensional moment problem
    • L.C. Petersen 1982 On the relation between the multidimensional moment problem and the one dimensional moment problem Math. Scand. 51 361 366
    • (1982) Math. Scand. , vol.51 , pp. 361-366
    • Petersen, L.C.1
  • 32
    • 0000135301 scopus 로고
    • Positive polynomials on compact semi-algebraic sets
    • M. Putinar 1993 Positive polynomials on compact semi-algebraic sets Indiana Univ. Math. J. 42 969 984
    • (1993) Indiana Univ. Math. J. , vol.42 , pp. 969-984
    • Putinar, M.1
  • 33
    • 34548058483 scopus 로고    scopus 로고
    • Solving nonlinear polynomial system via symbolic-numeric elimination method
    • Faugère, J.C., Rouillier, F. (eds.)
    • Reid, G., Zhi, L.: Solving nonlinear polynomial system via symbolic-numeric elimination method. In: Faugère, J.C., Rouillier, F. (eds.) Proceedings of the International Conference on Polynomial System Solving, pp. 50-53 (2004)
    • (2004) Proceedings of the International Conference on Polynomial System Solving , pp. 50-53
    • Reid, G.1    Zhi, L.2
  • 34
    • 0033131339 scopus 로고    scopus 로고
    • Solving zero-dimensional systems through the rational univariate representation
    • F. Rouillier 1999 Solving zero-dimensional systems through the rational univariate representation J. Appl. Alg. Eng. Comm. Comp. 9 433 461
    • (1999) J. Appl. Alg. Eng. Comm. Comp. , vol.9 , pp. 433-461
    • Rouillier, F.1
  • 35
    • 0001250675 scopus 로고
    • The K-moment problem for compact semi-algebraic sets
    • K. Schmüdgen 1991 The K-moment problem for compact semi-algebraic sets Math. Ann. 289 203 206
    • (1991) Math. Ann. , vol.289 , pp. 203-206
    • Schmüdgen, K.1
  • 37
    • 0033296299 scopus 로고    scopus 로고
    • Using SeDuMi 1.02, a Matlab Toolbox for optimization over symmetric cones
    • J.F. Sturm 1999 Using SeDuMi 1.02, a Matlab Toolbox for optimization over symmetric cones Optim. Meth. Softw. 11-12 625 653
    • (1999) Optim. Meth. Softw. , vol.1112 , pp. 625-653
    • Sturm, J.F.1
  • 38
    • 0001656792 scopus 로고    scopus 로고
    • Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
    • J. Verschelde 1999 Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation ACM Trans. Math. Softw. 25 251 276
    • (1999) ACM Trans. Math. Softw. , vol.25 , pp. 251-276
    • Verschelde, J.1
  • 39
    • 33847158335 scopus 로고    scopus 로고
    • Sums of squares and semidefinite programming relaxations for polynomial optimization problems witth structured sparsity
    • S. Waki S. Kim M. Kojima M. Maramatsu 2006 Sums of squares and semidefinite programming relaxations for polynomial optimization problems witth structured sparsity SIAM J. Optim. 17 218 242
    • (2006) SIAM J. Optim. , vol.17 , pp. 218-242
    • Waki, S.1    Kim, S.2    Kojima, M.3    Maramatsu, M.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.