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Volumn 96, Issue 2, 2003, Pages 293-320

Semidefinite programming relaxations for semialgebraic problems

Author keywords

Convex optimization; Polynomial equations real algebraic geometry; Semidefinite programming; Sums of squares

Indexed keywords

ALGEBRA; GEOMETRY; NONLINEAR EQUATIONS; POLYNOMIALS; PROBLEM SOLVING;

EID: 1542266056     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10107-003-0387-5     Document Type: Article
Times cited : (1412)

References (55)
  • 2
    • 0001559061 scopus 로고
    • A quadratic form representation of polynomials of several variables and its applications
    • [BL68]
    • [BL68] Bose, N.K., Li, C.C.: A quadratic form representation of polynomials of several variables and its applications. IEEE Transactions on Automatic Control 14, 447-448 (1968)
    • (1968) IEEE Transactions on Automatic Control , vol.14 , pp. 447-448
    • Bose, N.K.1    Li, C.C.2
  • 4
    • 0030291482 scopus 로고    scopus 로고
    • On the combinatorial and algebraic complexity of quantifier elimination
    • [BPR96]
    • [BPR96] Basu, S., Pollack, R., Roy, M.-F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43(6), 1002-1045 (1996)
    • (1996) J. ACM , vol.43 , Issue.6 , pp. 1002-1045
    • Basu, S.1    Pollack, R.2    Roy, M.-F.3
  • 5
    • 0001259424 scopus 로고
    • Bounds for the degrees in the Nullstellensatz
    • [Bro87]
    • [Bro87] Brownawell, W.D.: Bounds for the degrees in the Nullstellensatz. Annals of Mathematics 126, 577-591 (1987)
    • (1987) Annals of Mathematics , vol.126 , pp. 577-591
    • Brownawell, W.D.1
  • 6
    • 0000892961 scopus 로고
    • Recent improvements in the complexity of the effective Nullstellensatz
    • [BS91]
    • [BS91] Berenstein, C.A., Struppa, D.C.: Recent improvements in the complexity of the effective Nullstellensatz. Linear Algebra and its Applications 157(2), 203-215 (1991)
    • (1991) Linear Algebra and its Applications , vol.157 , Issue.2 , pp. 203-215
    • Berenstein, C.A.1    Struppa, D.C.2
  • 11
    • 0002106815 scopus 로고
    • On non-negative forms in real variables some or all of which are non-negative
    • [Dia62]
    • [Dia62] Diananda, P.H.: On non-negative forms in real variables some or all of which are non-negative. Proceedings of the Cambridge Philosophical Society 58, 17-25 (1962)
    • (1962) Proceedings of the Cambridge Philosophical Society , vol.58 , pp. 17-25
    • Diananda, P.H.1
  • 12
    • 0036433723 scopus 로고    scopus 로고
    • Approximating the stability number of a graph via copositive programming
    • [dKP02]
    • [dKP02] de Klerk, E., Pasechnik, D.V.: Approximating the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875-892 (2002)
    • (2002) SIAM J. Optim. , vol.12 , Issue.4 , pp. 875-892
    • De Klerk, E.1    Pasechnik, D.V.2
  • 13
    • 0003053246 scopus 로고    scopus 로고
    • Hilbert's 17th problem and best dual bounds in quadratic minimization
    • [Fer98]
    • [Fer98] Ferrier, Ch.: Hilbert's 17th problem and best dual bounds in quadratic minimization. Cybernetics and Systems Analysis 34(5), 696-709 (1998)
    • (1998) Cybernetics and Systems Analysis , vol.34 , Issue.5 , pp. 696-709
    • Ferrier, Ch.1
  • 14
    • 0032186918 scopus 로고    scopus 로고
    • Comments on A procedure for the positive definiteness of forms of even order
    • [Fu98]
    • [Fu98] Fu, M.: Comments on A procedure for the positive definiteness of forms of even order. IEEE Transactions on Automatic Control 43(10), 1430 (1998)
    • (1998) IEEE Transactions on Automatic Control , vol.43 , Issue.10 , pp. 1430
    • Fu, M.1
  • 16
    • 0035960943 scopus 로고    scopus 로고
    • Complexity of Null- And positivstellensatz proofs
    • [GV02]
    • [GV02] Grigoriev, D., Vorobjov, N.: Complexity of Null- and Positivstellensatz proofs. Annals of Pure and Applied Logic 113(1-3), 153-160 (2002)
    • (2002) Annals of Pure and Applied Logic , vol.113 , Issue.1-3 , pp. 153-160
    • Grigoriev, D.1    Vorobjov, N.2
  • 17
    • 84893574327 scopus 로고
    • Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
    • [GW95]
    • [GW95] Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115-1145 (1995)
    • (1995) J. ACM , vol.42 , Issue.6 , pp. 1115-1145
    • Goemans, M.X.1    Williamson, D.P.2
  • 18
    • 0030127026 scopus 로고    scopus 로고
    • A procedure for the positive definiteness of forms of even order
    • [HH96]
    • [HH96] Hasan, M.A., Hasan, A.A.: A procedure for the positive definiteness of forms of even order. IEEE Transactions on Automatic Control 41(4), 615-617 (1996)
    • (1996) IEEE Transactions on Automatic Control , vol.41 , Issue.4 , pp. 615-617
    • Hasan, M.A.1    Hasan, A.A.2
  • 19
    • 0040962243 scopus 로고
    • The discriminant of the characteristic polynomial of a normal matrix
    • [Ily92]
    • [Ily92] Ilyushechkin, N.V.: The discriminant of the characteristic polynomial of a normal matrix. Math. Zameski 51, 16-23 (1992)
    • (1992) Math. Zameski , vol.51 , pp. 16-23
    • Ilyushechkin, N.V.1
  • 20
    • 0040368135 scopus 로고
    • ; Englis translation in Math. Notes 51, 230-235 (1992)
    • (1992) Math. Notes , vol.51 , pp. 230-235
  • 21
    • 84968474102 scopus 로고
    • Sharp effective nullstellensatz
    • [Kol88]
    • [Kol88] Kollár, J.: Sharp effective Nullstellensatz. J. Amer. Math. Soc. 1, 963-975 (1988)
    • (1988) J. Amer. Math. Soc. , vol.1 , pp. 963-975
    • Kollár, J.1
  • 22
    • 0035238864 scopus 로고    scopus 로고
    • Global optimization with polynomials and the problem of moments
    • [Las01]
    • [Las01] Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796-817 (2001)
    • (2001) SIAM J. Optim. , vol.11 , Issue.3 , pp. 796-817
    • Lasserre, J.B.1
  • 24
    • 0032332990 scopus 로고    scopus 로고
    • On the discriminant of real symmetric matrices
    • [Lax98]
    • [Lax98] Lax, P.D.: On the discriminant of real symmetric matrices. Communications on Pure and Applied Mathematics LI, 1387-1396 (1998)
    • (1998) Communications on Pure and Applied Mathematics , vol.LI , pp. 1387-1396
    • Lax, P.D.1
  • 25
    • 38149146720 scopus 로고
    • Stable sets and polynomials
    • [Lov94]
    • [Lov94] Lovász, L.: Stable sets and polynomials. Discrete mathematics 124, 137-153 (1994)
    • (1994) Discrete Mathematics , vol.124 , pp. 137-153
    • Lovász, L.1
  • 26
    • 0001154274 scopus 로고
    • Cones of matrices and set-functions and 0-1 optimization
    • [LS91]
    • [LS91] Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1(2), 166-190 (1991)
    • (1991) SIAM J. Optim. , vol.1 , Issue.2 , pp. 166-190
    • Lovász, L.1    Schrijver, A.2
  • 28
    • 0023452095 scopus 로고
    • Some NP-complete problems in quadratic and nonlinear programming
    • [MK87]
    • [MK87] Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Mathematical Programming 39, 117-129 (1987)
    • (1987) Mathematical Programming , vol.39 , pp. 117-129
    • Murty, K.G.1    Kabadi, S.N.2
  • 29
    • 85036767048 scopus 로고    scopus 로고
    • The use of symbolic methods in control system analysis and design
    • [Mun99] Munro, N.: editor. IEE Books
    • [Mun99] Munro, N.: editor. The Use of Symbolic Methods in Control System Analysis and Design, volume 56 of Control engineering series. IEE Books, 1999
    • (1999) Control Engineering Series , vol.56
  • 30
    • 0001776805 scopus 로고    scopus 로고
    • Squared functional systems and optimization problems
    • [Nes00]. J.B.G. Frenk, C. Roos, T. Terlaky, S. Zhang, editors. Kluwer Academic Publishers
    • [Nes00] Nesterov, Y.: Squared functional systems and optimization problems. In: J.B.G. Frenk, C. Roos, T. Terlaky, S. Zhang, editors, High Performance Optimization, pages 405-440. Kluwer Academic Publishers, 2000
    • (2000) High Performance Optimization , pp. 405-440
    • Nesterov, Y.1
  • 33
    • 1542296013 scopus 로고    scopus 로고
    • An explicit construction of distinguished representations of polynomials nonnegative over finite sets
    • [Par02a]
    • [Par02a] Parrilo, P.A.: An explicit construction of distinguished representations of polynomials nonnegative over finite sets. Technical Report IfA Technical Report AUT02-02. Available from http://www.aut.ee.ethz.ch/ ~parrilo, ETH Zürich, 2002
    • (2002) Technical Report IfA Technical Report , vol.AUT02-02
    • Parrilo, P.A.1
  • 36
    • 0001276275 scopus 로고    scopus 로고
    • A new bound for Pólya's theorem with applications to polynomials positive on polyhedra
    • [PR01]
    • [PR01] Powers, V., Reznick, B.: A new bound for Pólya's theorem with applications to polynomials positive on polyhedra. J. Pure Appl. Algebra 164(1-2), 221-229 (2001)
    • (2001) J. Pure Appl. Algebra , vol.164 , Issue.1-2 , pp. 221-229
    • Powers, V.1    Reznick, B.2
  • 38
    • 0032064831 scopus 로고    scopus 로고
    • An algorithm for sums of squares of real polynomials
    • [PW98]
    • [PW98] Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. Journal of pure and applied algebra 127, 99-104 (1998)
    • (1998) Journal of Pure and Applied Algebra , vol.127 , pp. 99-104
    • Powers, V.1    Wörmann, T.2
  • 41
    • 84972566433 scopus 로고
    • Extremal PSD forms with few terms
    • [Rez78]
    • [Rez78] Reznick, B.: Extremal PSD forms with few terms. Duke Mathematical Journal 45(2), 363-374 (1978)
    • (1978) Duke Mathematical Journal , vol.45 , Issue.2 , pp. 363-374
    • Reznick, B.1
  • 42
    • 51249169659 scopus 로고
    • Uniform denominators in Hilbert's seventeenth problem
    • [Rez95]
    • [Rez95] Reznick, B.: Uniform denominators in Hilbert's seventeenth problem. Math Z. 220, 75-97 (1995)
    • (1995) Math Z. , vol.220 , pp. 75-97
    • Reznick, B.1
  • 43
    • 0001157581 scopus 로고    scopus 로고
    • Some concrete aspects of Hilbert's 17th problem
    • [Rez00]. American Mathematical Society
    • [Rez00] Reznick, B.: Some concrete aspects of Hilbert's 17th problem. In: Contemporary Mathematics, volume 253, pages 251-272. American Mathematical Society, 2000
    • (2000) Contemporary Mathematics , vol.253 , pp. 251-272
    • Reznick, B.1
  • 44
    • 0001321687 scopus 로고
    • A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems
    • [SA90]
    • [SA90] Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Disc. Math. 3(3), 411-430 (1990)
    • (1990) SIAM J. Disc. Math. , vol.3 , Issue.3 , pp. 411-430
    • Sherali, H.D.1    Adams, W.P.2
  • 46
    • 0001250675 scopus 로고
    • The k-moment problem for compact semialgebraic sets
    • [Sch91]
    • [Sch91] Schmüdgen, K.: The k-moment problem for compact semialgebraic sets. Math. Ann. 289, 203-206 (1991)
    • (1991) Math. Ann. , vol.289 , pp. 203-206
    • Schmüdgen, K.1
  • 47
    • 0000604097 scopus 로고
    • Class of global minimum bounds of polynomial functions
    • [Sho87]
    • [Sho87] Shor, N.Z.: Class of global minimum bounds of polynomial functions. Cybernetics 23(6), 731-734 (1987);
    • (1987) Cybernetics , vol.23 , Issue.6 , pp. 731-734
    • Shor, N.Z.1
  • 48
    • 1842743052 scopus 로고
    • Russian orig.: Kibernetika, 6, 9-11 (1987)
    • (1987) Kibernetika , vol.6 , pp. 9-11
  • 50
    • 34250416099 scopus 로고
    • A nullstellensatz and a positivstellensatz in semialgebraic geometry
    • [Ste74]
    • [Ste74] Stengle, G.: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87-97(1974)
    • (1974) Math. Ann. , vol.207 , pp. 87-97
    • Stengle, G.1
  • 51
    • 0030162881 scopus 로고    scopus 로고
    • Complexity estimates for the schmüdgen positivstellensatz
    • [Ste96]
    • [Ste96] Stengle, G.: Complexity estimates for the Schmüdgen Positivstellensatz. J. Complexity 12(2), 167-174 (1996)
    • (1996) J. Complexity , vol.12 , Issue.2 , pp. 167-174
    • Stengle, G.1
  • 52
    • 0004943866 scopus 로고    scopus 로고
    • [Stu99] September
    • [Stu99] Sturm, J.: SeDuMi version 1.03. September 1999. Available from http://www.unim-aas.nl/~sturm/software/sedumi.html
    • (1999) SeDuMi Version 1.03
    • Sturm, J.1
  • 53
  • 54
    • 0030106462 scopus 로고    scopus 로고
    • Semidefinite programming
    • [VB96]
    • [VB96] Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49-95 (1996)
    • (1996) SIAM Review , vol.38 , Issue.1 , pp. 49-95
    • Vandenberghe, L.1    Boyd, S.2
  • 55
    • 0003395487 scopus 로고    scopus 로고
    • [WSV00] Wolkowicz, H., Saigal, R., Vandenberghe, L.: editors. Kluwer
    • [WSV00] Wolkowicz, H., Saigal, R., Vandenberghe, L.: editors. Handbook of Semidefinite Programming. Kluwer, 2000
    • (2000) Handbook of Semidefinite Programming


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.