메뉴 건너뛰기




Volumn 137, Issue 1-2, 2013, Pages 453-476

NP-hardness of deciding convexity of quartic polynomials and related problems

Author keywords

68Q25 Analysis of algorithms problem complexity; 90C60 Abstract computational complexity for mathematical programming problems; Mathematics Subject Classification (2000): 90C25 Convex programming

Indexed keywords

CLASSIFICATION (OF INFORMATION); CONVEX OPTIMIZATION; POLYNOMIAL APPROXIMATION;

EID: 84878602784     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-011-0499-2     Document Type: Article
Times cited : (112)

References (43)
  • 4
    • 84866243985 scopus 로고    scopus 로고
    • A convex polynomial that is not sos-convex
    • doi: 10.1007/s10107-011-0457-z
    • Ahmadi, A.A.; Parrilo, P.A.: A convex polynomial that is not sos-convex. Math. Program. (2011) doi: 10.1007/s10107-011-0457-z
    • (2011) Math. Program.
    • Ahmadi, A.A.1    Parrilo, P.A.2
  • 5
    • 0001033538 scopus 로고
    • Quasi-concave programming
    • 138509 10.2307/1911819 0104.14302
    • Arrow K.J.; Enthoven A.C.: Quasi-concave programming. Econometrica 29(4), 779-800 (1961)
    • (1961) Econometrica , vol.29 , Issue.4 , pp. 779-800
    • Arrow, K.J.1    Enthoven, A.C.2
  • 8
    • 0034264701 scopus 로고    scopus 로고
    • A survey of computational complexity results in systems and control
    • 1834719 10.1016/S0005-1098(00)00050-9 0989.93006
    • Blondel V.D.; Tsitsiklis J.N.: A survey of computational complexity results in systems and control. Automatica 36(9), 1249-1274 (2000)
    • (2000) Automatica , vol.36 , Issue.9 , pp. 1249-1274
    • Blondel, V.D.1    Tsitsiklis, J.N.2
  • 11
    • 56549098854 scopus 로고    scopus 로고
    • Establishing convexity of polynomial Lyapunov functions and their sublevel sets
    • 2466603 10.1109/TAC.2008.2007857
    • Chesi G.; Hung Y.S.: Establishing convexity of polynomial Lyapunov functions and their sublevel sets. IEEE Trans. Automat. Control 53(10), 2431-2436 (2008)
    • (2008) IEEE Trans. Automat. Control , vol.53 , Issue.10 , pp. 2431-2436
    • Chesi, G.1    Hung, Y.S.2
  • 12
    • 0003143974 scopus 로고
    • Positive semidefinite biquadratic forms
    • 10.1016/0024-3795(75)90058-0 0336.15014
    • Choi M.D.: Positive semidefinite biquadratic forms. Linear Algebra Appl. 12, 95-100 (1975)
    • (1975) Linear Algebra Appl. , vol.12 , pp. 95-100
    • Choi, M.D.1
  • 13
    • 34250471395 scopus 로고
    • On pseudo-convex functions of nonnegative variables
    • 290801 10.1007/BF01584075 0227.90040
    • Cottle R.W.; Ferland J.A.: On pseudo-convex functions of nonnegative variables. Math. Program. 1(1), 95-101 (1971)
    • (1971) Math. Program. , vol.1 , Issue.1 , pp. 95-101
    • Cottle, R.W.1    Ferland, J.A.2
  • 14
    • 34250226591 scopus 로고
    • Criteria for quasiconvexity and pseudoconvexity: Relationships and comparisons
    • 657079 10.1007/BF01583788 0479.90067
    • Crouzeix J.P.; Ferland J.A.: Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons. Math. Program. 23(2), 193-205 (1982)
    • (1982) Math. Program. , vol.23 , Issue.2 , pp. 193-205
    • Crouzeix, J.P.1    Ferland, J.A.2
  • 16
    • 44349154854 scopus 로고    scopus 로고
    • The complexity of optimizing over a simplex, hypercube or sphere: A short survey
    • 2407041 10.1007/s10100-007-0052-9 1152.90607
    • de Klerk E.: The complexity of optimizing over a simplex, hypercube or sphere: a short survey. CEJOR Cent. Eur. J. Oper. Res. 16(2), 111-125 (2008)
    • (2008) CEJOR Cent. Eur. J. Oper. Res. , vol.16 , Issue.2 , pp. 111-125
    • De Klerk, E.1
  • 18
    • 0039574143 scopus 로고
    • Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions
    • 636024 10.1016/0024-3795(81)90007-0 0468.15008
    • Ferland J.A.: Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions. Linear Algebra Appl. 38, 51-63 (1981)
    • (1981) Linear Algebra Appl. , vol.38 , pp. 51-63
    • Ferland, J.A.1
  • 21
    • 2842612385 scopus 로고    scopus 로고
    • On the difficulty of deciding the convexity of polynomials over simplexes
    • 10.1142/S0218195996000150 0854.68106
    • Guo B.: On the difficulty of deciding the convexity of polynomials over simplexes. Int. J. Comput. Geometry Appl. 6(2), 227-229 (1996)
    • (1996) Int. J. Comput. Geometry Appl. , vol.6 , Issue.2 , pp. 227-229
    • Guo, B.1
  • 22
    • 0041345998 scopus 로고    scopus 로고
    • Classical deterministic complexity of Edmonds' problem and quantum entanglement
    • (electronic). ACM, New York
    • Gurvits, L.: Classical deterministic complexity of Edmonds' problem and quantum entanglement. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 10-19 (electronic). ACM, New York (2003)
    • (2003) Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing , pp. 10-19
    • Gurvits, L.1
  • 23
    • 68449087853 scopus 로고    scopus 로고
    • Semidefinite representation of convex sets
    • SER. A 2533752 10.1007/s10107-008-0240-y 1192.90143
    • Helton J.W.; Nie J.: Semidefinite representation of convex sets. Math. Program. 122(1(Ser. A)), 21-64 (2010)
    • (2010) Math. Program. , vol.122 , Issue.1 , pp. 21-64
    • Helton, J.W.1    Nie, J.2
  • 25
    • 70450219137 scopus 로고    scopus 로고
    • Convexity in semialgebraic geometry and polynomial optimization
    • 2496762 10.1137/080728214
    • Lasserre J.B.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4), 1995-2014 (2008)
    • (2008) SIAM J. Optim. , vol.19 , Issue.4 , pp. 1995-2014
    • Lasserre, J.B.1
  • 26
    • 50849098824 scopus 로고    scopus 로고
    • Representation of nonnegative convex polynomials
    • 2430795 10.1007/s00013-008-2687-8 1159.14033
    • Lasserre J.B.: Representation of nonnegative convex polynomials. Arch. Math. 91(2), 126-130 (2008)
    • (2008) Arch. Math. , vol.91 , Issue.2 , pp. 126-130
    • Lasserre, J.B.1
  • 27
    • 84860278873 scopus 로고    scopus 로고
    • Certificates of convexity for basic semi-algebraic sets
    • 2651473 10.1016/j.aml.2010.04.009 1192.90136
    • Lasserre J.B.: Certificates of convexity for basic semi-algebraic sets. Appl. Math. Lett. 23(8), 912-916 (2010)
    • (2010) Appl. Math. Lett. , vol.23 , Issue.8 , pp. 912-916
    • Lasserre, J.B.1
  • 28
    • 0003364572 scopus 로고
    • Testing and imposing monotonicity, convexity and quasiconvexity constraints
    • M.A. Fuss D.L. McFadden (eds) North-Holland Publishing Company New York
    • Lau L.J.: Testing and imposing monotonicity, convexity and quasiconvexity constraints. In: Fuss, M.A.; McFadden, D.L. (eds.) Production Economics: A Dual Approach to Theory and Applications, pp. 409-453. North-Holland Publishing Company, New York (1978)
    • (1978) Production Economics: A Dual Approach to Theory and Applications , pp. 409-453
    • Lau, L.J.1
  • 29
    • 73249147288 scopus 로고    scopus 로고
    • Biquadratic optimization over unit spheres and semidefinite programming relaxations
    • 2546345 10.1137/080729104 1221.90074
    • Ling C.; Nie J.; Qi L.; Ye Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20(3), 1286-1310 (2009)
    • (2009) SIAM J. Optim. , vol.20 , Issue.3 , pp. 1286-1310
    • Ling, C.1    Nie, J.2    Qi, L.3    Ye, Y.4
  • 33
    • 0016084160 scopus 로고
    • Second order conditions for pseudo-convex functions
    • 372132 10.1137/0127011 0254.26019
    • Mereau P.; Paquet J.G.: Second order conditions for pseudo-convex functions. SIAM J. Appl. Math. 27, 131-137 (1974)
    • (1974) SIAM J. Appl. Math. , vol.27 , pp. 131-137
    • Mereau, P.1    Paquet, J.G.2
  • 34
    • 0000230999 scopus 로고
    • Maxima for graphs and a new proof of a theorem of Turán
    • 175813 10.4153/CJM-1965-053-6 0129.39902
    • Motzkin T.S.; Straus E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533-540 (1965)
    • (1965) Can. J. Math. , vol.17 , pp. 533-540
    • Motzkin, T.S.1    Straus, E.G.2
  • 35
    • 0023452095 scopus 로고
    • Some NP-complete problems in quadratic and nonlinear programming
    • 916001 10.1007/BF02592948 0637.90078
    • Murty K.G.; Kabadi S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117-129 (1987)
    • (1987) Math. Program. , vol.39 , pp. 117-129
    • Murty, K.G.1    Kabadi, S.N.2
  • 38
    • 34249841086 scopus 로고
    • Open questions in complexity theory for numerical optimization
    • 1195031 10.1007/BF01581088 0784.90102
    • Pardalos P.M.; Vavasis S.A.: Open questions in complexity theory for numerical optimization. Math. Program. 57(2), 337-339 (1992)
    • (1992) Math. Program. , vol.57 , Issue.2 , pp. 337-339
    • Pardalos, P.M.1    Vavasis, S.A.2
  • 41
    • 0027606675 scopus 로고
    • Lagrange multipliers and optimality
    • 1220880 10.1137/1035044 0779.49024
    • Rockafellar R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183-238 (1993)
    • (1993) SIAM Rev. , vol.35 , pp. 183-238
    • Rockafellar, R.T.1
  • 42
    • 0001096069 scopus 로고
    • A new decision method for elementary algebra
    • 63994 10.2307/1969640 0056.01804
    • Seidenberg A.: A new decision method for elementary algebra. Ann. Math. (2) 60, 365-374 (1954)
    • (1954) Ann. Math. (2) , vol.60 , pp. 365-374
    • Seidenberg, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.