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Volumn 45, Issue 2, 2002, Pages 169-184

Permanently going back and forth between the "quadratic world" and the "convexity world" in optimization

Author keywords

Joint positive definiteness; Quadratic functions; Range convexity

Indexed keywords

CONFORMAL MAPPING; FUNCTIONS; GEOMETRY; LINEAR EQUATIONS; MATRIX ALGEBRA; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

EID: 0141938582     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00245-001-0034-6     Document Type: Article
Times cited : (48)

References (26)
  • 1
    • 22844457303 scopus 로고    scopus 로고
    • Regular zeros of a quadratic mapping and local controllability of nonlinear systems
    • A. V. Arutyunov and V. N. Rozova, Regular zeros of a quadratic mapping and local controllability of nonlinear systems, Differential Equations, 35 (1999), 723-728.
    • (1999) Differential Equations , vol.35 , pp. 723-728
    • Arutyunov, A.V.1    Rozova, V.N.2
  • 2
    • 51249170448 scopus 로고
    • Problems of distance geometry and convex properties of quadratic maps
    • A. I. Barvinok, Problems of distance geometry and convex properties of quadratic maps, Discrete Comput. Geom., 13 (1995), 189-202.
    • (1995) Discrete Comput. Geom. , vol.13 , pp. 189-202
    • Barvinok, A.I.1
  • 3
    • 0141947819 scopus 로고    scopus 로고
    • Convexity, duality and optimization
    • Lecture Notes, University of Michigan
    • A. I. Barvinok, Convexity, Duality and Optimization, Lecture Notes, University of Michigan, 1998.
    • (1998)
    • Barvinok, A.I.1
  • 4
    • 24444481168 scopus 로고    scopus 로고
    • On convex properties of the quadratic image of the sphere
    • Preprint, University of Michigan
    • A. I. Barvinok, On convex properties of the quadratic image of the sphere, Preprint, University of Michigan, 1999.
    • (1999)
    • Barvinok, A.I.1
  • 5
    • 0004217842 scopus 로고    scopus 로고
    • Joint positiveness of matrices
    • Unpublished manuscript
    • H. F. Bohnenblust, Joint positiveness of matrices, Unpublished manuscript.
    • Bohnenblust, H.F.1
  • 6
    • 0010120775 scopus 로고
    • On the fields of values of a matrix
    • L. Brickman, On the fields of values of a matrix, Proc. Amer. Math. Soc., 12 (1961), 61-66.
    • (1961) Proc. Amer. Math. Soc. , vol.12 , pp. 61-66
    • Brickman, L.1
  • 7
    • 84966261323 scopus 로고
    • Linear systems of real quadratic forms
    • E. Calabi, Linear systems of real quadratic forms, Proc. Amer. Math. Soc., 15 (1964), 844-846.
    • (1964) Proc. Amer. Math. Soc. , vol.15 , pp. 844-846
    • Calabi, E.1
  • 8
    • 0141913419 scopus 로고    scopus 로고
    • A note on quadratic forms
    • X. Chen and Y. Yuan, A note on quadratic forms, Math. Programming, 86 (1999), 187-197.
    • (1999) Math. Programming , vol.86 , pp. 187-197
    • Chen, X.1    Yuan, Y.2
  • 10
    • 0037958862 scopus 로고
    • On the mapping of quadratic forms
    • L. L. Dines, On the mapping of quadratic forms, Bull. Amer. Math. Soc., 47 (1941), 494-498.
    • (1941) Bull. Amer. Math. Soc. , vol.47 , pp. 494-498
    • Dines, L.L.1
  • 11
    • 0001641947 scopus 로고
    • Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen
    • P. Finsler, Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen, Comment. Math. Helv. 9 (1936/37), 188-192.
    • (1936) Comment. Math. Helv. , pp. 188-192
    • Finsler, P.1
  • 12
    • 0004290782 scopus 로고
    • Springer-Verlag, 1958; Heidelberger Taschenbücker
    • W. Greub, Linear Algebra, 1st edn., Springer-Verlag, 1958; Heidelberger Taschenbücker, Bd. 179, 1976.
    • (1976) Linear Algebra, 1st Edn. , vol.179
    • Greub, W.1
  • 13
    • 0033181498 scopus 로고    scopus 로고
    • Two extensions to Finsler's recurring theorem
    • C. Hamburger, Two extensions to Finsler's recurring theorem, Appl. Math. Optim., 40 (1999), 183-190.
    • (1999) Appl. Math. Optim. , vol.40 , pp. 183-190
    • Hamburger, C.1
  • 15
  • 16
    • 0141947815 scopus 로고
    • A generalization of Finsler's theorem for quadratic inequalities and equalities
    • D. H. Jacobson, A generalization of Finsler's theorem for quadratic inequalities and equalities, Questiones Math., 1 (1976), 19-28.
    • (1976) Quaestiones Math. , vol.1 , pp. 19-28
    • Jacobson, D.H.1
  • 17
    • 0041856546 scopus 로고    scopus 로고
    • Eigenvalue-constrained faces
    • A. S. Lewis, Eigenvalue-constrained faces, Linear Algebra Appl., 269 (1998), 159-181.
    • (1998) Linear Algebra Appl. , vol.269 , pp. 159-181
    • Lewis, A.S.1
  • 18
    • 0141947812 scopus 로고    scopus 로고
    • Lagrange duality in nonconvex optimization theory and modifications of the Toeplitz-Hausdorff theorem
    • A. Matveev, Lagrange duality in nonconvex Optimization theory and modifications of the Toeplitz-Hausdorff theorem, St Petersburg Math. J., 7 (1996), 787-815.
    • (1996) St Petersburg Math. J. , vol.7 , pp. 787-815
    • Matveev, A.1
  • 19
    • 0141982395 scopus 로고    scopus 로고
    • On the convexity of the ranges of quadratic mappings
    • A. S. Matveev, On the convexity of the ranges of quadratic mappings, St Petersburg Math. J., 10 (1999), 343-372.
    • (1999) St Petersburg Math. J. , vol.10 , pp. 343-372
    • Matveev, A.S.1
  • 20
    • 0001253761 scopus 로고
    • Power distribution inequalities in optimization and robustness of uncertain systems
    • A. Megretsky and S. Treil, Power distribution inequalities in Optimization and robustness of uncertain systems, Math. Systems Estimation Control, 3 (1993), 301-319.
    • (1993) Math. Systems Estimation Control , vol.3 , pp. 301-319
    • Megretsky, A.1    Treil, S.2
  • 21
    • 0032259651 scopus 로고    scopus 로고
    • Convexity of quadratic transformations and its use in control and optimization
    • B. T. Polyak, Convexity of quadratic transformations and its use in Control and Optimization, J. Optim. Theory Appl., 99 (1998), 553-583.
    • (1998) J. Optim. Theory Appl. , vol.99 , pp. 553-583
    • Polyak, B.T.1
  • 22
    • 21744437382 scopus 로고    scopus 로고
    • Generalized numerical ranges, joint positive definitness and multiple eigenvalues
    • Y.T. Poon, Generalized numerical ranges, joint positive definitness and multiple eigenvalues, Proc. Amer. Math. Soc., 125 (1997), 1625-1634.
    • (1997) Proc. Amer. Math. Soc. , vol.125 , pp. 1625-1634
    • Poon, Y.T.1
  • 23
    • 0003582391 scopus 로고
    • Convexity and optimization in finite dimension I
    • Springer-Verlag, New York
    • J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimension I, Springer-Verlag, New York, 1970.
    • (1970)
    • Stoer, J.1    Witzgall, C.2
  • 24
    • 0001175776 scopus 로고
    • A recurring theorem about pairs of quadratic forms and extension: A survey
    • F. Uhlig, A recurring theorem about pairs of quadratic forms and extension: a survey, Linear Algebra Appl., 25 (1979), 219-237.
    • (1979) Linear Algebra Appl. , vol.25 , pp. 219-237
    • Uhlig, F.1
  • 25
    • 0026895462 scopus 로고
    • Nonconvex optimization problem: The infinite-horizon linear-quadratic control problem with quadratic constraints
    • V. A. Yakubovich, Nonconvex optimization problem: the infinite-horizon linear-quadratic control problem with quadratic constraints, Systems Control Lett., 19 (1992), 13-22.
    • (1992) Systems Control Lett. , vol.19 , pp. 13-22
    • Yakubovich, V.A.1
  • 26
    • 0002765829 scopus 로고
    • On a subproblem of trust region algorithms for constrained optimization
    • Y. Yuan, On a subproblem of trust region algorithms for constrained optimization, Math. Programming, 47 (1990), 53-63.
    • (1990) Math. Programming , vol.47 , pp. 53-63
    • Yuan, Y.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.