메뉴 건너뛰기




Volumn 31, Issue 3, 2009, Pages 1432-1457

Calibrating least squares semidefinite programming with equality and inequality constraints

Author keywords

Covariance matrix; Quadratic convergence; Smoothing Newton method

Indexed keywords

APPROXIMATE SOLUTION; BI-CGSTAB; INEQUALITY CONSTRAINT; ITERATIVE SOLVERS; LEAST SQUARE; METRIC PROJECTION OPERATOR; NUMERICAL EXPERIMENTS; QUADRATIC CONVERGENCE; SEMI-DEFINITE PROGRAMMING; SEMISMOOTH; SEMISMOOTH EQUATIONS; SMOOTHING NEWTON METHODS;

EID: 84861325244     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/080727075     Document Type: Article
Times cited : (55)

References (52)
  • 1
    • 0345116314 scopus 로고    scopus 로고
    • Interest rate model calibration using semidefinite programming
    • A. d'Aspremont, Interest rate model calibration using semidefinite programming, Appl. Math. Finance, 10 (2003), pp. 183-213.
    • (2003) Appl. Math. Finance , vol.10 , pp. 183-213
    • D'aspremont, A.1
  • 2
    • 84861314023 scopus 로고    scopus 로고
    • Risk-management method for the Libor market model using semidefinite programming
    • A. d'Aspremont, Risk-management method for the Libor market model using semidefinite programming, J. Comput. Finance, 8 (2005), pp. 77-99.
    • (2005) J. Comput. Finance , vol.8 , pp. 77-99
    • D'aspremont, A.1
  • 3
    • 0031141136 scopus 로고    scopus 로고
    • Complementarity and nondegeneracy in semidefinite programming
    • PII S0025561096000792
    • F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton, Complementarity and nondegeneracy in semidefinite programming, Math. Program., 77 (1997), pp. 111-128. (Pubitemid 127412627)
    • (1997) Mathematical Programming, Series B , vol.77 , Issue.2 , pp. 111-128
    • Alizadeh, F.1    Haeberly, J.-P.A.2    Overton, M.L.3
  • 4
  • 5
    • 84927914894 scopus 로고
    • On matrices depending on parameters
    • V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp. 29-43.
    • (1971) Russian Math. Surveys , vol.26 , pp. 29-43
    • Arnold, V.I.1
  • 7
    • 34547344235 scopus 로고    scopus 로고
    • Implementation of a primal-dual method for SDP on a shared memory parallel architecture
    • DOI 10.1007/s10589-007-9030-3
    • B. Borchers and J. G. Young, Implementation of a primal-dual method for SDP on a shared memory parallel architecture, Comput. Optim. Appl., 37 (2007), pp. 355-369. (Pubitemid 47139046)
    • (2007) Computational Optimization and Applications , vol.37 , Issue.3 , pp. 355-369
    • Borchers, B.1    Young, J.G.2
  • 8
    • 85120507955 scopus 로고    scopus 로고
    • A preconditioned Newton algorithm for the nearest correlation matrix
    • to appear
    • R. Borsdorf and N. J. Higham, A preconditioned Newton algorithm for the nearest correlation matrix, IMA J. Numer. Anal., to appear.
    • IMA J. Numer. Anal.
    • Borsdorf, R.1    Higham, N.J.2
  • 9
    • 0026869613 scopus 로고
    • Partially finite convex programming, part II: Explicit lattice models
    • J. Borwein and A. S. Lewis, Partially finite convex programming, part II: Explicit lattice models, Math. Program., 57 (1992), pp. 49-83.
    • (1992) Math. Program. , vol.57 , pp. 49-83
    • Borwein, J.1    Lewis, A.S.2
  • 10
    • 33646249113 scopus 로고    scopus 로고
    • Least-squares covariance matrix adjustment
    • S. Boyd and L. Xiao, Least-squares covariance matrix adjustment, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 532-546.
    • (2005) SIAM J. Matrix Anal. Appl. , vol.27 , pp. 532-546
    • Boyd, S.1    Xiao, L.2
  • 11
    • 58049197110 scopus 로고    scopus 로고
    • Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming
    • Z. X. Chan and D. Sun, Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming, SIAM J. Optim., 19 (2008), pp. 370-396.
    • (2008) SIAM J. Optim. , vol.19 , pp. 370-396
    • Chan, Z.X.1    Sun, D.2
  • 12
    • 0030284902 scopus 로고    scopus 로고
    • 1 convex constrained optimization
    • X. Chen, Convergence of the BFGS method for LC1 convex constrained optimization, SIAM J. Control Optim., 34 (1996), pp. 2051-2063. (Pubitemid 126630028)
    • (1996) SIAM Journal on Control and Optimization , vol.34 , Issue.6 , pp. 2051-2063
    • Chen, X.1
  • 15
    • 8644260616 scopus 로고    scopus 로고
    • Best approximation in inner product spaces
    • Springer, New York
    • F. Deutsch, Best Approximation in Inner Product Spaces, CMS Books Math. Ouvrages Math. SMC 7, Springer, New York, 2001.
    • (2001) CMS Books Math. Ouvrages Math. SMC , vol.7
    • Deutsch, F.1
  • 16
    • 0001669049 scopus 로고    scopus 로고
    • A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space
    • DOI 10.1006/jath.1996.3082, PII S0021904596930828
    • F. Deutsch, W. Li, and J. D. Ward, A dual approach to constrained interpolation from a convex subset of Hilbert space, J. Approx. Theory, 90 (1997), pp. 385-414. (Pubitemid 127344252)
    • (1997) Journal of Approximation Theory , vol.90 , Issue.3 , pp. 385-414
    • Deutsch, F.1    Li, W.2    Ward, J.D.3
  • 17
    • 84963450943 scopus 로고
    • Duality and well-posedness in convex interpolation
    • A. L. Dontchev and B. D. Kalchev, Duality and well-posedness in convex interpolation, Numer. Funct. Anal. Optim., 10 (1989), pp. 673-689.
    • (1989) Numer. Funct. Anal. Optim. , vol.10 , pp. 673-689
    • Dontchev, A.L.1    Kalchev, B.D.2
  • 18
    • 0035595452 scopus 로고    scopus 로고
    • Convergence of Newton's method for convex best interpolation
    • A. L. Dontchev, H.-D. Qi, and L. Qi, Convergence of Newton's method for convex best interpolation, Numer. Math., 87 (2001), pp. 435-456.
    • (2001) Numer. Math. , vol.87 , pp. 435-456
    • Dontchev, A.L.1    Qi, H.-D.2    Qi, L.3
  • 19
    • 0013103572 scopus 로고    scopus 로고
    • Quadratic convergence of Newton's method for convex interpolation and smoothing
    • A. L. Dontchev, H.-D. Qi, and L. Qi, Quadratic convergence of Newton's method for convex interpolation and smoothing, Constr. Approx., 19 (2003), pp. 123-143.
    • (2003) Constr. Approx. , vol.19 , pp. 123-143
    • Dontchev, A.L.1    Qi, H.-D.2    Qi, L.3
  • 20
    • 84948499836 scopus 로고
    • An algorithm for restricted least squares regression
    • R. L. Dykstra, An algorithm for restricted least squares regression, J. Amer. Statist. Assoc., 78 (1983), pp. 837-842.
    • (1983) J. Amer. Statist. Assoc. , vol.78 , pp. 837-842
    • Dykstra, R.L.1
  • 21
    • 0001918140 scopus 로고
    • On the basic theorem for complementarity
    • B. C. Eaves, On the basic theorem for complementarity, Math. Program., 1 (1971), pp. 68-75.
    • (1971) Math. Program. , vol.1 , pp. 68-75
    • Eaves, B.C.1
  • 23
    • 0000554369 scopus 로고    scopus 로고
    • Solution of monotone complementarity problems with locally Lipschitzian functions
    • PII S0025561096000597
    • A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Program., 76 (1997), pp. 513-532. (Pubitemid 127380006)
    • (1997) Mathematical Programming, Series B , vol.76 , Issue.3 , pp. 513-532
    • Fischer, A.1
  • 25
    • 0344153904 scopus 로고    scopus 로고
    • Computing the nearest correlation matrix - A problem from finance
    • DOI 10.1093/imanum/22.3.329
    • N. J. Higham, Computing the nearest correlation matrix-a problem from finance, IMA J. Numer. Anal., 22 (2002), pp. 329-343. (Pubitemid 37445900)
    • (2002) IMA Journal of Numerical Analysis , vol.22 , Issue.3 , pp. 329-343
    • Higham, N.J.1
  • 26
    • 0001701112 scopus 로고
    • Constrained interpolation and smoothing
    • L. D. Irvine, S. P. Marin, and P. W. Smith, Constrained interpolation and smoothing, Constr. Approx., 2 (1986), pp. 129-151.
    • (1986) Constr. Approx. , vol.2 , pp. 129-151
    • Irvine, L.D.1    Marin, S.P.2    Smith, P.W.3
  • 27
    • 85153127913 scopus 로고
    • Newton's method for nondifferentiable functions
    • Akademie-Verlag, Berlin
    • B. Kummer, Newton's method for nondifferentiable functions, in Advances in Mathematical Optimization, Math. Res. 45, Akademie-Verlag, Berlin, 1988, pp. 114-125.
    • (1988) Advances in Mathematical Optimization, Math. Res. , vol.45 , pp. 114-125
    • Kummer, B.1
  • 28
    • 34250967722 scopus 로고
    • Über monotone matrixfunktionen
    • K. Löwner, Über monotone matrixfunktionen, Math. Z., 38 (1934), pp. 177-216.
    • (1934) Math. Z. , vol.38 , pp. 177-216
    • Löwner, K.1
  • 29
    • 14544268199 scopus 로고    scopus 로고
    • A dual approach to semidefinite least-squares problems
    • J. Malick, A dual approach to semidefinite least-squares problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 272-284.
    • (2004) SIAM J. Matrix Anal. Appl. , vol.26 , pp. 272-284
    • Malick, J.1
  • 31
    • 0001036636 scopus 로고
    • Smoothing and interpolation in a convex subset of a Hilbert space
    • C. A. Micchelli and F. I. Utreras, Smoothing and interpolation in a convex subset of a Hilbert space, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 728-746
    • (1988) SIAM J. Sci. Stat. Comput. , vol.9 , pp. 728-746
    • Micchelli, C.A.1    Utreras, F.I.2
  • 32
    • 55549093536 scopus 로고    scopus 로고
    • Some theoretical aspects on Newton's method for constrained best interpolation
    • Springer, New York
    • H.-D. Qi, Some theoretical aspects on Newton's method for constrained best interpolation, in Continuous Optimization: Current Trends and Modern Applications, Springer, New York, 2005, pp. 23-49.
    • (2005) Continuous Optimization: Current Trends and Modern Applications , pp. 23-49
    • Qi, H.-D.1
  • 33
    • 58049191300 scopus 로고    scopus 로고
    • Positive semidefinite matrix completions on chordal graphs and the constraint nondegeneracy in semidefinite programming
    • H.-D. Qi, Positive semidefinite matrix completions on chordal graphs and the constraint nondegeneracy in semidefinite programming, Linear Algebra Appl., 430 (2009), pp. 1151-1164.
    • (2009) Linear Algebra Appl. , vol.430 , pp. 1151-1164
    • Qi, H.-D.1
  • 34
    • 34247364678 scopus 로고    scopus 로고
    • A quadratically convergent newton method for computing the nearest correlation matrix
    • DOI 10.1137/050624509
    • H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 360-385. (Pubitemid 46639147)
    • (2006) SIAM Journal on Matrix Analysis and Applications , vol.28 , Issue.2 , pp. 360-385
    • Qi, H.1    Sun, D.2
  • 35
    • 77950517553 scopus 로고    scopus 로고
    • Correlation stress testing for value-at-risk: An unconstrained convex optimization approach
    • to appear
    • H.-D. Qi and D. F. Sun, Correlation stress testing for value-at-risk: An unconstrained convex optimization approach, Comput. Optim. Appl., to appear.
    • Comput. Optim. Appl.
    • Qi, H.-D.1    Sun, D.F.2
  • 36
    • 0000607524 scopus 로고
    • Convergence analysis of some algorithms for solving nonsmooth equations
    • L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
    • (1993) Math. Oper. Res. , vol.18 , pp. 227-244
    • Qi, L.1
  • 37
    • 0002287357 scopus 로고    scopus 로고
    • A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities
    • L. Qi, D. F. Sun, and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), pp. 1-35.
    • (2000) Math. Program. , vol.87 , pp. 1-35
    • Qi, L.1    Sun, D.F.2    Zhou, G.3
  • 38
    • 0027543961 scopus 로고
    • A nonsmooth version of Newton's method
    • L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), pp. 353-367.
    • (1993) Math. Program. , vol.58 , pp. 353-367
    • Qi, L.1    Sun, J.2
  • 39
    • 0035238733 scopus 로고    scopus 로고
    • Regularization of P0-functions in box variational inequality problems
    • G. Ravindran and M. S. Gowda, Regularization of P0-functions in box variational inequality problems, SIAM J. Optim., 11 (2001), pp. 748-760.
    • (2001) SIAM J. Optim. , vol.11 , pp. 748-760
    • Ravindran, G.1    Gowda, M.S.2
  • 40
    • 0020901152 scopus 로고
    • Local structure of feasible sets in nonlinear programming, Part II: Nondegeneracy
    • S. M. Robinson, Local structure of feasible sets in nonlinear programming, Part II: Nondegeneracy, Math. Program. Study, 22 (1984), pp. 217-230.
    • (1984) Math. Program. Study , vol.22 , pp. 217-230
    • Robinson, S.M.1
  • 41
    • 0004267646 scopus 로고
    • Princeton University Press, Princeton
    • R. T. Rockafellar, Convex Analyis, Princeton University Press, Princeton, 1970.
    • (1970) Convex Analyis
    • Rockafellar, R.T.1
  • 43
    • 35248812683 scopus 로고    scopus 로고
    • The most general methodology to create a valid correlation matrix for risk management and option pricing purposes
    • R. Rebonato and P. Jäckel, The most general methodology to create a valid correlation matrix for risk management and option pricing purposes, J. Risk, 2 (1999), pp. 17-27.
    • (1999) J. Risk , vol.2 , pp. 17-27
    • Rebonato, R.1    Jäckel, P.2
  • 44
    • 0003432569 scopus 로고
    • Smoothing an indefinite variance-covariance matrix
    • N. C. Schwertman and D. M. Allen, Smoothing an indefinite variance-covariance matrix, J. Stat. Comput. Simul., 9 (1979), pp. 183-194.
    • (1979) J. Stat. Comput. Simul. , vol.9 , pp. 183-194
    • Schwertman, N.C.1    Allen, D.M.2
  • 45
    • 0033296299 scopus 로고    scopus 로고
    • Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones
    • J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11/12 (1999), pp. 625-653.
    • (1999) Optim. Methods Softw. , vol.11-12 , pp. 625-653
    • Sturm, J.F.1
  • 46
    • 0036474487 scopus 로고    scopus 로고
    • Semismooth matrix valued functions
    • D. F. Sun and J. Sun, Semismooth matrix valued functions, Math. Oper. Res., 27 (2002), pp. 150-169.
    • (2002) Math. Oper. Res. , vol.27 , pp. 150-169
    • Sun, D.F.1    Sun, J.2
  • 47
    • 55549112664 scopus 로고    scopus 로고
    • Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems
    • K. C. Toh, R. H. Tütüncü, and M. J. Todd, Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems, Pacific J. Optim., 3 (2007), pp. 135-164.
    • (2007) Pacific J. Optim. , vol.3 , pp. 135-164
    • Toh, K.C.1    Tütüncü, R.H.2    Todd, M.J.3
  • 48
    • 3142743708 scopus 로고    scopus 로고
    • Solving semidefinite-quadratic-linear programs using SDPT3
    • DOI 10.1007/s10107-002-0347-5
    • R. H. Tütüncü, K. C. Toh, and M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3, Math. Program., 95 (2003), pp. 189-217. (Pubitemid 44757853)
    • (2003) Mathematical Programming, Series B , vol.95 , Issue.2 , pp. 189-217
    • Tutuncu, R.H.1    Toh, K.C.2    Todd, M.J.3
  • 49
    • 0000005482 scopus 로고
    • Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems
    • H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 631-644.
    • (1992) SIAM J. Sci. Stat. Comput. , vol.13 , pp. 631-644
    • Van Der Vorst, H.A.1
  • 51
    • 0002130882 scopus 로고
    • Projections on convex sets in Hilbert space and spectral theory i and II
    • E. H. Zarantonello, ed., Academic Press, New York
    • E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory I and II, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed., Academic Press, New York, 1971, pp. 237-424.
    • (1971) Contributions to Nonlinear Functional Analysis , pp. 237-424
    • Zarantonello, E.H.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.