Skip to main content
SpringerLink
Log in

Global infimum of strictly convex quadratic functions with bounded perturbations

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

The problem of minimizing \({\tilde f=f+p}\) over some convex subset of a Euclidean space is investigated, where f(x) = x T Ax + b T x is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function \({\tilde f}\) is strictly outer γ-convex for any γ > γ*, where γ* is determined by s and the smallest eigenvalue of A. As consequence, a γ*-local minimal solution of \({\tilde f}\) is its global minimal solution and the diameter of the set of global minimal solutions of \({\tilde f}\) is less than or equal to γ*. Especially, the distance between the global minimal solution of f and any global minimal solution of \({\tilde f}\) is less than or equal to γ*/2. This property is used to prove a roughly generalized support property of \({\tilde f}\) and some generalized optimality conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Al-Othman AK, Al-Sumait JS, Sykulski JK (2007) Application of pattern search method to power system valve-point economic dispatch. Int J Electr Power Energy Syst 29: 720–730

    Article  Google Scholar 

  • Canovas MJ, Hantoute A, Lopez M, Marco A (2008) Lipschitz behavior of convex semi-infinite optimization problems: a variational approach. J Global Optim 41: 1–13

    Article  MATH  MathSciNet  Google Scholar 

  • Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Academic Press, Boston

    MATH  Google Scholar 

  • Crouzeix M, Philippe B, Sadkane M (1994) The Davidson method. SIAM J Sci Comput 15: 62–76

    Article  MATH  MathSciNet  Google Scholar 

  • Cullum J (1978) The simultaneous computation of a few of the algebraically largest and smallest eigenvalues of a large, sparse, symmetric matrix. BIT 18: 265–275

    Article  MATH  MathSciNet  Google Scholar 

  • Danaraj RMS, Gajendran F (2005) Quadratic programming solution to emission and economic dispatch problem. IE(I) Journal-EL 86: 129–132

    Google Scholar 

  • Daniel JW (1973) Stability of the solution of definite quadratic programs. Math Program 5: 41–53

    Article  MATH  MathSciNet  Google Scholar 

  • Davidson ER (1975) The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J Comput Phys 17: 87–94

    Article  Google Scholar 

  • Guddat J, Röhmisch W, Schultz R (1992) Some applications of mathematical programming techniques on optimal power dispatch. Computing 49: 193–200

    Article  MATH  MathSciNet  Google Scholar 

  • Jansen B, Roos C, Terlaky T (1996) Interior point methods, a decade after Karmarkar—a survey, with application to the smallest eigenvalue problem. Stat Neerlandica 50: 146–170

    Article  MATH  MathSciNet  Google Scholar 

  • Klatte D (1979) On the lower semicontinuity of optimal sets in convex parametric optimization. Point-to-set maps and mathematical programming. Math Program Stud 10:104–109

    MATH  MathSciNet  Google Scholar 

  • Klatte D (1997) Lower semicontinuity of the minimum in parametric convex programs. J Optim Theory Appl 94:511–517

    Article  MATH  MathSciNet  Google Scholar 

  • Kummer B (1984) Stability of generalized equations and Kuhn-Tucker points of perturbed convex programs. System modelling and optimization (Copenhagen, 1983) Lecture Notes in Control and Information Sciences, vol. 59. Springer, Berlin, pp 213–218

  • Lanczos C (1950) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J Res Nat Bur Stand 45: 255–282

    MathSciNet  Google Scholar 

  • Lee GM, Tam NN, Yen ND (2005) On the optimal value function of a linearly perturbed quadratic program. J Global Optim 32: 119–134

    Article  MATH  MathSciNet  Google Scholar 

  • Lee GM, Tam NN, Yen ND (2006) Continuity of the solution map in quadratic programs under linear perturbations. J Optim Theory Appl 129: 415–423

    Article  MATH  MathSciNet  Google Scholar 

  • Ma EM, Zarowski Ch (1995) On lower bounds for the smallest eigenvalue of a Hermitian positive-definite matrix. IEEE Trans Inform Theory 41: 539–540

    Article  MATH  MathSciNet  Google Scholar 

  • Markowitz HM (1952) Portfolio selection. J Finance 7: 77–91

    Article  Google Scholar 

  • Markowitz HM (1959) Portfolio selection. Wiley, New York

    Google Scholar 

  • Mirnia K, Ghaffari-Hadigheh A (2007) Support set expansion sensitivity analysis in convex quadratic optimization. Optim Methods Softw 22: 601–616

    Article  MATH  MathSciNet  Google Scholar 

  • Phu HX (2003) Strictly and roughly convexlike functions. J Optim Theory Appl 117: 139–156

    Article  MATH  MathSciNet  Google Scholar 

  • Phu HX (2007a) Outer γ-convexity and inner γ-convexity of disturbed functions. Vietnam J Math 35: 107–119

    MATH  MathSciNet  Google Scholar 

  • Phu HX (2007b) Some properties of solution sets to nonconvex quadratic programming problems. Optimization 56: 369–383

    Article  MATH  MathSciNet  Google Scholar 

  • Phu HX (2008) Outer Γ-convexity in vector spaces. Numer Funct Anal Optim 29: 835–854

    Article  MATH  MathSciNet  Google Scholar 

  • Phu HX (2010) Minimizing convex functions with bounded perturbations. SIAM J Optim 20: 2709–2729

    Article  Google Scholar 

  • Phu HX, An PT (1999) Outer γ-convexity in normed linear spaces. Vietnam J Math 27: 323–334

    MATH  MathSciNet  Google Scholar 

  • Phu HX, Bock HG, Pickenhain S (2000) Rough stability of solutions to nonconvex optimization problems. In: Dockner EJ, Hartl RF, Luptačik M, Sorger G (eds) Optimization, dynamics, and economic analysis. Physica-Verlag, Heidelberg New York, pp 22–35

    Google Scholar 

  • Phu HX, Yen ND (2001) On the stability of solutions to quadratic programming problems. Math Program Ser A 89: 385–394

    Article  MATH  MathSciNet  Google Scholar 

  • Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Schultz R (1984) Subdifferentials of perturbed convex functions. 16th annual conference on mathematical optimization (Sellin, 1984) Seminarberichte, vol. 64, Humboldt University, Berlin, pp 106–113

  • Schultz R (1988) Estimates for Kuhn-Tucker points of perturbed convex programs. Optimization 19: 29–43

    Article  MATH  MathSciNet  Google Scholar 

  • Singer I (1981) Duality theorems for perturbed convex optimization. J Math Anal Appl 81: 437–452

    Article  MATH  MathSciNet  Google Scholar 

  • Sleijpen GL, van der Vorst HA (1996) A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J Matrix Anal Appl 17: 401–425

    Article  MATH  MathSciNet  Google Scholar 

  • Trudzik LI (1985) Perturbed convex programming in reflexive Banach spaces. Nonlinear Anal 9: 61–78

    Article  MATH  MathSciNet  Google Scholar 

  • van den Bosch PPJ, Lootsma FA (1987) Scheduling of power generation via large-scale nonlinear optimization. J Optim Theory Appl 55: 313–326

    Article  MATH  MathSciNet  Google Scholar 

  • Voss H (2001) A variant of the inverted Lanczos method. BIT 41: 1111–1120

    Article  MathSciNet  Google Scholar 

  • Walters DC, Sheble GB (1993) Genetic algorithm solution of economic dispatch with valve-point loading. IEEE Trans Power Syst 8: 1325–1332

    Article  Google Scholar 

  • Wood AJ, Wollenberg BF (1984) Power generation operation and control. Wiley, New York

    Google Scholar 

  • Zhu Y, Tomsovic K (2007) Optimal distribution power flow for systems with distributed energy resources. Int J Electr Power Energy Syst 29: 260–267

    Article  Google Scholar 

  • Zlobec S (1983) Characterizing an optimal input in perturbed convex programming. Math Program 25: 109–121

    Article  MATH  MathSciNet  Google Scholar 

  • Zlobec S, Gardner R, Ben-Israel A (1982) Regions of stability for arbitrarily perturbed convex programs. Mathematical programming with data perturbations, I (Washington, D.C., 1979) Lecture Notes in Pure and Applied Mathematics, vol. 73, Dekker, New York, pp 69–89

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, Hanoi, Vietnam

    Hoang Xuan Phu

  2. Faculty of Information Technology, Le Qui Don University, 100 Hoang Quoc Viet Road, Hanoi, Vietnam

    Vo Minh Pho

Authors
  1. Hoang Xuan Phu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vo Minh Pho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hoang Xuan Phu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Phu, H.X., Pho, V.M. Global infimum of strictly convex quadratic functions with bounded perturbations. Math Meth Oper Res 72, 327–345 (2010). https://doi.org/10.1007/s00186-010-0324-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-010-0324-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation