Advertisement

On Extensions of the Frank-Wolfe Theorems

  • Zhi-Quan Luo
  • Shuzhong Zhang
Article

Abstract

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.

convex quadratic system existence of optimal solutions quadratically constrained quadratic programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Auslender, “How to deal with the unbounded in optimization: Theory and algorithms,” Mathematical Programming Series B, vol. 79, pp. 3-8, 1997.Google Scholar
  2. 2.
    M. Avriel, W.E. Diewert, S. Schaible, and I. Zang, Generalized Concavity, Plenum Press: New York, 1988.Google Scholar
  3. 3.
    E. Blum and W. Oettli, “Direct proof of the existence theorem in quadratic programming,” Operations Research, vol. 20, pp. 165-167, 1972.Google Scholar
  4. 4.
    V. Chvátal, Linear Programming, W.H. Freeman and Company: New York, 1983.Google Scholar
  5. 5.
    B.C. Eaves, “On quadratic programming,” Management Science, vol. 17, pp. 698-711, 1971.Google Scholar
  6. 6.
    C.A. Floudas and V. Visweswaran, “Quadratic optimization,” in Handbook of Global Optimization, R. Horn and P.M. Pardalos (Eds.), Kluwer Academic Publishers: The Netherlands, 1995, pp. 217-269.Google Scholar
  7. 7.
    M. Frank and P. Wolfe, “An algorithm for quadratic programming,” Naval Research Logistics Quarterly, vol. 3, pp. 95-110, 1956.Google Scholar
  8. 8.
    A.J. Hoffman, “On approximate solutions of systems of linear inequalities,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 263-265, 1952.Google Scholar
  9. 9.
    Z.-Q. Luo, “On the solution set continuity of the convex quadratic feasibility problem,” unpublished manuscript, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada, September 1995.Google Scholar
  10. 10.
    Z.-Q. Luo, J.F. Sturm, and S. Zhang, “Duality results for conic convex programming,” Report 9719/A, Econometric Institute, Erasmus University Rotterdam, 1997.Google Scholar
  11. 11.
    Z.-Q. Luo and P.Y. Tseng, “Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem,” SIAM J. Optimization, vol. 2, pp. 43-54, 1992.Google Scholar
  12. 12.
    O.L. Mangasarian and L.L. Schumaker, “Discrete splines via mathematical programming,” SIAM Journal on Control, vol. 9, pp. 174-183, 1971.Google Scholar
  13. 13.
    T. Matsui, “NP-hardness of linear multiplicative programming and related problems,” Report METR95-13, University of Tokyo, Japan, 1995.Google Scholar
  14. 14.
    A.F. Perold, “A generalization of the Frank-Wolfe theorem,” Mathematical Programming, vol. 18, pp. 215-227, 1980.Google Scholar
  15. 15.
    S.M. Robinson, “Some continuity properties of polyhedral multifunctions,” Mathematical Programming Study, vol. 14, pp. 206-214, 1981.Google Scholar
  16. 16.
    T. Terlaky, “On lp programming,” European Journal of Operations Research, vol. 22, pp. 70-100, 1985.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Zhi-Quan Luo
    • 1
  • Shuzhong Zhang
    • 2
  1. 1.Communication Research Lab, Department of Electrical and Computer EngineeringMcMaster UniversityHamiltonCanada
  2. 2.Econometric InstituteErasmus University RotterdamThe Netherlands

Personalised recommendations