Mathematical Programming

, Volume 135, Issue 1–2, pp 195–220 | Cite as

An optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications

  • Zdeněk Dostál
  • Tomáš Kozubek
Full Length Paper Series A


We propose a modification of our MPGP algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints. Our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections, possibly with the superrelaxation steplength. The error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the Hessian matrix in a given positive interval, then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded. We also show how to use the algorithm for the solution of separable and equality constraints. The power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with Tresca friction discretized by almost twelve millions nodal variables.


QPQC with separable constraints Spherical constraints Rate of convergence 

Mathematics Subject Classification (2000)

65K10 90C20 90C25 90C90 


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  1. 1.
    Anitescu M.: A superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming. SIAM J. Optim. 12, 949–978 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ben-Tal A., Nemirovski A.: Lectures on Modern Convex Optimization. SIAM, Philadelphia (2001)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bertsekas D.P.: Nonlinear Optimization. Athena Scientific, Belmont (1999)Google Scholar
  4. 4.
    Bouchala J., Dostál Z., Sadowská M.: Scalable total BETI based algorithm for 3D coercive contact problems of linear elastostatics. Computing 85(3), 189–217 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bouchala, J., Dostál, Z., Vodstrčil, P.: Separable Spherical Constraints and the Decrease of a Quadratic Function in the Gradient Projection (submitted)Google Scholar
  6. 6.
    Conn A.R., Gould N.I.M., Toint Ph.L.: Testing a class of algorithms for solving minimization problems with simple bounds on the variables. Math. Comput. 50, 399–430 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Conn A.R., Gould N.I.M., Toint Ph.L.: Trust Region Methods. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    Dostál Z.: Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7(3), 871–887 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dostál Z.: Inexact semimonotonic augmented lagrangians with optimal feasibility Convergence for convex bound and equality constrained quadratic programming. SIAM J. Numer. Anal. 43, 96–115 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dostál Z.: An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum. Computing 78, 311–328 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dostál Z.: On the decrease of a quadratic function along the projected–gradient path. ETNA 31, 25–59 (2008)zbMATHGoogle Scholar
  12. 12.
    Dostál, Z.: Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities, 1st edition. SOIA 23. Springer US, New York (2009)Google Scholar
  13. 13.
    Dostál Z., Domorádová M., Sadowská M.: Superrelaxation in minimizing quadratic functions subject to bound constraints. Comput Optim Appl. 48, 23–44 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dostál Z., Friedlander A., Santos S.A.: Augmented Lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints. SIAM J. Optim. 13, 1120–1140 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dostál, Z., Horák, D.: Scalability and FETI based algorithm for large discretized variational inequalities. Math. Comput. Simul. 61, 3–6, 347–357 (2003)Google Scholar
  16. 16.
    Dostál, Z., Kozubek, T., Markopoulos, A., Brzobohatý, T., Vondrák, V., Horyl, P.: Theoretically supported scalable TFETI algorithm for the solution of multibody 3D contact problems with friction (2011). doi: 10.1016/j.cma.2011.02.015
  17. 17.
    Dostál Z., Kozubek T., Vondrák V., Brzobohatý T., Markopoulos A.: Scalable TFETI algorithm for the solution of multibody contact problems of elasticity. Int. J. Numer. Meth. Engng. 82(11), 1384–1405 (2010)zbMATHGoogle Scholar
  18. 18.
    Dostál Z., Kučera R.: Algorithm for minimization of quadratic functions with bounded spectrum subject to separable convex inequality and linear equality constraints. SIAM J. Optim. 20(6), 2913–2938 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Dostál Z., Schöberl J.: Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination. Comput. Optim. Appl. 30(1), 23–44 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ecker G., Niemi R.D.: Dual method for quadratic programs with quadratic constraints. SIAM J. Appl. Math. 28, 568–576 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Friedlander A., Martínez J.M.: On the maximization of a concave quadratic function with box constraints. SIAM J. Optim. 4, 177–192 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Friedlander A., Martínez J.M., Raydan M.: A new method for large scale box constrained quadratic minimization problems. Optim. Methods Softw. 5, 57–74 (1995)CrossRefGoogle Scholar
  23. 23.
    Kozubek, T., Markopoulos, A., Brzobohatý, T., Kučera, R., Vondrák, V., Dostál, Z.: MatSol—MATLAB efficient solvers for problems in engineering.
  24. 24.
    Kučera R.: Minimizing quadratic functions with separable quadratic constraints. Optim. Methods Softw. 22, 453–467 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kučera R.: Convergence rate of an optimization algorithm for minimizing quadratic functions with separable convex constraints. SIAM J. Optim. 19, 846–862 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Luo Z.-Q., Tseng P.: On the linear convergence of descent methods for convex essentially smooth minimization. SIAM J. Control Optim. 30(2), 408–425 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Luo Z.-Q., Tseng P.: Error bounds and convergence analysis of feasible descent methods: a general approach. Ann. Oper. Res. 46, 157–178 (1993)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Martínez J.M.: Local minimizers of quadratic functions on Euclidean balls and spheres. SIAM J. Optim. 4, 159–176 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Mehrotra S., Sun J.: A method of analytic centers for quadratically constrained convex quadratic programs. SIAM J. Numer. Anal. 28, 529–544 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Panagiotopoulos P.D.: A non-linear programming approach to the unilateral contact- and friction boundary value problem. Ingenieur Archiv. 44, 421–432 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Saad Y.: Iterative Methods for Large Linear Systems. SIAM, Philadelphia (2002)Google Scholar
  32. 32.
    Schöberl J.: Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60(4), 323–344 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Sorensen D.C.: Minimization of a large scale quadratic function subject to spherical constraints. SIAM J. Optim. 7(1), 141–161 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  1. 1.FEECS VŠB–Technical University of OstravaOstravaCzech Republic

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