Computational Optimization and Applications

, Volume 72, Issue 3, pp 727–768 | Cite as

A constraint-reduced MPC algorithm for convex quadratic programming, with a modified active set identification scheme

  • M. Paul LaiuEmail author
  • André L. Tits


A constraint-reduced Mehrotra-predictor-corrector algorithm for convex quadratic programming is proposed. (At each iteration, such algorithms use only a subset of the inequality constraints in constructing the search direction, resulting in CPU savings.) The proposed algorithm makes use of a regularization scheme to cater to cases where the reduced constraint matrix is rank deficient. Global and local convergence properties are established under arbitrary working-set selection rules subject to satisfaction of a general condition. A modified active-set identification scheme that fulfills this condition is introduced. Numerical tests show great promise for the proposed algorithm, in particular for its active-set identification scheme. While the focus of the present paper is on dense systems, application of the main ideas to large sparse systems is briefly discussed.


Convex quadratic programming Constraint reduction Primal-dual interior-point method Mehrotra’s predictor-corrector Regularization Active constraints identification 



  1. 1.
    Altman, A., Gondzio, J.: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optim. Methods Softw. 11(1–4), 275–302 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertsimas, D., Tsitsiklis, J.: Introduction to Linear Optimization. Athena, Austin (1997)Google Scholar
  3. 3.
    Cartis, C., Yan, Y.: Active-set prediction for interior point methods using controlled perturbations. Comput. Optim. Appl. 63(3), 639–684 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castro, J., Cuesta, J.: Quadratic regularizations in an interior-point method for primal block-angular problems. Math. Program. 130(2), 415–445 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, L., Wang, Y., He, G.: A feasible active set QP-free method for nonlinear programming. SIAM J. Optim. 17(2), 401–429 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dantzig, G.B., Ye, Y.: A build-up interior-point method for linear programming: affine scaling form. Technical report, University of Iowa, Iowa City (1991)Google Scholar
  7. 7.
    Drummond, L., Svaiter, B.: On well definedness of the central path. J. Optim. Theory Appl. 102(2), 223–237 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J Optim. 9(1), 14–32 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gill, P.E., Murray, W., Ponceleón, D.B., Saunders, M.A.: Solving reduced KKT systems in barrier methods for linear programming. In: Watson, G.A., Griffiths, D. (eds.) Numerical Analysis 1993. Pitman Research Notes in Mathematics 303, pp. 89–104. Longmans Press, New York (1994)Google Scholar
  10. 10.
    Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pp. 95–110. Springer, Berlin (2008)Google Scholar
  11. 11.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1 (2014). Accessed 27 Feb 2019
  12. 12.
    Hager, W.W., Seetharama Gowda, M.: Stability in the presence of degeneracy and error estimation. Math. Program. 85(1), 181–192 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    He, M.: Infeasible constraint reduction for linear and convex quadratic optimization. Ph.D. thesis, University of Maryland (2011). Accessed 27 Feb 2019
  14. 14.
    He, M.Y., Tits, A.L.: Infeasible constraint-reduced interior-point methods for linear optimization. Optim. Methods Softw. 27(4–5), 801–825 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hertog, D., Roos, C., Terlaky, T.: Adding and deleting constraints in the logarithmic barrier method for LP. In: Du, D.Z., Sun, J. (eds.) Advances in Optimization and Approximation, pp. 166–185. Kluwer Academic Publishers, Dordrecht (1994)CrossRefGoogle Scholar
  16. 16.
    Jung, J.H.: Adaptive constraint reduction for convex quadratic programming and training support vector machines. Ph.D. thesis, University of Maryland (2008). Accessed 27 Feb 2019
  17. 17.
    Jung, J.H., O’Leary, D.P., Tits, A.L.: Adaptive constraint reduction for training support vector machines. Electron. Trans. Numer. Anal. 31, 156–177 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jung, J.H., O’Leary, D.P., Tits, A.L.: Adaptive constraint reduction for convex quadratic programming. Comput. Optim. Appl. 51(1), 125–157 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Laiu, M.P.: Positive filtered P\(_{N}\) method for linear transport equations and the associated optimization algorithm. Ph.D. thesis, University of Maryland (2016). Accessed 27 Feb 2019
  20. 20.
    Laiu, M.P., Hauck, C.D., McClarren, R.G., O’Leary, D.P., Tits, A.L.: Positive filtered P\(_{N}\) moment closures for linear kinetic equations. SIAM J. Numer. Anal. 54(6), 3214–3238 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)zbMATHGoogle Scholar
  23. 23.
    Park, S.: A constraint-reduced algorithm for semidefinite optimization problems with superlinear convergence. J. Optim. Theory Appl. 170(2), 512–527 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Park, S., O’Leary, D.P.: A polynomial time constraint-reduced algorithm for semidefinite optimization problems. J. Optim. Theory Appl. 166(2), 558–571 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Saunders, M.A., Tomlin, J.A.: Solving regularized linear programs using barrier methods and KKT systems. Technical report, SOL 96-4. Department of Operations Research, Stanford University (1996)Google Scholar
  26. 26.
    Sturm, J.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)Google Scholar
  27. 27.
    Tits, A., Wächter, A., Bakhtiari, S., Urban, T., Lawrence, C.: A primal–dual interior-point method for nonlinear programming with strong global and local convergence properties. SIAM J. Optim. 14(1), 173–199 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tits, A.L., Absil, P.A., Woessner, W.P.: Constraint reduction for linear programs with many inequality constraints. SIAM J. Optim. 17(1), 119–146 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tits, A.L., Zhou, J.L.: A simple, quadratically convergent algorithm for linear and convex quadratic programming. In: Hager, W., Hearn, D., Pardalos, P. (eds.) Large Scale Optimization: State of the Art, pp. 411–427. Kluwer Academic Publishers, Dordrecht (1994)CrossRefGoogle Scholar
  30. 30.
    Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3: A Matlab software package for semidefinite programming, version 1.3. Optim. Methods Softw. 11(1–4), 545–581 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tone, K.: An active-set strategy in an interior point method for linear programming. Math. Program. 59(1), 345–360 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95(2), 189–217 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Winternitz, L.: Primal–dual interior-point algorithms for linear programming problems with many inequality constraints. Ph.D. thesis, University of Maryland (2010). Accessed 27 Feb 2019
  34. 34.
    Winternitz, L.B., Nicholls, S.O., Tits, A.L., O’Leary, D.P.: A constraint-reduced variant of Mehrotra’s predictor–corrector algorithm. Comput. Optim. Appl. 51(1), 1001–1036 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Winternitz, L.B., Tits, A.L., Absil, P.A.: Addressing rank degeneracy in constraint-reduced interior-point methods for linear optimization. J. Optim. Theory App. 160(1), 127–157 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wright, S.J.: Primal–Dual Interior-Point Methods. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  37. 37.
    Wright, S.J.: Modifying SQP for degenerate problems. SIAM J. Optim. 13(2), 470–497 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ye, Y.: A “build-down” scheme for linear programming. Math. Program. 46(1), 61–72 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zhang, Y., Zhang, D.: On polynomiality of the Mehrotra-type predictor–corrector interior-point algorithms. Math. Program. 68(1), 303–318 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

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© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Computational and Applied Mathematics Group, Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  2. 2.Department of Electrical and Computer Engineering & Institute for Systems ResearchUniversity of MarylandCollege ParkUSA

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