\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints

  • * Corresponding author: Li-Ping Pang

    * Corresponding author: Li-Ping Pang 

The first author is supported by Huzhou science and technology plan on No.2016GY03

Abstract Full Text(HTML) Figure(0) / Table(2) Related Papers Cited by
  • We study the convergence of the log-exponential regularization method for mathematical programs with vertical complementarity constraints (MPVCC). The previous paper assume that the sequence of Lagrange multipliers are bounded and it can be found by software. However, the KKT points can not be computed via Matlab subroutines exactly in many cases. We note that it is realistic to compute inexact KKT points from a numerical point of view. We prove that, under the MPVCC-MFCQ assumption, the accumulation point of the inexact KKT points is Clarke (C-) stationary point. The idea of inexact KKT conditions can be used to define stopping criteria for many practical algorithms. Furthermore, we introduce a feasible strategy that guarantees inexact KKT conditions and provide some numerical examples to certify the reliability of the approach. It means that we can apply the inexact regularization method to solve MPVCC and show the advantages of the improvement.

    Mathematics Subject Classification: Primary: 90C30, 90C33; Secondary: 90C46.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  The numerical results for Example 2

    t $(y^t_1, y^t_2, y^t_3, y^t_4, z^t_1, z^t_2)$ $ f^t $
    0.2 (0.0592, -0.4868, 0.3863, 0.2741, 1.0058, 0.4937) 1.7496
    0.01 (-0.0000, -0.4999, 0.4011, 0.1994, 1.0001, 0.4999) 1.6929
    0.005 ( 0.0000, -0.5000, 0.3997, 0.1998, 1.0000, 0.5000) 1.6901
     | Show Table
    DownLoad: CSV

    Table 2.  The numerical results for Example 4, 5

    Example Algorithm $z$ $f$ $Gap$
    4 Algorithm 2 (0.0000, 2.0000) 0.0000 100 %
    fmincon (0.0004, 2.0000) 0.0000 99.98 %
    ADH (0.0000, 1.9988) 0.0000 99.94 %
    AH (-0.0000, 1.9999) 0.0000 100 %
    $Polak^1$ (0.0000, 1.8708) 0.0167 93.54 %
    5 Algorithm 2 (0.7500, 0.0000) 0.0625 100 %
    fmincon (0.7500, 0.0003) 0.0625 99.96 %
    ADH (0.7500, 0.0000) 0.0625 100 %
    AH (0.7500, 0.0000) 0.0625 100 %
    $Polak^1$ (0.7500, 0.0000) 0.0625 100 %
     | Show Table
    DownLoad: CSV
  •   R. Andreani , J. M. Martínez , L. T. Santos  and  B. F. Svaiter , On the behaviour of constrained optimization methods when Lagrange multipliers do not exist, Optimization Methods and Software, 29 (2014) , 646-657.  doi: 10.1080/10556788.2013.841692.
      R. Andreani , J. M. Martínez  and  B. F. Svaiter , A new sequential optimality condition for constrained optimization and algorithmic consequences, SIAM Journal on Optimization, 20 (2010) , 3533-3554.  doi: 10.1137/090777189.
      M. S. Bazaraa and C. M. Shetty, Foundations of Optimization, 122 Springer-Verlag, Berlin/New York, 1976.
      B. Byrd , M. E. Hribar  and  J. Nocedal , An interior point algorithm for large-scale nonlinear programming, SIAM Journal on Optimization, 9 (1999) , 877-900.  doi: 10.1137/S1052623497325107.
      R. Fletcher  and  S. Leyffer , Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002) , 239-269.  doi: 10.1007/s101070100244.
      P. E. Gill , W. Murray  and  M. A. Saunders , SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM Journal on Optimization, 12 (2002) , 979-1006.  doi: 10.1137/S1052623499350013.
      M. S. Gowda  and  R. Sznajder , A generalization of the Nash equilibrium theorem on bimatrix games, International Journal of Game Theory, 25 (1996) , 1-12.  doi: 10.1007/BF01254380.
      T. Hoheisel , C. Kanzow  and  A. Schwartz , Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints, Mathematical Programming Series A, 137 (2013) , 257-288.  doi: 10.1007/s10107-011-0488-5.
      C. Kanzow , On the multiplier-penalty-approach for quasi-variational inequalities, Mathematical Programming Series A, 160 (2016) , 33-63.  doi: 10.1007/s10107-015-0973-3.
      C. Kanzow  and  A. Schwartz , A new regularization method for mathematical programs with complementarity constraints with strong convergence properties, SIAM Journal on Optimization, 23 (2013) , 770-798.  doi: 10.1137/100802487.
      C. Kanzow  and  A. Schwartz , Convergence properties of the inexact Lin-Fukushima relaxation method for mathematical programs with complementarity constraints, Computational Optimization and Applications, 59 (2014) , 249-262.  doi: 10.1007/s10589-013-9575-2.
      C. Kanzow  and  A. Schwartz , The price of inexactness: Convergence properties of relaxation methods for mathematical programs with complementarity constraints revisited, Mathematical Programming Series A, 40 (2015) , 253-275.  doi: 10.1287/moor.2014.0667.
      Y. Li , T. Tan  and  X. Li , A log-exponential smoothing method for mathematical programs with complementarity constraints, Applied Mathematics and Computation, 218 (2012) , 5900-5909.  doi: 10.1016/j.amc.2011.11.046.
      Y. C. Liang, Some Studies on Optimization Problems with Equilibrium Constraints, Ph. D thesis, Dalian University of Technology in Dalian, 2006.
      Y. C. Liang  and  G. H. Lin , Stationaruty conditions and their reformulations for mathematical programs with vertical complementarity constraints, Journal of Optimization Theory and Applications, 154 (2012) , 54-70.  doi: 10.1007/s10957-012-9992-x.
      G. H. Lin  and  M. Fukushima , A modified relaxation scheme for mathematical programs with complementarity constraints, Annals of Operations Research, 133 (2005) , 63-84.  doi: 10.1007/s10479-004-5024-z.
      O. L. Mangasarian , Equivalence of the complementarity problem to a system of nonlinear equations, SIAM Journal on Applied Mathematics, 31 (1976) , 89-92.  doi: 10.1137/0131009.
      J. M. Peng  and  Z. H. Lin , A non-interior continuation method for generalized linear complementarity problems, Mathematical Programming, 86 (1999) , 533-563.  doi: 10.1007/s101070050104.
      H. D. Qi  and  L. Z. Liao , A smoothing Newton method for extended vertical linear complementarity problems, SIAM Journal on Matrix Analysis and Applications, 21 (1999) , 45-66.  doi: 10.1137/S0895479897329837.
      L. Qi  and  Z. Wei , On the constant positive linear dependence condition and its application to SQP methods, SIAM Journal on Optimization, 10 (2000) , 963-981.  doi: 10.1137/S1052623497326629.
      H. Scheel  and  S. Scholtes , Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Mathematics of Operations Ressearch, 25 (2000) , 1-22.  doi: 10.1287/moor.25.1.1.15213.
      S. Scholtes , Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11 (2001) , 918-936.  doi: 10.1137/S1052623499361233.
      J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions, 1nd edition, Springer-Verlag, New York, 1970.
      A. Wächter  and  L. T. Biegler , On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006) , 25-57.  doi: 10.1007/s10107-004-0559-y.
      H. J. Xiong  and  B. Yu , An aggergate deformation homotopy method for min-max-min problems with max-min constraints, Computational Optimization and Applications, 47 (2010) , 501-527.  doi: 10.1007/s10589-008-9229-y.
      H. Yin  and  J. Zhang , Global convergence of a smooth approximation method for mathematical programs with complementarity constraints, Mathematical Methods of Operations Research, 64 (2006) , 255-269.  doi: 10.1007/s00186-006-0076-2.
      J. Zhang , S. Lin  and  L. W. Zhang , A log-exponential regularization method for a mathematical program with general vertical complementarity constraints, Journal of Industrial and management Optimization, 9 (2013) , 561-577.  doi: 10.3934/jimo.2013.9.561.
  • 加载中

Tables(2)

SHARE

Article Metrics

HTML views(3360) PDF downloads(359) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return