A study of the difference-of-convex approach for solving linear programs with complementarity constraints
- 239 Downloads
This paper studies the difference-of-convex (DC) penalty formulations and the associated difference-of-convex algorithm (DCA) for computing stationary solutions of linear programs with complementarity constraints (LPCCs). We focus on three such formulations and establish connections between their stationary solutions and those of the LPCC. Improvements of the DCA are proposed to remedy some drawbacks in a straightforward adaptation of the DCA to these formulations. Extensive numerical results, including comparisons with an existing nonlinear programming solver and the mixed-integer formulation, are presented to elucidate the effectiveness of the overall DC approach.
KeywordsDifference-of-convex Complementarity constraints Penalty functions Bilevel programming
Mathematics Subject Classification90C05 Linear programming 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions)
We thank a referee for very helpful comments that have improved the presentation of the paper.
- 2.Burdakov, O., Kanzow, Ch., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type constraints and a regularization method. Preprint 324, Institute of Mathematics, University of Würzburg, Germany (2014) (last revised February 2015)Google Scholar
- 4.Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM classics in applied mathematics 60, Philadelphia (2009) [Originally published by Academic Press, Boston (1992)]Google Scholar
- 6.Feng, M., Mitchell, J.E., Pang, J.S., Wächter, A., Shen, X.: Complementarity formulations of \(\ell _0\)-norm optimization problems. Pac. J. Optim. (accepted August 2016)Google Scholar
- 20.Le Thi, H.A., Pham Dinh, T.: Recent advances in DC programming and DCA. Trans. Comput. Collect. Intell. 8342, 1–37 (2014)Google Scholar
- 21.Le Thi, H.A., Pham Dinh, T.: The state of the art in DC programming and DCA. Research Report, Lorraine University (2013)Google Scholar
- 24.Leyffer, S., Munson, T.S.: A globally convergent Filter method for MPECs. Preprint ANL/MCSP1457-0907, Argonne National Laboratory, Mathematics and Computer Science Division (revised April 2009)Google Scholar
- 26.Leyffer, S.: MacMPEC: AMPL collection of MPECs (2000). http://www.mcs.anl.gov/~leyffer/MacMPEC/
- 30.Pang, J.S., Razaviyayn, M., Alvarado, A.: Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. (2016). https://doi.org/10.1287/moor.2016.0795
- 34.Yu, B.: A branch and cut approach to linear programs with linear complementarity constraints. Ph.D. thesis. Department of Decision Sciences and Engineering Systems, Rensselaer Polytechic Institute (2011)Google Scholar
- 35.Yu, B., Mitchell, J.E., Pang, J.S.: Obtaining tighter relaxations of mathematical programs with complementarity constraints. In: Terlaky, T., Curtis, F. (eds.) Modeling and Optimization: Theory and Applications, pp. 1–23. Springer Proceedings in Mathematics and Statistics, New York (2012)Google Scholar