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Convergence analysis of sparse quasiNewton updates with positive definite matrix completion for twodimensional functions
Improved convergence properties of the LinFukushimaRegularization method for mathematical programs with complementarity constraints
1.  University of Würzburg, Institute of Mathematics, Am Hubland, 97074 Würzburg, Germany, Germany, Germany 
References:
[1] 
M. S. Bazaraa and C. M. Shetty, "Foundations of Optimization,'', Lecture Notes in Economics and Mathematical Systems, (1976). 
[2] 
Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions,, Optimization, 32 (1995), 193. doi: 10.1080/02331939508844048. 
[3] 
A. V. Demiguel, M. P. Friedlander, F. J. Nogales and S. Scholtes, A twosided relaxation scheme for mathematical programs with equilibrium constraints,, SIAM Journal on Optimization, 16 (2005), 587. doi: 10.1137/04060754x. 
[4] 
S. Dempe, "Foundations of Bilevel Programming, Nonconvex Optimization and Its Applications,", 61 (2002), 61 (2002). 
[5] 
M. L. Flegel and C. Kanzow, On the Guignard constraint qualification for mathematical programs with equilibrium constraints,, Optimization, 54 (2005), 517. doi: 10.1080/02331930500342591. 
[6] 
M. L. Flegel and C. Kanzow, A direct proof for Mstationarity under MPECACQ for mathematical programs with equilibrium constraints,, In, (2006), 111. doi: 10.1007/0387342214_6. 
[7] 
T. Hoheisel, C. Kanzow and A. Schwartz, Convergence of a local regularization approach for mathematical programs with complementarity or vanishing constraints,, Preprint 293, (2010). 
[8] 
T. Hoheisel, C. Kanzow and A. Schwartz, Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints,, Preprint 299, (2010). 
[9] 
A. Kadrani, J. P. Dussault and A. Benchakroun, A new regularization scheme for mathematical programs with complementarity constraints,, SIAM Journal on Optimization, 20 (2009), 78. doi: 10.1137/070705490. 
[10] 
C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties,, Preprint 296, (2010). 
[11] 
S. Leyffer, MacMPEC: AMPL collection of MPECs,, , (2000). 
[12] 
G. H. Lin and M. Fukushima, A modified relaxation scheme for mathematical programs with complementarity constraints,, Annals of Operations Research, 133 (2005), 63. doi: 10.1007/s104790045024z. 
[13] 
, www.netlib.org/ampl/solvers, /examples/amplfunc.c, (). 
[14] 
Z. Q. Luo, J. S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,'', Cambridge University Press, (1996). 
[15] 
O. L. Mangasarian, "Nonlinear Programming,'', McGrawHill, (1969). 
[16] 
J. V. Outrata, M. Kočvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints,'', Nonconvex Optimization and its Applications, (1998). 
[17] 
L. Qi and Z. Wei, On the constant positive linear dependence condition and its applications to SQP methods,, SIAM Journal on Optimization, 10 (2000), 963. doi: 10.1137/S1052623497326629. 
[18] 
H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity,, Mathematics of Operations Research, 25 (2000), 1. doi: 10.1287/moor.25.1.1.15213. 
[19] 
S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,, SIAM Journal on Optimization, 11 (2001), 918. doi: 10.1137/S1052623499361233. 
[20] 
S. Steffensen and M. Ulbrich, A new relaxation scheme for mathematical programs with equilibrium constraints,, SIAM Journal on Optimization, 20 (2010), 2504. doi: 10.1137/090748883. 
[21] 
J. J. Ye, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints,, SIAM Journal on Optimization, 10 (2000), 943. doi: 10.1137/S105262349834847X. 
[22] 
J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems,, Optimization, 33 (1995), 9. doi: 10.1080/02331939508844060. 
show all references
References:
[1] 
M. S. Bazaraa and C. M. Shetty, "Foundations of Optimization,'', Lecture Notes in Economics and Mathematical Systems, (1976). 
[2] 
Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions,, Optimization, 32 (1995), 193. doi: 10.1080/02331939508844048. 
[3] 
A. V. Demiguel, M. P. Friedlander, F. J. Nogales and S. Scholtes, A twosided relaxation scheme for mathematical programs with equilibrium constraints,, SIAM Journal on Optimization, 16 (2005), 587. doi: 10.1137/04060754x. 
[4] 
S. Dempe, "Foundations of Bilevel Programming, Nonconvex Optimization and Its Applications,", 61 (2002), 61 (2002). 
[5] 
M. L. Flegel and C. Kanzow, On the Guignard constraint qualification for mathematical programs with equilibrium constraints,, Optimization, 54 (2005), 517. doi: 10.1080/02331930500342591. 
[6] 
M. L. Flegel and C. Kanzow, A direct proof for Mstationarity under MPECACQ for mathematical programs with equilibrium constraints,, In, (2006), 111. doi: 10.1007/0387342214_6. 
[7] 
T. Hoheisel, C. Kanzow and A. Schwartz, Convergence of a local regularization approach for mathematical programs with complementarity or vanishing constraints,, Preprint 293, (2010). 
[8] 
T. Hoheisel, C. Kanzow and A. Schwartz, Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints,, Preprint 299, (2010). 
[9] 
A. Kadrani, J. P. Dussault and A. Benchakroun, A new regularization scheme for mathematical programs with complementarity constraints,, SIAM Journal on Optimization, 20 (2009), 78. doi: 10.1137/070705490. 
[10] 
C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties,, Preprint 296, (2010). 
[11] 
S. Leyffer, MacMPEC: AMPL collection of MPECs,, , (2000). 
[12] 
G. H. Lin and M. Fukushima, A modified relaxation scheme for mathematical programs with complementarity constraints,, Annals of Operations Research, 133 (2005), 63. doi: 10.1007/s104790045024z. 
[13] 
, www.netlib.org/ampl/solvers, /examples/amplfunc.c, (). 
[14] 
Z. Q. Luo, J. S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,'', Cambridge University Press, (1996). 
[15] 
O. L. Mangasarian, "Nonlinear Programming,'', McGrawHill, (1969). 
[16] 
J. V. Outrata, M. Kočvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints,'', Nonconvex Optimization and its Applications, (1998). 
[17] 
L. Qi and Z. Wei, On the constant positive linear dependence condition and its applications to SQP methods,, SIAM Journal on Optimization, 10 (2000), 963. doi: 10.1137/S1052623497326629. 
[18] 
H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity,, Mathematics of Operations Research, 25 (2000), 1. doi: 10.1287/moor.25.1.1.15213. 
[19] 
S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,, SIAM Journal on Optimization, 11 (2001), 918. doi: 10.1137/S1052623499361233. 
[20] 
S. Steffensen and M. Ulbrich, A new relaxation scheme for mathematical programs with equilibrium constraints,, SIAM Journal on Optimization, 20 (2010), 2504. doi: 10.1137/090748883. 
[21] 
J. J. Ye, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints,, SIAM Journal on Optimization, 10 (2000), 943. doi: 10.1137/S105262349834847X. 
[22] 
J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems,, Optimization, 33 (1995), 9. doi: 10.1080/02331939508844060. 
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