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On constraint qualifications: Motivation, design and interrelations
1.  Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695, United States 
2.  Department of Industrial and System Engineering, North Carolina State University, Raleigh, NC 27695 
3.  Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 
References:
[1] 
J. Abadie, On the KuhnTucker theorem, in "Nonlinear Programming" (ed. J. Abadie), NorthHolland Pub. Co., (1967), 1936. 
[2] 
R. Andreani, G. Haeser, M. L. Schuverdt and P. J. S. Silva, A relaxed constant positive linear dependence constraint qualification and applications, Mathematical Programming, 135 (2012), 255273. doi: 10.1007/s1010701104560. 
[3] 
R. Andreani, J. M. Martinez and M. L. Schuverdt, On the relation between constant positive linear dependence condition and quasinormality constraint qualification, Journal of Optimization Theory and Applications, 125 (2005), 473483. doi: 10.1007/s1095700418619. 
[4] 
K. J. Arrow, L. Hurwicz and H. Uzawa, Constraint qualifications in maximization problems, Naval Research Logistics Quarterly, 8 (1961), 175191. doi: 10.1002/nav.3800080206. 
[5] 
M. S. Bazaraa, J. J. Goode and C. M. Shetty, Constraint qualifications revisited, Management Science, 18 (1972), 567573. doi: 10.1287/mnsc.18.9.567. 
[6] 
M. S. Bazaraa, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," $3^{rd}$ edition, WileyInterscience, Hoboken, NJ, 2006. doi: 10.1002/0471787779. 
[7] 
D. P. Bertsekas, "Nonlinear Programming," $2^{nd}$ edition, Athena Scientific, 1999. 
[8] 
D. P. Bertsekas, A. Nedić and A. E. Ozdaglar, "Convex Analysis and Optimization," Athena Scientific, Belmont, MA, 2003. 
[9] 
D. P. Bertsekas and A. E. Ozdaglar, Pseudonormality and a Lagrange multiplier theory for constrained optimization, Journal of Optimization Theory and Applications, 114 (2002), 287343. doi: 10.1023/A:1016083601322. 
[10] 
J. M. Borwein and H. Wolkowicz, A simple constraint qualification in infinite dimensional programming, Mathematical Programming, 35 (1986), 8396. doi: 10.1007/BF01589443. 
[11] 
M. Canon, C. Cullum and E. Polak, Constrained minimization problems in finitedimensional spaces, SIAM Journal on Control, 4 (1966), 528547. doi: 10.1137/0304041. 
[12] 
R. W. Cottle, A theorem of Fritz John in mathematical programming, RAND Memorandum RM3858PR, RAND Corporation, 1963. 
[13] 
W. Fenchel, "Convex Cones, Sets and Functions," Princeton University Press, Princeton, 1953. 
[14] 
D. Gale, "The Theory of Linear Economic Models," McGrawHill Book Company, Inc., New YorkTorontoLondon, 1960. 
[15] 
G. Giorgi and A. Guerraggio, On the notion of tangent cone in mathematical programming, Optimization, 25 (1992), 1123. doi: 10.1080/02331939208843804. 
[16] 
F. J. Gould and J. W. Tolle, A necessary and sufficient qualification for constrained optimization, SIAM Journal on Applied Mathematics, 20 (1971), 164172. doi: 10.1137/0120021. 
[17] 
F. J. Gould and J. W. Tolle, Geometry of optimality conditions and constraint qualifications, Mathematical Programming, 2 (1972), 118. doi: 10.1007/BF01584534. 
[18] 
M. Gowda and M. Teboulle, A comparison of constraint qualifications in infinitedimensional convex programming, SIAM Journal on Control and Optimization, 28 (1990), 925935. doi: 10.1137/0328051. 
[19] 
M. Guignard, Generalized KuhnTucker conditions for mathematical programming problems in a Banach space, SIAM Journal on Control, 7 (1969), 232241. doi: 10.1137/0307016. 
[20] 
M. R. Hestenes, "Calculus of Variations and Optimal Control Theory," John Wiley & Sons, Inc., New YorkLondonSydney, 1966. 
[21] 
M. R. Hestenes, "Optimization Theory: The Finite Dimensional Case," Pure and Applied Mathematics, WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1975. 
[22] 
L. Hurwicz, Programming in linear spaces, in "Studies in Linear and Nonlinear Programming" (eds. K. J. Arrow, L. Hurwicz and H. Uzawa), Stanford University Press, (1958), 38102. 
[23] 
R. Janin, Directional derivative of the marginal function in nonlinear programming. Sensitivity, stability and parametric analysis, Mathematical Programming Studies, 21 (1984), 110126. doi: 10.1007/BFb0121214. 
[24] 
F. John, Extremum problems with inequalities as subsidiary conditions, in "Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948" (eds. K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker), WileyInterscience New York, (1948), 187204. 
[25] 
A. Jourani, Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems, Journal of Optimization Theory and Applications, 81 (1994), 533548. doi: 10.1007/BF02193099. 
[26] 
S. Karlin, "Mathematical Methods and Theory in Games, Programming, and Economics," AddisonWesley Publishing Co., Inc., Reading, Mass.London, 1959. 
[27] 
H. W. Kuhn and A. W. Tucker, Nonlinear programming, in "Second Berkeley Symposium on Mathematical Statistics and Probability," University of California Press, Berkeley and Los Angeles, (1951), 481492. 
[28] 
L. Kuntz, Constraint qualifications in quasidifferentiable optimization, Mathematical Programming, 60 (1993), 339347. doi: 10.1007/BF01580618. 
[29] 
L. Kuntz and S. Scholtes, A nonsmooth variant of the MangasarianFromovitz constraint qualification, Journal of Optimization Theory and Applications, 82 (1994), 5975. doi: 10.1007/BF02191779. 
[30] 
S. Lu, Implications of the constant rank constraint qualification, Mathematical Programming, 126 (2011), 365392. doi: 10.1007/s1010700902883. 
[31] 
D. Luenberger and Y. Ye, "Linear and Nonlinear Programming," $3^{rd}$ edition, International Series in Operations Research & Management Science, 116, Springer, 2008. 
[32] 
C. Ma, X. Li, K. F. C. Yiu, Y. Yang and L. Zhang, On an exact penalty function method for semiinfinite programming problems, Journal of Industrial and Management Optimization, 8 (2012), 705726. doi: 10.3934/jimo.2012.8.705. 
[33] 
O. L. Mangasarian, "Nonlinear Programming," McGrawHill Book Company, Inc., New YorkLondonSydney, 1969. 
[34] 
O. L. Mangasarian and S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, Journal of Mathematical Analysis and Applications, 17 (1967), 3747. doi: 10.1016/0022247X(67)901631. 
[35] 
R. R. Merkovsky and D. E. Ward, General constraint qualifications in nondifferentiable programming, Mathematical Programming, 47 (1990), 389405. doi: 10.1007/BF01580871. 
[36] 
L. Minchenko and S. Stakhovski, On relaxed constant rank regularity condition in mathematical programming, Optimization, 60 (2011), 429440. doi: 10.1080/02331930902971377. 
[37] 
A. E. Ozdaglar and D. P. Bertsekas, The relation between pseudonormality and quasiregularity in constrained optimization, Optimization Methods and Software, 19 (2004), 493506. doi: 10.1080/10556780410001709420. 
[38] 
D. W. Peterson, A review of constraint qualifications in finitedimensional spaces, SIAM Review, 15 (1973), 639654. doi: 10.1137/1015075. 
[39] 
L. Qi and Z. Wei, On the constant positive linear dependence condition and its application to SQP methods, SIAM Journal on Optimization, 10 (2000), 963981. doi: 10.1137/S1052623497326629. 
[40] 
K. Ritter, Optimization theory in linear spaces, Mathematische Annalen, 184 (1970), 133154. doi: 10.1007/BF01350314. 
[41] 
R. T. Rockafellar, "Conjugate Duality and Optimization," Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. doi: 10.1137/1.9781611970524. 
[42] 
R. T. Rockafellar, Lagrange multipliers and optimality, SIAM Review, 35 (1993), 183238. doi: 10.1137/1035044. 
[43] 
R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, SpringerVerlag, Berlin, 1998. doi: 10.1007/9783642024313. 
[44] 
M. Slater, Lagrange multipliers revisited, Cowles Foundation Discussion Paper No. 80, Cowles Foundation for Research in Economics, Yale University, 1950. 
[45] 
M. V. Solodov, Constraint qualifications, in "Wiley Encyclopedia of Operations Research and Management Science," John Wiley & Sons, Inc., 2010. doi: 10.1002/9780470400531.eorms0978. 
[46] 
O. Stein, On constraint qualifications in nonsmooth optimization, Journal of Optimization Theory and Applications, 121 (2004), 647671. doi: 10.1023/B:JOTA.0000037607.48762.45. 
[47] 
P. Varaiya, Nonlinear programming in Banach space, SIAM Journal on Applied Mathematics, 15 (1967), 284293. doi: 10.1137/0115028. 
[48] 
W. I. Zangwill, "Nonlinear Programming: A Unified Approach," PrenticeHall International Series in Management, PrenticeHall, Inc., Englewood Cliffs, NJ, 1969. 
show all references
References:
[1] 
J. Abadie, On the KuhnTucker theorem, in "Nonlinear Programming" (ed. J. Abadie), NorthHolland Pub. Co., (1967), 1936. 
[2] 
R. Andreani, G. Haeser, M. L. Schuverdt and P. J. S. Silva, A relaxed constant positive linear dependence constraint qualification and applications, Mathematical Programming, 135 (2012), 255273. doi: 10.1007/s1010701104560. 
[3] 
R. Andreani, J. M. Martinez and M. L. Schuverdt, On the relation between constant positive linear dependence condition and quasinormality constraint qualification, Journal of Optimization Theory and Applications, 125 (2005), 473483. doi: 10.1007/s1095700418619. 
[4] 
K. J. Arrow, L. Hurwicz and H. Uzawa, Constraint qualifications in maximization problems, Naval Research Logistics Quarterly, 8 (1961), 175191. doi: 10.1002/nav.3800080206. 
[5] 
M. S. Bazaraa, J. J. Goode and C. M. Shetty, Constraint qualifications revisited, Management Science, 18 (1972), 567573. doi: 10.1287/mnsc.18.9.567. 
[6] 
M. S. Bazaraa, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," $3^{rd}$ edition, WileyInterscience, Hoboken, NJ, 2006. doi: 10.1002/0471787779. 
[7] 
D. P. Bertsekas, "Nonlinear Programming," $2^{nd}$ edition, Athena Scientific, 1999. 
[8] 
D. P. Bertsekas, A. Nedić and A. E. Ozdaglar, "Convex Analysis and Optimization," Athena Scientific, Belmont, MA, 2003. 
[9] 
D. P. Bertsekas and A. E. Ozdaglar, Pseudonormality and a Lagrange multiplier theory for constrained optimization, Journal of Optimization Theory and Applications, 114 (2002), 287343. doi: 10.1023/A:1016083601322. 
[10] 
J. M. Borwein and H. Wolkowicz, A simple constraint qualification in infinite dimensional programming, Mathematical Programming, 35 (1986), 8396. doi: 10.1007/BF01589443. 
[11] 
M. Canon, C. Cullum and E. Polak, Constrained minimization problems in finitedimensional spaces, SIAM Journal on Control, 4 (1966), 528547. doi: 10.1137/0304041. 
[12] 
R. W. Cottle, A theorem of Fritz John in mathematical programming, RAND Memorandum RM3858PR, RAND Corporation, 1963. 
[13] 
W. Fenchel, "Convex Cones, Sets and Functions," Princeton University Press, Princeton, 1953. 
[14] 
D. Gale, "The Theory of Linear Economic Models," McGrawHill Book Company, Inc., New YorkTorontoLondon, 1960. 
[15] 
G. Giorgi and A. Guerraggio, On the notion of tangent cone in mathematical programming, Optimization, 25 (1992), 1123. doi: 10.1080/02331939208843804. 
[16] 
F. J. Gould and J. W. Tolle, A necessary and sufficient qualification for constrained optimization, SIAM Journal on Applied Mathematics, 20 (1971), 164172. doi: 10.1137/0120021. 
[17] 
F. J. Gould and J. W. Tolle, Geometry of optimality conditions and constraint qualifications, Mathematical Programming, 2 (1972), 118. doi: 10.1007/BF01584534. 
[18] 
M. Gowda and M. Teboulle, A comparison of constraint qualifications in infinitedimensional convex programming, SIAM Journal on Control and Optimization, 28 (1990), 925935. doi: 10.1137/0328051. 
[19] 
M. Guignard, Generalized KuhnTucker conditions for mathematical programming problems in a Banach space, SIAM Journal on Control, 7 (1969), 232241. doi: 10.1137/0307016. 
[20] 
M. R. Hestenes, "Calculus of Variations and Optimal Control Theory," John Wiley & Sons, Inc., New YorkLondonSydney, 1966. 
[21] 
M. R. Hestenes, "Optimization Theory: The Finite Dimensional Case," Pure and Applied Mathematics, WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1975. 
[22] 
L. Hurwicz, Programming in linear spaces, in "Studies in Linear and Nonlinear Programming" (eds. K. J. Arrow, L. Hurwicz and H. Uzawa), Stanford University Press, (1958), 38102. 
[23] 
R. Janin, Directional derivative of the marginal function in nonlinear programming. Sensitivity, stability and parametric analysis, Mathematical Programming Studies, 21 (1984), 110126. doi: 10.1007/BFb0121214. 
[24] 
F. John, Extremum problems with inequalities as subsidiary conditions, in "Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948" (eds. K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker), WileyInterscience New York, (1948), 187204. 
[25] 
A. Jourani, Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems, Journal of Optimization Theory and Applications, 81 (1994), 533548. doi: 10.1007/BF02193099. 
[26] 
S. Karlin, "Mathematical Methods and Theory in Games, Programming, and Economics," AddisonWesley Publishing Co., Inc., Reading, Mass.London, 1959. 
[27] 
H. W. Kuhn and A. W. Tucker, Nonlinear programming, in "Second Berkeley Symposium on Mathematical Statistics and Probability," University of California Press, Berkeley and Los Angeles, (1951), 481492. 
[28] 
L. Kuntz, Constraint qualifications in quasidifferentiable optimization, Mathematical Programming, 60 (1993), 339347. doi: 10.1007/BF01580618. 
[29] 
L. Kuntz and S. Scholtes, A nonsmooth variant of the MangasarianFromovitz constraint qualification, Journal of Optimization Theory and Applications, 82 (1994), 5975. doi: 10.1007/BF02191779. 
[30] 
S. Lu, Implications of the constant rank constraint qualification, Mathematical Programming, 126 (2011), 365392. doi: 10.1007/s1010700902883. 
[31] 
D. Luenberger and Y. Ye, "Linear and Nonlinear Programming," $3^{rd}$ edition, International Series in Operations Research & Management Science, 116, Springer, 2008. 
[32] 
C. Ma, X. Li, K. F. C. Yiu, Y. Yang and L. Zhang, On an exact penalty function method for semiinfinite programming problems, Journal of Industrial and Management Optimization, 8 (2012), 705726. doi: 10.3934/jimo.2012.8.705. 
[33] 
O. L. Mangasarian, "Nonlinear Programming," McGrawHill Book Company, Inc., New YorkLondonSydney, 1969. 
[34] 
O. L. Mangasarian and S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, Journal of Mathematical Analysis and Applications, 17 (1967), 3747. doi: 10.1016/0022247X(67)901631. 
[35] 
R. R. Merkovsky and D. E. Ward, General constraint qualifications in nondifferentiable programming, Mathematical Programming, 47 (1990), 389405. doi: 10.1007/BF01580871. 
[36] 
L. Minchenko and S. Stakhovski, On relaxed constant rank regularity condition in mathematical programming, Optimization, 60 (2011), 429440. doi: 10.1080/02331930902971377. 
[37] 
A. E. Ozdaglar and D. P. Bertsekas, The relation between pseudonormality and quasiregularity in constrained optimization, Optimization Methods and Software, 19 (2004), 493506. doi: 10.1080/10556780410001709420. 
[38] 
D. W. Peterson, A review of constraint qualifications in finitedimensional spaces, SIAM Review, 15 (1973), 639654. doi: 10.1137/1015075. 
[39] 
L. Qi and Z. Wei, On the constant positive linear dependence condition and its application to SQP methods, SIAM Journal on Optimization, 10 (2000), 963981. doi: 10.1137/S1052623497326629. 
[40] 
K. Ritter, Optimization theory in linear spaces, Mathematische Annalen, 184 (1970), 133154. doi: 10.1007/BF01350314. 
[41] 
R. T. Rockafellar, "Conjugate Duality and Optimization," Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. doi: 10.1137/1.9781611970524. 
[42] 
R. T. Rockafellar, Lagrange multipliers and optimality, SIAM Review, 35 (1993), 183238. doi: 10.1137/1035044. 
[43] 
R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, SpringerVerlag, Berlin, 1998. doi: 10.1007/9783642024313. 
[44] 
M. Slater, Lagrange multipliers revisited, Cowles Foundation Discussion Paper No. 80, Cowles Foundation for Research in Economics, Yale University, 1950. 
[45] 
M. V. Solodov, Constraint qualifications, in "Wiley Encyclopedia of Operations Research and Management Science," John Wiley & Sons, Inc., 2010. doi: 10.1002/9780470400531.eorms0978. 
[46] 
O. Stein, On constraint qualifications in nonsmooth optimization, Journal of Optimization Theory and Applications, 121 (2004), 647671. doi: 10.1023/B:JOTA.0000037607.48762.45. 
[47] 
P. Varaiya, Nonlinear programming in Banach space, SIAM Journal on Applied Mathematics, 15 (1967), 284293. doi: 10.1137/0115028. 
[48] 
W. I. Zangwill, "Nonlinear Programming: A Unified Approach," PrenticeHall International Series in Management, PrenticeHall, Inc., Englewood Cliffs, NJ, 1969. 
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