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Channel coordination mechanism with retailers having fairness preference ---An improved quantity discount mechanism
On constraint qualifications: Motivation, design and inter-relations
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 Kuhn-Tucker theorem,, in, (1967), 19.
|
[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), 255.
doi: 10.1007/s10107-011-0456-0. |
[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), 473.
doi: 10.1007/s10957-004-1861-9. |
[4] |
K. J. Arrow, L. Hurwicz and H. Uzawa, Constraint qualifications in maximization problems,, Naval Research Logistics Quarterly, 8 (1961), 175.
doi: 10.1002/nav.3800080206. |
[5] |
M. S. Bazaraa, J. J. Goode and C. M. Shetty, Constraint qualifications revisited,, Management Science, 18 (1972), 567.
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, (2006).
doi: 10.1002/0471787779. |
[7] |
D. P. Bertsekas, "Nonlinear Programming,", $2^{nd}$ edition, (1999).
|
[8] |
D. P. Bertsekas, A. Nedić and A. E. Ozdaglar, "Convex Analysis and Optimization,", Athena Scientific, (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), 287.
doi: 10.1023/A:1016083601322. |
[10] |
J. M. Borwein and H. Wolkowicz, A simple constraint qualification in infinite dimensional programming,, Mathematical Programming, 35 (1986), 83.
doi: 10.1007/BF01589443. |
[11] |
M. Canon, C. Cullum and E. Polak, Constrained minimization problems in finite-dimensional spaces,, SIAM Journal on Control, 4 (1966), 528.
doi: 10.1137/0304041. |
[12] |
R. W. Cottle, A theorem of Fritz John in mathematical programming,, RAND Memorandum RM-3858-PR, (1963). Google Scholar |
[13] |
W. Fenchel, "Convex Cones, Sets and Functions,", Princeton University Press, (1953). Google Scholar |
[14] |
D. Gale, "The Theory of Linear Economic Models,", McGraw-Hill Book Company, (1960).
|
[15] |
G. Giorgi and A. Guerraggio, On the notion of tangent cone in mathematical programming,, Optimization, 25 (1992), 11.
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), 164.
doi: 10.1137/0120021. |
[17] |
F. J. Gould and J. W. Tolle, Geometry of optimality conditions and constraint qualifications,, Mathematical Programming, 2 (1972), 1.
doi: 10.1007/BF01584534. |
[18] |
M. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite-dimensional convex programming,, SIAM Journal on Control and Optimization, 28 (1990), 925.
doi: 10.1137/0328051. |
[19] |
M. Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space,, SIAM Journal on Control, 7 (1969), 232.
doi: 10.1137/0307016. |
[20] |
M. R. Hestenes, "Calculus of Variations and Optimal Control Theory,", John Wiley & Sons, (1966).
|
[21] |
M. R. Hestenes, "Optimization Theory: The Finite Dimensional Case,", Pure and Applied Mathematics, (1975).
|
[22] |
L. Hurwicz, Programming in linear spaces,, in, (1958), 38. Google Scholar |
[23] |
R. Janin, Directional derivative of the marginal function in nonlinear programming. Sensitivity, stability and parametric analysis,, Mathematical Programming Studies, 21 (1984), 110.
doi: 10.1007/BFb0121214. |
[24] |
F. John, Extremum problems with inequalities as subsidiary conditions,, in, (1948), 187.
|
[25] |
A. Jourani, Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems,, Journal of Optimization Theory and Applications, 81 (1994), 533.
doi: 10.1007/BF02193099. |
[26] |
S. Karlin, "Mathematical Methods and Theory in Games, Programming, and Economics,", Addison-Wesley Publishing Co., (1959).
|
[27] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming,, in, (1951), 481.
|
[28] |
L. Kuntz, Constraint qualifications in quasidifferentiable optimization,, Mathematical Programming, 60 (1993), 339.
doi: 10.1007/BF01580618. |
[29] |
L. Kuntz and S. Scholtes, A nonsmooth variant of the Mangasarian-Fromovitz constraint qualification,, Journal of Optimization Theory and Applications, 82 (1994), 59.
doi: 10.1007/BF02191779. |
[30] |
S. Lu, Implications of the constant rank constraint qualification,, Mathematical Programming, 126 (2011), 365.
doi: 10.1007/s10107-009-0288-3. |
[31] |
D. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", $3^{rd}$ edition, 116 (2008).
|
[32] |
C. Ma, X. Li, K. F. C. Yiu, Y. Yang and L. Zhang, On an exact penalty function method for semi-infinite programming problems,, Journal of Industrial and Management Optimization, 8 (2012), 705.
doi: 10.3934/jimo.2012.8.705. |
[33] |
O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill Book Company, (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), 37.
doi: 10.1016/0022-247X(67)90163-1. |
[35] |
R. R. Merkovsky and D. E. Ward, General constraint qualifications in nondifferentiable programming,, Mathematical Programming, 47 (1990), 389.
doi: 10.1007/BF01580871. |
[36] |
L. Minchenko and S. Stakhovski, On relaxed constant rank regularity condition in mathematical programming,, Optimization, 60 (2011), 429.
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), 493.
doi: 10.1080/10556780410001709420. |
[38] |
D. W. Peterson, A review of constraint qualifications in finite-dimensional spaces,, SIAM Review, 15 (1973), 639.
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), 963.
doi: 10.1137/S1052623497326629. |
[40] |
K. Ritter, Optimization theory in linear spaces,, Mathematische Annalen, 184 (1970), 133.
doi: 10.1007/BF01350314. |
[41] |
R. T. Rockafellar, "Conjugate Duality and Optimization,", Lectures given at the Johns Hopkins University, (1973).
doi: 10.1137/1.9781611970524. |
[42] |
R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183.
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 (1998).
doi: 10.1007/978-3-642-02431-3. |
[44] |
M. Slater, Lagrange multipliers revisited,, Cowles Foundation Discussion Paper No. 80, (1950). Google Scholar |
[45] |
M. V. Solodov, Constraint qualifications,, in, (2010).
doi: 10.1002/9780470400531.eorms0978. |
[46] |
O. Stein, On constraint qualifications in nonsmooth optimization,, Journal of Optimization Theory and Applications, 121 (2004), 647.
doi: 10.1023/B:JOTA.0000037607.48762.45. |
[47] |
P. Varaiya, Nonlinear programming in Banach space,, SIAM Journal on Applied Mathematics, 15 (1967), 284.
doi: 10.1137/0115028. |
[48] |
W. I. Zangwill, "Nonlinear Programming: A Unified Approach,", Prentice-Hall International Series in Management, (1969).
|
show all references
References:
[1] |
J. Abadie, On the Kuhn-Tucker theorem,, in, (1967), 19.
|
[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), 255.
doi: 10.1007/s10107-011-0456-0. |
[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), 473.
doi: 10.1007/s10957-004-1861-9. |
[4] |
K. J. Arrow, L. Hurwicz and H. Uzawa, Constraint qualifications in maximization problems,, Naval Research Logistics Quarterly, 8 (1961), 175.
doi: 10.1002/nav.3800080206. |
[5] |
M. S. Bazaraa, J. J. Goode and C. M. Shetty, Constraint qualifications revisited,, Management Science, 18 (1972), 567.
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, (2006).
doi: 10.1002/0471787779. |
[7] |
D. P. Bertsekas, "Nonlinear Programming,", $2^{nd}$ edition, (1999).
|
[8] |
D. P. Bertsekas, A. Nedić and A. E. Ozdaglar, "Convex Analysis and Optimization,", Athena Scientific, (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), 287.
doi: 10.1023/A:1016083601322. |
[10] |
J. M. Borwein and H. Wolkowicz, A simple constraint qualification in infinite dimensional programming,, Mathematical Programming, 35 (1986), 83.
doi: 10.1007/BF01589443. |
[11] |
M. Canon, C. Cullum and E. Polak, Constrained minimization problems in finite-dimensional spaces,, SIAM Journal on Control, 4 (1966), 528.
doi: 10.1137/0304041. |
[12] |
R. W. Cottle, A theorem of Fritz John in mathematical programming,, RAND Memorandum RM-3858-PR, (1963). Google Scholar |
[13] |
W. Fenchel, "Convex Cones, Sets and Functions,", Princeton University Press, (1953). Google Scholar |
[14] |
D. Gale, "The Theory of Linear Economic Models,", McGraw-Hill Book Company, (1960).
|
[15] |
G. Giorgi and A. Guerraggio, On the notion of tangent cone in mathematical programming,, Optimization, 25 (1992), 11.
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), 164.
doi: 10.1137/0120021. |
[17] |
F. J. Gould and J. W. Tolle, Geometry of optimality conditions and constraint qualifications,, Mathematical Programming, 2 (1972), 1.
doi: 10.1007/BF01584534. |
[18] |
M. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite-dimensional convex programming,, SIAM Journal on Control and Optimization, 28 (1990), 925.
doi: 10.1137/0328051. |
[19] |
M. Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space,, SIAM Journal on Control, 7 (1969), 232.
doi: 10.1137/0307016. |
[20] |
M. R. Hestenes, "Calculus of Variations and Optimal Control Theory,", John Wiley & Sons, (1966).
|
[21] |
M. R. Hestenes, "Optimization Theory: The Finite Dimensional Case,", Pure and Applied Mathematics, (1975).
|
[22] |
L. Hurwicz, Programming in linear spaces,, in, (1958), 38. Google Scholar |
[23] |
R. Janin, Directional derivative of the marginal function in nonlinear programming. Sensitivity, stability and parametric analysis,, Mathematical Programming Studies, 21 (1984), 110.
doi: 10.1007/BFb0121214. |
[24] |
F. John, Extremum problems with inequalities as subsidiary conditions,, in, (1948), 187.
|
[25] |
A. Jourani, Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems,, Journal of Optimization Theory and Applications, 81 (1994), 533.
doi: 10.1007/BF02193099. |
[26] |
S. Karlin, "Mathematical Methods and Theory in Games, Programming, and Economics,", Addison-Wesley Publishing Co., (1959).
|
[27] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming,, in, (1951), 481.
|
[28] |
L. Kuntz, Constraint qualifications in quasidifferentiable optimization,, Mathematical Programming, 60 (1993), 339.
doi: 10.1007/BF01580618. |
[29] |
L. Kuntz and S. Scholtes, A nonsmooth variant of the Mangasarian-Fromovitz constraint qualification,, Journal of Optimization Theory and Applications, 82 (1994), 59.
doi: 10.1007/BF02191779. |
[30] |
S. Lu, Implications of the constant rank constraint qualification,, Mathematical Programming, 126 (2011), 365.
doi: 10.1007/s10107-009-0288-3. |
[31] |
D. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", $3^{rd}$ edition, 116 (2008).
|
[32] |
C. Ma, X. Li, K. F. C. Yiu, Y. Yang and L. Zhang, On an exact penalty function method for semi-infinite programming problems,, Journal of Industrial and Management Optimization, 8 (2012), 705.
doi: 10.3934/jimo.2012.8.705. |
[33] |
O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill Book Company, (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), 37.
doi: 10.1016/0022-247X(67)90163-1. |
[35] |
R. R. Merkovsky and D. E. Ward, General constraint qualifications in nondifferentiable programming,, Mathematical Programming, 47 (1990), 389.
doi: 10.1007/BF01580871. |
[36] |
L. Minchenko and S. Stakhovski, On relaxed constant rank regularity condition in mathematical programming,, Optimization, 60 (2011), 429.
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), 493.
doi: 10.1080/10556780410001709420. |
[38] |
D. W. Peterson, A review of constraint qualifications in finite-dimensional spaces,, SIAM Review, 15 (1973), 639.
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), 963.
doi: 10.1137/S1052623497326629. |
[40] |
K. Ritter, Optimization theory in linear spaces,, Mathematische Annalen, 184 (1970), 133.
doi: 10.1007/BF01350314. |
[41] |
R. T. Rockafellar, "Conjugate Duality and Optimization,", Lectures given at the Johns Hopkins University, (1973).
doi: 10.1137/1.9781611970524. |
[42] |
R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183.
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 (1998).
doi: 10.1007/978-3-642-02431-3. |
[44] |
M. Slater, Lagrange multipliers revisited,, Cowles Foundation Discussion Paper No. 80, (1950). Google Scholar |
[45] |
M. V. Solodov, Constraint qualifications,, in, (2010).
doi: 10.1002/9780470400531.eorms0978. |
[46] |
O. Stein, On constraint qualifications in nonsmooth optimization,, Journal of Optimization Theory and Applications, 121 (2004), 647.
doi: 10.1023/B:JOTA.0000037607.48762.45. |
[47] |
P. Varaiya, Nonlinear programming in Banach space,, SIAM Journal on Applied Mathematics, 15 (1967), 284.
doi: 10.1137/0115028. |
[48] |
W. I. Zangwill, "Nonlinear Programming: A Unified Approach,", Prentice-Hall International Series in Management, (1969).
|
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