December  2011, 31(4): 1069-1096. doi: 10.3934/dcds.2011.31.1069

Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling

1. 

Department of Mathematics & Computer Science, Northern Michigan University, Marquette, MI 49855, United States

2. 

Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

Received  November 2009 Revised  June 2010 Published  September 2011

This paper concerns new developments on first-order necessary conditions in set-valued optimization with applications of the results obtained to deriving refined versions of the so-called second fundamental theorem of welfare economics. It is shown that equilibrium marginal prices at local Pareto-type optimal allocations of nonconvex economies are in fact adjoint elements/ multipliers in necessary conditions for fully localized minimizers of appropriate constrained set-valued optimization problems. The latter notions are new in multiobjective optimization and reduce to conventional notions of minima for scalar problems. Our approach is based on advanced tools of variational analysis and generalized differentiation.
Citation: Truong Q. Bao, Boris S. Mordukhovich. Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1069-1096. doi: 10.3934/dcds.2011.31.1069
References:
[1]

T. Q. Bao and B. S. Mordukhovich, Necessary conditions for super minimizers in constrained multiobjective optimization, J. Global Optim., 43 (2009), 533-552. doi: 10.1007/s10898-008-9336-4.

[2]

T. Q. Bao and B. S. Mordukhovich, Relative Pareto minimizers in multiobjective optimization: Existence and optimality conditions, Math. Program., 122 (2010), 301-347. doi: 10.1007/s10107-008-0249-2.

[3]

S. Bellaassali and A. Jourani, Lagrange multipliers for multiobjective programs with a general preference, Set-Valued Anal., 16 (2008), 229-243. doi: 10.1007/s11228-008-0078-8.

[4]

J.-M. Bonnisseau and B. Cornet, Valuation equilibrium and Pareto optimum in nonconvex economies. General equilibrium theory and increasing returns, J. Math. Econ., 17 (1988), 293-308.

[5]

J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 20, Springer-Verlag, New York, 2005.

[6]

B. Cornet, The second welfare theorem in nonconvex economies, CORE discussion paper # 8630, 1986.

[7]

S. Dempe and V. Kalashnikov, eds., "Optimization with Multivalued Mappings: Theory, Applications and Algorithms," Springer Optim. Appl., 2, Springer, New York, 2006.

[8]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Math. Program. doi: 10.1007/s10107-010-0342-1.

[9]

M. Florenzano, P. Gourdel and A. Jofré, Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies, J. Economic Theory, 29 (2006), 549-564. doi: 10.1007/s00199-005-0033-y.

[10]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 17, Springer-Verlag, New York, 2003.

[11]

A. D. Ioffe, Variational analysis and mathematical economics, I: Subdifferential calculus and the second theorem of welfare economics, Adv. Math. Econ., 12 (2009), 71-95. doi: 10.1007/978-4-431-92935-2_3.

[12]

J. Jahn, "Vector Optimization. Theory, Applications and Extensions," Series in Operations Research and Decision Theory, Springer-Verlag, Berlin, 2004.

[13]

A. Jofré, A second-welfare theorem in nonconvex economies, in "Constructive, Experimental, and Nonlinear Analysis" (Limoges, 1999), 175–-184, CMS Conf. Proc., 27, Amer. Math. Soc., Providence, RI, 2000.

[14]

A. Jofré and J. R. Cayupi, A nonconvex separation property and some applications, Math. Program., 108 (2006), 37-51. doi: 10.1007/s10107-006-0703-y.

[15]

M. A. Khan, Ioffe's normal cone and the foundation of welfare economics: The infinite-dimensional theory, J. Math. Anal. Appl., 161 (1991), 284-298. doi: 10.1016/0022-247X(91)90376-B.

[16]

M. A. Khan, The Mordukhovich normal cone and the foundations of welfare economics, J. Public Economic Theory, 1 (1999), 309-338. doi: 10.1111/1097-3923.00014.

[17]

D. T. L/duc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.

[18]

A. Mas-Colell, "The Theory of General Economic Equilibrium. A Differentiable Approach," Econometric Society Monographs, 9, Cambridge University Press, Cambridge, MA, 1989.

[19]

B. S. Mordukhovich, An abstract extremal principle with applications to welfare economics, J. Math. Anal. Appl., 251 (2000), 187-216. doi: 10.1006/jmaa.2000.7041.

[20]

B. S. Mordukhovich, Nonlinear prices in nonconvex economics with classical Pareto and strong Pareto allocations, Positivity, 9 (2005), 541-568. doi: 10.1007/s11117-004-8076-z.

[21]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I: Basic Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330, Springer-Verlag, Berlin, 2006.

[22]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, II: Applications," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 331, Springer-Verlag, Berlin, 2006.

[23]

B. S. Mordukhovich, J. S. Treiman and Q. J. Zhu, An extended extremal principle with applications to multiobjective optimization, SIAM J. Optim., 14 (2003), 359-379. doi: 10.1137/S1052623402414701.

[24]

J. Quirk and R. Saposnik, "Introduction to General Equilibrium Theory and Welfare Economics," Economics Handbook Series, McGraw-Hill, New York, 1968.

[25]

R. T. Rockafellar, Directional Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3), 39 (1979), 331-355. doi: 10.1112/plms/s3-39.2.331.

[26]

P. A. Samuelson, "Foundations of Economic Analysis," Harvard University Press, Cambridge, MA, 1947.

[27]

Q. J. Zhu, Nonconvex separation theorem for multifunctions, subdifferential calculus and applications, Set-Valued Anal., 12 (2004), 275-290. doi: 10.1023/B:SVAN.0000023401.51035.28.

show all references

References:
[1]

T. Q. Bao and B. S. Mordukhovich, Necessary conditions for super minimizers in constrained multiobjective optimization, J. Global Optim., 43 (2009), 533-552. doi: 10.1007/s10898-008-9336-4.

[2]

T. Q. Bao and B. S. Mordukhovich, Relative Pareto minimizers in multiobjective optimization: Existence and optimality conditions, Math. Program., 122 (2010), 301-347. doi: 10.1007/s10107-008-0249-2.

[3]

S. Bellaassali and A. Jourani, Lagrange multipliers for multiobjective programs with a general preference, Set-Valued Anal., 16 (2008), 229-243. doi: 10.1007/s11228-008-0078-8.

[4]

J.-M. Bonnisseau and B. Cornet, Valuation equilibrium and Pareto optimum in nonconvex economies. General equilibrium theory and increasing returns, J. Math. Econ., 17 (1988), 293-308.

[5]

J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 20, Springer-Verlag, New York, 2005.

[6]

B. Cornet, The second welfare theorem in nonconvex economies, CORE discussion paper # 8630, 1986.

[7]

S. Dempe and V. Kalashnikov, eds., "Optimization with Multivalued Mappings: Theory, Applications and Algorithms," Springer Optim. Appl., 2, Springer, New York, 2006.

[8]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Math. Program. doi: 10.1007/s10107-010-0342-1.

[9]

M. Florenzano, P. Gourdel and A. Jofré, Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies, J. Economic Theory, 29 (2006), 549-564. doi: 10.1007/s00199-005-0033-y.

[10]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 17, Springer-Verlag, New York, 2003.

[11]

A. D. Ioffe, Variational analysis and mathematical economics, I: Subdifferential calculus and the second theorem of welfare economics, Adv. Math. Econ., 12 (2009), 71-95. doi: 10.1007/978-4-431-92935-2_3.

[12]

J. Jahn, "Vector Optimization. Theory, Applications and Extensions," Series in Operations Research and Decision Theory, Springer-Verlag, Berlin, 2004.

[13]

A. Jofré, A second-welfare theorem in nonconvex economies, in "Constructive, Experimental, and Nonlinear Analysis" (Limoges, 1999), 175–-184, CMS Conf. Proc., 27, Amer. Math. Soc., Providence, RI, 2000.

[14]

A. Jofré and J. R. Cayupi, A nonconvex separation property and some applications, Math. Program., 108 (2006), 37-51. doi: 10.1007/s10107-006-0703-y.

[15]

M. A. Khan, Ioffe's normal cone and the foundation of welfare economics: The infinite-dimensional theory, J. Math. Anal. Appl., 161 (1991), 284-298. doi: 10.1016/0022-247X(91)90376-B.

[16]

M. A. Khan, The Mordukhovich normal cone and the foundations of welfare economics, J. Public Economic Theory, 1 (1999), 309-338. doi: 10.1111/1097-3923.00014.

[17]

D. T. L/duc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.

[18]

A. Mas-Colell, "The Theory of General Economic Equilibrium. A Differentiable Approach," Econometric Society Monographs, 9, Cambridge University Press, Cambridge, MA, 1989.

[19]

B. S. Mordukhovich, An abstract extremal principle with applications to welfare economics, J. Math. Anal. Appl., 251 (2000), 187-216. doi: 10.1006/jmaa.2000.7041.

[20]

B. S. Mordukhovich, Nonlinear prices in nonconvex economics with classical Pareto and strong Pareto allocations, Positivity, 9 (2005), 541-568. doi: 10.1007/s11117-004-8076-z.

[21]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I: Basic Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330, Springer-Verlag, Berlin, 2006.

[22]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, II: Applications," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 331, Springer-Verlag, Berlin, 2006.

[23]

B. S. Mordukhovich, J. S. Treiman and Q. J. Zhu, An extended extremal principle with applications to multiobjective optimization, SIAM J. Optim., 14 (2003), 359-379. doi: 10.1137/S1052623402414701.

[24]

J. Quirk and R. Saposnik, "Introduction to General Equilibrium Theory and Welfare Economics," Economics Handbook Series, McGraw-Hill, New York, 1968.

[25]

R. T. Rockafellar, Directional Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3), 39 (1979), 331-355. doi: 10.1112/plms/s3-39.2.331.

[26]

P. A. Samuelson, "Foundations of Economic Analysis," Harvard University Press, Cambridge, MA, 1947.

[27]

Q. J. Zhu, Nonconvex separation theorem for multifunctions, subdifferential calculus and applications, Set-Valued Anal., 12 (2004), 275-290. doi: 10.1023/B:SVAN.0000023401.51035.28.

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