June  2018, 38(6): 3169-3188. doi: 10.3934/dcds.2018138

Normality and uniqueness of Lagrange multipliers

IIMAS, Universidad Nacional Autónoma de México, Apartado Postal 20-126, CDMX 01000, México

Who wants to be normal when you can be unique?
Helena Bonham Carter

Received  May 2017 Published  April 2018

In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.

Citation: Karla L. Cortez, Javier F. Rosenblueth. Normality and uniqueness of Lagrange multipliers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3169-3188. doi: 10.3934/dcds.2018138
References:
[1]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, (1990). Google Scholar

[2]

M. S. BazaraaH. D. Sherali and C. M. Shetty, Nonlinear Programming – Theory and Algorithms, John Wiley, New York, (1993). Google Scholar

[3]

J. A. Becerril and J. F. Rosenblueth, Necessity for isoperimetric inequality constraints, Discrete and Continuous Dynamical Systems -A, 37 (2017), 1129-1158. Google Scholar

[4]

J. A. Becerril and J. F. Rosenblueth, The importance of being normal, regular and proper in the calculus of variations, Journal of Optimization Theory & Applications, 172 (2017), 759-773. Google Scholar

[5]

A. Ben-Tal, Second-order and related extremality conditions in nonlinear programming, Journal of Optimization Theory & Applications, 31 (1980), 143-165. Google Scholar

[6]

G. Bigi and M. Castellani, Uniqueness of KKT multipliers in multiobjective optimization, Applied Mathematics Letters, 17 (2004), 1285-1290. Google Scholar

[7]

J. F. Bonnans, Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Applied Mathematics & Optimization, 29 (1994), 161-186. Google Scholar

[8]

K. L. Cortez and J. F. Rosenblueth, A second order constraint qualification for certain classes of optimal control problems, WSEAS Transactions on Systems and Control, 11 (2016), 419-424. Google Scholar

[9]

K. L. Cortez and J. F. Rosenblueth, Extended critical directions for time-control constrained problems, International Journal of Circuits, Systems and Signal Processing, 11 (2017), 1-11. Google Scholar

[10]

O. FujiwaraS. P. Han and O. L. Mangasarian, Local duality of nonlinear programs, SIAM Journal of Control and Optimization, 22 (1984), 162-169. Google Scholar

[11]

E. G. Gilbert and D. S. Bernstein, Second-order necessary conditions in optimal control: Accessory-problem results without normality conditions, Journal of Optimization Theory & Applications, 41 (1983), 75-106. Google Scholar

[12]

G. GiorgiA. Guerraggio and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, (2004). Google Scholar

[13]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, (1966). Google Scholar

[14]

M. R. Hestenes, Optimization Theory, The Finite Dimensional Case, John Wiley, New York, (1975). Google Scholar

[15]

J. Kyparisis, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Mathematical Programming, 32 (1985), 242-246. Google Scholar

[16]

J. Kyparisis, Sensitivity analysis for nonlinear programs and variational inequalities with non-unique multipliers, Mathematics of Operations Research, 15 (1990), 286-298. Google Scholar

[17]

F. Lempio and H. Maurer, Differential stability in infinite-dimensional nonlinear programming, Applied Mathematics & Optimization, 20 (1980), 139-152. Google Scholar

[18]

E. J. McShane, The Lagrange Multiplier Rule, The American Mathematical Monthly, 80 (1973), 922-925. Google Scholar

[19]

L. W. Neustadt, Optimization. A Theory of Necessary Conditions, Princeton University Press, Princeton, (1976). Google Scholar

[20]

J. F. Rosenblueth, Convex cones and conjugacy for inequality control constraints, Journal of Convex Analysis, 14 (2007), 361-393. Google Scholar

[21]

A. Shapiro, On uniqueness of Lagrange multipliers in optimization problems subject to cone constraints, Journal of Optimization, 7 (1997), 508-518. Google Scholar

[22]

G. Wachsmuth, On LICQ and the uniqueness of Lagrange multipliers, Operations Research Letters, 41 (2013), 78-80. Google Scholar

show all references

References:
[1]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, (1990). Google Scholar

[2]

M. S. BazaraaH. D. Sherali and C. M. Shetty, Nonlinear Programming – Theory and Algorithms, John Wiley, New York, (1993). Google Scholar

[3]

J. A. Becerril and J. F. Rosenblueth, Necessity for isoperimetric inequality constraints, Discrete and Continuous Dynamical Systems -A, 37 (2017), 1129-1158. Google Scholar

[4]

J. A. Becerril and J. F. Rosenblueth, The importance of being normal, regular and proper in the calculus of variations, Journal of Optimization Theory & Applications, 172 (2017), 759-773. Google Scholar

[5]

A. Ben-Tal, Second-order and related extremality conditions in nonlinear programming, Journal of Optimization Theory & Applications, 31 (1980), 143-165. Google Scholar

[6]

G. Bigi and M. Castellani, Uniqueness of KKT multipliers in multiobjective optimization, Applied Mathematics Letters, 17 (2004), 1285-1290. Google Scholar

[7]

J. F. Bonnans, Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Applied Mathematics & Optimization, 29 (1994), 161-186. Google Scholar

[8]

K. L. Cortez and J. F. Rosenblueth, A second order constraint qualification for certain classes of optimal control problems, WSEAS Transactions on Systems and Control, 11 (2016), 419-424. Google Scholar

[9]

K. L. Cortez and J. F. Rosenblueth, Extended critical directions for time-control constrained problems, International Journal of Circuits, Systems and Signal Processing, 11 (2017), 1-11. Google Scholar

[10]

O. FujiwaraS. P. Han and O. L. Mangasarian, Local duality of nonlinear programs, SIAM Journal of Control and Optimization, 22 (1984), 162-169. Google Scholar

[11]

E. G. Gilbert and D. S. Bernstein, Second-order necessary conditions in optimal control: Accessory-problem results without normality conditions, Journal of Optimization Theory & Applications, 41 (1983), 75-106. Google Scholar

[12]

G. GiorgiA. Guerraggio and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, (2004). Google Scholar

[13]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, (1966). Google Scholar

[14]

M. R. Hestenes, Optimization Theory, The Finite Dimensional Case, John Wiley, New York, (1975). Google Scholar

[15]

J. Kyparisis, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Mathematical Programming, 32 (1985), 242-246. Google Scholar

[16]

J. Kyparisis, Sensitivity analysis for nonlinear programs and variational inequalities with non-unique multipliers, Mathematics of Operations Research, 15 (1990), 286-298. Google Scholar

[17]

F. Lempio and H. Maurer, Differential stability in infinite-dimensional nonlinear programming, Applied Mathematics & Optimization, 20 (1980), 139-152. Google Scholar

[18]

E. J. McShane, The Lagrange Multiplier Rule, The American Mathematical Monthly, 80 (1973), 922-925. Google Scholar

[19]

L. W. Neustadt, Optimization. A Theory of Necessary Conditions, Princeton University Press, Princeton, (1976). Google Scholar

[20]

J. F. Rosenblueth, Convex cones and conjugacy for inequality control constraints, Journal of Convex Analysis, 14 (2007), 361-393. Google Scholar

[21]

A. Shapiro, On uniqueness of Lagrange multipliers in optimization problems subject to cone constraints, Journal of Optimization, 7 (1997), 508-518. Google Scholar

[22]

G. Wachsmuth, On LICQ and the uniqueness of Lagrange multipliers, Operations Research Letters, 41 (2013), 78-80. Google Scholar

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