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On the Second Order Optimality Conditions for Optimization Problems with Inequality Constraints

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DOI: 10.4236/ojop.2013.24014    3,355 Downloads   6,864 Views  
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A nonlinear optimization problem (P) with inequality constraints can be converted into a new optimization problem (PE) with equality constraints only. This is a Valentine method for finite dimensional optimization. We review second order optimality conditions for (PE) in connection with those of (P). A strictly complementary slackness condition can be made to get the property that sufficient optimality conditions for (P) imply the same property for (PE). We give some new results (see Theorems 3.1, 3.2 and 3.3) .Without any assumption, a counterexample is given to show that these conditions are not equivalent.

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The authors declare no conflicts of interest.

Cite this paper

M. Naffouti, "On the Second Order Optimality Conditions for Optimization Problems with Inequality Constraints," Open Journal of Optimization, Vol. 2 No. 4, 2013, pp. 109-115. doi: 10.4236/ojop.2013.24014.


[1] F. Jhon, “Extremum Problems with Inequalities as Side Conditions, Studies and Essays, Courant Anniversary Volume,” Wiley-Interscience, Hoboken, 1948.
[2] E. S. Levitin, A. A. Milyutin and N. P. Osmolovskii, “Conditions of High Order for a Local Minimum in Problems with Constraints,” Russian Mathematical Surveys, Vol. 33, No. 6, 1978, pp. 97-168.
[3] A. Ioffe, “Necessary and Sufficient Conditions for a Local Minimum. 3: Second Order Conditions and Augmented Duality,” SIAM Journal of Control and Optimization, Vol. 17, No. 2, 1979, pp. 266-288.
[4] F. A. Valentine, “The Problem of Lagrange with Differentiable Inequalities as Side Consrtaints, Contribution to the Calculus of Variation 1933-1937,” University of Chicago Press, Chicago, 1937, pp. 407-448.
[5] J. B. Hiriart-Urruty, “Optimisation et Analyse Convexe, Exercices Corrigées,” EDP Sciences, 2009.
[6] L. D. Berkovitz, “Variational Methods of Control and Programming,” Journal of Mathematical Analysis and Applications, Vol. 3, No. 1, 1961, pp. 145-169.
[7] K. G. Murty and S. N. Kabadi, “Some NP-Complete Problems in Quadratic and Nonlinear Programming,” Mathematical Programming, Vol. 39, No. 2, 1987, pp. 117-129.
[8] Y. Chabrillac and J. P. Crouzeix, “Definiteness and Semidefiniteness of Quadratic Forms Revisited,” Linear Algebra and Its Applications, Vol. 63, No. 1, 1984, pp. 283-292.
[9] A. Baccari, “On the Classical Necessary Second-Order Optimality Conditions,” Journal of Optimization Theory and Applications, Vol. 123, No. 1, 2004, pp. 213-221.
[10] A. Baccari and A. Trad, “On the Classical Necessary Second-Order Optimality Conditions in The Presence of Equality and Inequality Constraints,” SIAM. Journal of Optimization, Vol. 15, No. 2, 2004, pp. 394-408.
[11] R. Andreani, J. M. Martinez and M. L. Schuverdt, “On Second-Order Optimality Conditions for Nonliear Programming,” Optimization, Vol. 56, No. 5-6, 2007, pp. 529-542.
[12] M. Daldoul and A. Baccari, “An Application of Matrix Computations to Classical Second-Order Optimality Conditions,” Optimization Letters, Vol. 3, No. 4, 2009, pp. 547-557.
[13] A. Ben-Tal and J. Zowe, “A Unified Theory of First and Second Order Conditions for Extremum Problems in Topological Vector Spaces,” Mathematical Programming Study, Vol. 19, 1982, pp. 39-76.
[14] J. F. Bonnans and A. Shapiro, “Perturbation Analysis of Optimization Problems,” Springer, Berlin, 2000.
[15] 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, Vol. 17, 1967, pp. 37-47.
[16] D. P. Bertsekas, “Constrained Optimization and Lagrange Multiplier Methods,” Academic Press, Cambridge, 1982.

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