Abstract
In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Gárciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118, 117–133 (2003)
Schuverdt, M.L.: Métodos de Lagrangiano aumentado com convergência utilizando a condição de dependência linear positiva constante. Ph.D. Thesis, UNICAMP (2006)
Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization (2010). doi:10.1080/02331930903578700
Haeser, G.: Condições sequenciais de otimalidade. Ph.D. Thesis, UNICAMP (2009)
Martínez, J.M., Pilotta, E.A.: Inexact restoration algorithms for constrained optimization. J. Optim. Theory Appl. 104, 135–163 (2000)
Martínez, J.M., Pilotta, E.A.: Inexact restoration methods for nonlinear programming: advances and perspectives. In: Qi, L.Q., Teo, K.L., Yang, X.Q. (eds.) Optimization and Control with Applications, pp. 271–292. Springer, Berlin (2005)
Andreani, R., Martínez, J.M., Schuverdt, M.L.: On the relation between the Constant Positive Linear Dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)
Andreani, R., Martínez, J.M., Schuverdt, M.L.: On second-order optimality conditions for nonlinear programming. Optimization 56, 529–542 (2007)
Gárciga-Otero, R., Svaiter, B.F.: A new condition characterizing solutions of variational inequality problems. J. Optim. Theory Appl. 137, 89–98 (2008)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower–level constraints. SIAM J. Control Optim. 18, 1286–1309 (2007)
Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Control Optim. 10, 963–981 (2000)
Janin, R.: Direction derivative of the marginal function in nonlinear programming. Math. Program. Stud. 21, 127–138 (1984)
Mangasarian, O.L., Fromovitz, S.: The Fritz-John necessary optimality conditions in presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)
Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problem. Part 1: Sufficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009)
Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problem. Part 2: Necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009)
Giannessi, F.: Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)
Hestenes, M.R.: Optimization Theory—The Finite-Dimensional Case. Wiley, New York (1975)
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Abadie, J.: On the Kuhn-Tucker theorem. In: Abadie, J. (ed.) Nonlinear Programming, pp. 21–36. North-Holland, Amsterdam (1967)
Iusem, A.N., Nasri, M.: Augmented Lagrangian methods for variational inequality problems. RAIRO. Rech. Opér. 44, 5–25 (2010)
Auslender, A., Teboulle, M.: Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Control Optim. 10, 1097–1115 (1999)
Martínez, J.M.: Inexact restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming. J. Optim. Theory Appl. 111, 39–58 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Di Pillo.
We are indebted to two anonymous referees for insightful comments and suggestions.
G. Haeser was supported by CNPq Grant 503328/2009-0 and FAPESP Grant 09/09414-7.
M.L. Schuverdt was supported by PRONEX-CNPq/FAPERJ Grant E-26/171.164/2003.
Rights and permissions
About this article
Cite this article
Haeser, G., Schuverdt, M.L. On Approximate KKT Condition and its Extension to Continuous Variational Inequalities. J Optim Theory Appl 149, 528–539 (2011). https://doi.org/10.1007/s10957-011-9802-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9802-x