Abstract
This work is concerned with differentiable constrained vector optimization problems. It focus on the intrinsic connection between positive linearly dependent gradient sets and the distinct notions of regularity that come to play in this context. The main aspect of this contribution is the development of regularity conditions, based on the positive linear dependence or independence of gradient sets, for problems with general nonlinear constraints, without any convexity hypothesis. Being easy to verify, these conditions might be useful to define termination criteria in the development of algorithms.
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References
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)
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)
Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer, London (1999)
Maeda, T.: Constraint qualifications in multiobjective optimization problems: Differentiable case. J. Optim. Theory Appl. 80, 483–500 (1994)
Castellani, M., Mastroeni, G., Pappalardo, M.: Separation of sets, Lagrange multipliers and totally regular extremum problems. J. Optim. Theory Appl. 92, 249–261 (1997)
Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)
Bigi, G., Pappalardo, M.: Regularity conditions for the linear separation of sets. J. Optim. Theory Appl. 99, 533–540 (1998)
Bigi, G., Pappalardo, M.: Regularity conditions in vector optimization. J. Optim. Theory Appl. 102, 83–96 (1999)
Preda, V., Chitescu, I.: On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl. 100, 417–433 (1999)
Giorgi, G., Jiménez, B., Novo, V.: On constraint qualifications in directionally differentiable multiobjective optimization problems. RAIRO Oper. Res. 38, 255–274 (2004)
Davis, Ch.: Theory of positive linear dependence. Am. J. Math. 76, 733–746 (1954)
Stoer, J., Witzgall, Ch.: Convexity and Optimization in Finite Dimensions I. Springer, New York (1970)
Da Cunha, N.O., Polak, E.: Constraint minimization under vector-value criteria in finite dimensional spaces. J. Math. Anal. Appl. 19, 103–124 (1967)
Bigi, G.: Optimality and Lagrangian regularity in vector optimization. Ph.D. thesis, Dipartimento di Matematica, Università di Pisa, Italy. http://www.di.unipi.it/~bigig/mat/abstracts/thesis.html (1999)
Robinson, S., Meyer, R.: Lower semicontinuity of multivalued linearization mapping. SIAM J. Control 11, 525–533 (1973)
Robinson, S.: First order conditions for general nonlinear optimization. SIAM J. Appl.. Math. 30, 597–607 (1976)
Mangasarian, O.L.: Nonlinear Programming. SIAM, Philadelphia (1994)
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)
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Communicated by F. Giannessi.
This work was supported by Fundación Antorchas, Grant 13900/4; by Southern National University, Grant UNS 24/L069; by Comahue National University, Grant E060/04; by CNPq-Brazil (Grants 302412/2004-2, 473586/2005-5 and 303465/2007-7) and by FAPESP-Brazil (06/53768-0).
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Maciel, M.C., Santos, S.A. & Sottosanto, G.N. Regularity Conditions in Differentiable Vector Optimization Revisited. J Optim Theory Appl 142, 385–398 (2009). https://doi.org/10.1007/s10957-009-9519-2
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DOI: https://doi.org/10.1007/s10957-009-9519-2