Second-order conditions for constrained vector optimization problems with \(\ell \)-stable data. (English) Zbl 1237.90221

Ginchev, Ivan; Guerraggio, Angelo

Optimization 60, No. 1-3, 179-199 (2011).

Summary: We consider the constrained vector optimization problem min\(_C f(x), g(x) \in -K\), where \(C \subset \mathbb R^m\) and \(K \subset \mathbb R^p\) are pointed closed convex cones, and \(f : \mathbb R^n \rightarrow \mathbb R^m\) and \(g : \mathbb R^n \rightarrow \mathbb R^p\) are \(\ell \)-stable at a point \(x^{0} \in \mathbb R^n\). We give second-order sufficient and necessary conditions \(x^{0}\) to be an \(i\)-minimizer (isolated minimizer) of order two, and second-order necessary conditions \(x^{0}\) to be a \(w\)-minimizer (weakly efficient point). The obtained results improve the ones of D. Bednařík and K. Pastor [Math. Program. 113, No. 2 (A), 283–298 (2008; Zbl 1211.90276)] (from unconstrained scalar problems to constrained vector problems) and I. Ginchev, A. Guerraggio and M. Rocca [Math. Program. 104, No. 2–3 (B), 389–405 (2005; Zbl 1102.90058)] (from problems with \(C^{1,1}\) data to problems with \(\ell \)-stable data). In fact, the former paper introduces and studies the notion of a \(\ell \)-stable at a point scalar function, and shows some possible applications. Here we generalize this notion to vector functions.

MSC:

90C29	Multi-objective and goal programming
90C30	Nonlinear programming
90C46	Optimality conditions and duality in mathematical programming
49J52	Nonsmooth analysis

Keywords:

vector optimization; \(\ell \)-stable functions; second-order conditions

Citations:

Zbl 1211.90276; Zbl 1102.90058

References:

[1]	Aubin J-P, Set-valued Analysis (1990)
[2]	DOI: 10.1137/S0363012903424174 · Zbl 1089.49023
[3]	DOI: 10.1007/s10107-007-0094-8 · Zbl 1211.90276
[4]	DOI: 10.1016/j.na.2007.09.006 · Zbl 1146.49017
[5]	DOI: 10.1016/j.na.2008.03.011 · Zbl 1158.49022
[6]	DOI: 10.1016/j.aml.2009.01.019 · Zbl 1162.49018
[7]	DOI: 10.1007/BF01585095 · Zbl 0488.90059
[8]	DOI: 10.1137/0328045 · Zbl 0714.49020
[9]	Diewert WE, Generalized Concavity in Optimization and Economics pp 51– (1981)
[10]	DOI: 10.1007/BF00425960 · Zbl 0864.49012
[11]	DOI: 10.1007/0-387-24276-7_27
[12]	DOI: 10.1007/s10107-005-0621-4 · Zbl 1102.90058
[13]	DOI: 10.1007/s10492-006-0002-1 · Zbl 1164.90399
[14]	DOI: 10.1023/A:1017519922669 · Zbl 1038.49027
[15]	DOI: 10.1023/A:1022114319999 · Zbl 1030.90115
[16]	DOI: 10.1007/BF01442169 · Zbl 0542.49011
[17]	DOI: 10.1016/S0022-247X(03)00337-8 · Zbl 1033.90120
[18]	DOI: 10.1080/02331938808843333 · Zbl 0647.49014
[19]	Liu L, J. Syst. Sci. Math. Sci. 4 pp 128– (1991)
[20]	DOI: 10.1023/A:1023068513188 · Zbl 0903.90152
[21]	DOI: 10.1023/A:1022272728208 · Zbl 0995.90085
[22]	Mangasarian OL, Classics in Applied Mathematics 10 (1994)
[23]	DOI: 10.1080/01630569308816543 · Zbl 0804.90114
[24]	DOI: 10.1080/02331939208843851 · Zbl 0814.49012