Abstract
In this paper we give necessary conditions for Hartley proper efficiency in a vector optimization problem whose objectives and constraints are described by nonconvex locally Lipchitz set-valued maps. The obtained necessary conditions are written in terms of a Lagrange multiplier rule. Our approach is based on a reduction theorem which leads the problem of studying proper efficiency to a scalar optimization problem whose objective is given by a function of max-type. Sufficient conditions for Hartley proper efficiency are also considered.
Similar content being viewed by others
References
A. Auslender (1984) ArticleTitleStability in mathematical programming with nondifferentiable data SIAM Journal on Control and Optimization 22 239–254 Occurrence Handle10.1137/0322017
H.P. Benson (1979) ArticleTitleAn improved definition of proper efficiency for vector maximization with respect to cones Journal of Mathematical Analysis and Applications 71 232–241 Occurrence Handle10.1016/0022-247X(79)90226-9
J.M. Borwein (1977) ArticleTitleProper efficient points for maximizations with respect to cones SIAM Journal on Control and Optimization 15 57–63 Occurrence Handle10.1137/0315004
J.M. Borwein (1980) ArticleTitleThe geometry of Pareto efficiency over cones Mathematische Operationforschung und Statistik, Serie Optimization 11 235–248
J.M. Borwein D.M. Zhuang (1991) ArticleTitleSuper efficiency in convex vector optimization Mathematical Methods of Operations Research 35 175–184 Occurrence Handle10.1007/BF01415905
J.M. Borwein D.M. Zhuang (1993) ArticleTitleSuper efficiency in vector optimization Transactions of the American Mathematical Society 338 105–122 Occurrence Handle10.2307/2154446
F.H. Clarke (1983) Optimization and Nonsmooth Analysis Wiley-Interscience New York
Dien, P.H. (1983), On locally Lipschitz maps and general extremal problems with inclusion constraints, PhD Thesis, Hanoi Institute of Mathematics.
P.H. Dien (1983) ArticleTitleLocally Lipschitz set-valued mappings and generalized extremal problems Acta Mathematica Vietnamica 8 IssueID2 109–122
A.M. Geoffrion (1968) ArticleTitleProper efficiency and the theory of vector maximization Journal of Mathematical Analysis and Applications 22 618–630 Occurrence Handle10.1016/0022-247X(68)90201-1
X.H. Gong H.B. Dong S.Y. Wang (2003) ArticleTitleOptimality conditions for proper efficient solutions of vector set-valued optimization Journal of Mathematical Analysis and Applications 284 332–350 Occurrence Handle10.1016/S0022-247X(03)00360-3
A. Guerraggio E. Molho A. Zaffaroni (1994) ArticleTitleOn the notion of proper efficiency in vector optimization Journal of Optimization Theory and Applications 82 1–21 Occurrence Handle10.1007/BF02191776
R. Hartley (1978) ArticleTitleOn cone-efficiency, cone-convexity, and cone-compactness SIAM Journal on Applied Mathematics 34 211–222 Occurrence Handle10.1137/0134018
M.I. Henig (1982) ArticleTitleProper efficiency with respect to cones Journal of Optimization Theory and Applications 36 387–407 Occurrence Handle10.1007/BF00934353
X.H. Huang X.Q. Yang (2002) ArticleTitleOn characterizations of proper efficiency for nonconvex multiobjective optimization Journal of Global Optimization 23 213–231 Occurrence Handle10.1023/A:1016522528364
D.S. Kim G.M. Lee P.H. Sach (2004) ArticleTitleHartley proper efficiency in multifunction optimization Journal of Optimization Theory and Applications 120 129–145 Occurrence Handle10.1023/B:JOTA.0000012736.02360.58
H.W. Kuhn A.W. Tucker (1950) Nonlinear Programming, Proc. Second Berkeley Sympesium on Mathematical Statistics and Probability University of California Press Berkeley, California 481–492
G.M. Lee D.S. Kim P.H. Sach (2005) ArticleTitleCharacterizations of Hartley proper efficiency in nonconvex vector optimization Journal of Global Optimization 33 273–298 Occurrence Handle10.1007/s10898-004-1935-0
Z.F. Li (1998) ArticleTitleBenson proper efficiency in the vector optimization of set-valued maps Journal of Optimization Theory and Applications 98 623–649 Occurrence Handle10.1023/A:1022676013609
A. Mehra (2002) ArticleTitleSupper efficiency in vector optimization with nearly convexlike set-valued maps Journal of Mathematical Analysis and Applications 276 815–832 Occurrence Handle10.1016/S0022-247X(02)00452-3
W.D. Rong Y.N. Wu (1998) ArticleTitleCharacterization of supper efficiency in cone-convexlike vector optimization with set-valued maps Mathematical Methods of Operations Research 48 247–258 Occurrence Handle10.1007/s001860050026
P.H. Sach (2003) ArticleTitleNearly subconvexlike set-valued maps and vector optimization problems Journal of Optimization Theory and Applications 119 335–356 Occurrence Handle10.1023/B:JOTA.0000005449.20614.41
P.H. Sach B.D. Craven (1991) ArticleTitleInvex multifunctions and duality Numerical Functional Analysis and Optimizations 12 575–591
P.H. Sach G.M. Lee D.S. Kim (2003) ArticleTitleInfine functions, nonsmooth alternative theorems and vector optimization problems Journal of Global Optimization 27 51–81 Occurrence Handle10.1023/A:1024698418606
Y. Sawaragi H. Nakayama T. Tanino (1985) Theory of Multiobjective Optimization Academic Press New York
W. Song (1998) ArticleTitleDuality in set-valued optimization Dissertations Mathematicae 375 1–68
X.M. Yang D. Li S.Y. Wang (2001) ArticleTitleNear-Subconvexlikeness in vector optimization with set-valued functions Journal of Optimization Theory and Applications 110 413–427 Occurrence Handle10.1023/A:1017535631418
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sach, P.H. Hartley Proper Efficiency in Multiobjective Optimization Problems with Locally Lipschitz Set-valued Objectives and Constraints. J Glob Optim 35, 1–25 (2006). https://doi.org/10.1007/s10898-005-1652-3
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-005-1652-3