J Glob Optim (2014) 59:191–205 DOI 10.1007/s10898-013-0103-9 Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems C. S. Lalitha · Prashanto Chatterjee Received: 11 October 2012 / Accepted: 19 August 2013 / Published online: 31 August 2013 © Springer Science+Business Media New York 2013 Abstract In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is intro- duced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and conti- nuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set. Keywords Levitin–Polyak well-posedness · Quasiconvexity · Efficiency · Upper semicontinuity · Hausdorff convergence Mathematics Subject Classification 49K40 · 90C26 · 90C29 Research of C. S. Lalitha was supported by R&D Doctoral Research Programme funds for university faculty. C. S. Lalitha Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road, New Delhi 110021, India e-mail: cslalitha@maths.du.ac.in P. Chatterjee (B ) Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi 110007, India e-mail: chatterjee.prashanto9@gmail.com 123