  Second-order optimality conditions for vector nonlinear programming problems
with inequality constraints are studied in this paper. We introduce a new
second-order constraint qualification, which includes Mangasarian-Fromovitz
constraint qualification as a particular case. We obtain necessary and
sufficient conditions for weak efficiency of problems with a second-order
pseudoconvex vector objective function and quasiconvex constraints. We also
derive Lagrange multiplier characterizations of the solution set of a scalar
problem with a second-order pseudoconvex objective function and quasiconvex
inequality constraints, provided that one of the solutions and the Lagrange
multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we
introduce a notion of a second-order KKT-pseudoconvex problem with inequality
constraints. We derive sufficient and also necessary conditions for efficiency
of second-order KKT-pseudoconvex problems. Three examples are presented.
