Summary: We investigate sufficient optimality conditions for a general multiobjective optimization problem with both inequality constraints and equality contraints. Three concepts on vector generalized convexity are introduced which can be useful for extending sufficient optimality conditions (previously restricted only to convex programs) to larger classes of multiobjective optimization problems.
We introduce a new dass of functions for which the Kuhn-Tucker conditions are sufficient for efficiency. We give counterexamples to show that the sufficient optimality Theorems 3.1 and 3.3 given recently in A. A. K. Majumdar [J. Optimization Theory Appl. 92, No. 2, 419–427 (1997; Zbl 0886.90122)] does not hold. Based on these remarks, three sufficient optimality conditions are established for nonlinear multiobjective optimization problems involving this new class of functions. The first one extends the result of [C. Singh, J. Optimization Theory Appl. 53, 115–123 (1987; Zbl 0593.90071) Theorem 3.2]. The second theorem, requires the generalized invexity of the functions involved extends the first one. The later theorem extends in fact, the result of [I. Marusciac, Math., Rev. Anal. Numér. Theor. Approx. 11, 109–114 (1982; Zbl 0501.90081)].