Multiobjective programming with semilocally convex functions
Multiobjective programming, in recent times, has become an important area of investigation. We try to show naturally some of the uses of the class of semilocally convex functions in dealing with optimality conditions and duality theory as applied for treating a multiobjective mathematical program. The concept of semilocal convexity was introduced by G. M. Ewing (1977) who applied it to derive sufficient optimality conditions for variational and control problems. Semilocally convex functions have certain important convex type properties, e.g., local minima of semilocally convex functions defined on locally star-shaped sets are also global minima, and nonnegative linear combinations of semilocally convex functions are also semilocally convex. In Section 2 we introduce certain notations and mention the key lemmas which were relevant for single objective programming problems [see \textit{T. Weir}, ibid. 168, 1-12 (1992; Zbl 0762.90064)]. In the same section, some lemmas are given as statements in a different format. Finally, we state the alternative theorem derived by Weir which is an important proposition for deducing necessary optimality conditions. In Section 3 we introduce a multiobjective programming problem and we derive necessary saddle-point criteria for optimality for a semilocally convex program. Section 4 deals with Kuhn-Tucker necessary optimality conditions, sufficient optimality conditions, and duality for a semilocally convex multiobjective program. Some applications have been given in Section 5 for a multiobjective program with fractional objectives.
Kuhn-Tucker necessary optimality conditions
scientific article; zbMATH DE number 902550