Invexity in Nonlinear Programming
- PDF / 382,956 Bytes
- 41 Pages / 439.192 x 666.032 pts Page_size
- 120 Downloads / 198 Views
5.1 Invexity in Necessary and Sufficient Optimality Conditions Hanson’s [83] introduction of invex functions was motivated by the purpose to weaken further the class of mathematical programming problems for which the necessary optimality conditions are also sufficient. There was also the related problem of finding the widest class of functions for which weak and strong duality hold for the dual problems, such as the Wolfe dual problem or the Mond–Weir dual problem. Let us consider the following basic nonlinear programming problem: (P) Minimize f (x), x∈K K = {x : x ∈ C, g(x) ≤ 0}, where f : C → R and g : C → Rm are (Frechet) differentiable on the open set C ⊆ Rn (if we have a problem with equality constraints, of the type h(x) = 0, h : C → Rp , we could re-write these constraints as h(x) ≤ 0, −h(x) ≤ 0). It is well known that under certain regularity assumptions on the vector function g (constraint qualifications) the Karush–Kuhn–Tucker conditions are necessary for optimality in (P), that is, if x∗ is a solution of (P) or even if it is a point of local minimum of f on K, then there exists a vector λ∗ ∈ Rm such that ∇f (x∗ ) + λ∗ T ∇g(x∗ ) = 0
(5.1)
λ∗ T g(x∗ ) = 0
(5.2)
λ∗ T ≥ 0.
(5.3)
It is also well known that relations (5.1)–(5.3) become sufficient for optimality if some (generalized) convexity assumption is made on f and g.
74
5 Invexity in Nonlinear Programming
More precisely, if (x∗ , λ∗ ), with x∗ ∈ K, satisfies (5.1)–(5.3), then x∗ is optimal for (P), provided one of the following assumptions is imposed: (a) f (x) and gi (x) convex, i = 1, . . . , m [133] (b) f (x) pseudo-convex and gi (x) quasi-convex, with i ∈ I = {i : gi (x) = 0} the set of the active or effective constraints at x∗ ∈ K [142, 143] (c) f (x) pseudo-convex and λ∗ T g(x) quasi-convex [169] (d) f (x) + λ∗ T g(x) pseudo-convex [169] Hanson [83] noted that the (generalized) convexity requirements appearing in the (a)–(d) above can be further weakened as in the related proofs of the sufficiency for problem (P) there is no explicit dependence on the linear term (x − m), appearing in the definition of differentiable convex, pseudo-convex and quasi-convex functions. Thus this linear term can be substituted with an arbitrary vector-valued function. More precisely, if x∗ ∈ K and (x∗ , λ∗ ) satisfies (5.1)–(5.3), then x∗ solves (P) if any one of the following conditions is satisfied: (a) f (x) and every gi (x), i ∈ I, are invex with respect to the same η. (b) f (x) is pseudo-invex and every gi (x), i ∈ I, is quasi-invex with respect to the same η. (c) f (x) is pseudo-invex and λ∗ g(x) is quasi-invex with respect to the same η. (d) The Lagrangian function f (x) + λ∗ T g(x) is pseudo-invex with respect to an arbitrary η (i.e., the Lagrangian function f (x) + λ∗ T g(x) is invex). The proofs are easy. We give only the proof for (a), the original result of Hanson [83]: For any x ∈ C satisfying g(x) ≤ 0, we have f (x) − f (x∗ ) ≤ η(x, x∗ )T ∇f (x∗ ) = −η(x, x∗ )T ∇(λ∗ T g(x)) ≥ −λ∗ T (g(x) − g(x∗ )) = −λ∗ T g(x) ≥ 0. So x∗ is minimal. We stress the fact that f and
Data Loading...
