Optimality results for a specific fractional problem

In this paper, one minimizes a fractional function over a compact set. Using an exact separation theorem, one gives necessary optimality conditions for strict optimal solutions in terms of Frechet subdifferentials. All data are assumed locally Lipschitz.

1. Introduction. The separation theorems for convex sets play a key role in functional analysis and optimization theory. In fact, many crucial results with their proofs are based on separation arguments which are applied to convex sets ( see [6] ). In [8], Zheng, Yang and Zou proposed a related approach (an exact separation theorem) which can be considered as a generalization of the convex separation theorem to nonconvex sets and used as a powerful tool for deducing optimality conditions in nonconvex optimization. In order to use it, one should have an empty intersection between the sets and each set is considered near one of its elements; which is not the case in the extremal principle in [2].
In this paper, we are concerned with the following nonconvex fractional programming problem (P ) : g (x) Subject to : x ∈ Ω where X is an Asplund space, Ω is a compact subset of X, f : X → R and g : X → R are Lipshitz continuous functions such that g (x) = 0, for all x ∈ Ω.
The point x ∈ Ω is said to be a strict local optimal solution of (P ) iff there exists a neighborhood U of x such that With the help of the mentioned exact separation theorem [8] (see Theorem 2.3 below), one gives necessary optimality conditions for (P ) in terms of Fréchet subdifferentials. Throughout this work, we use standard notations. We denote by X * the topological dual of X with the canonical dual pairing ·, · ; (x, y) := x + y is the l 1 -norm of (x, y) ∈ X × X; B X and B X * stand for the closed unit balls in the space and dual space; and w * denotes the weak * topology on the dual space. For a multifunction F : X ⇒ X * , the expressions lim sup ∀k ∈ N signify, respectively, the sequential Painlevé-Kuratowski upper/outer and lower/inner limits in the norm topology in X and the weak * topology in X * ; N := {1, 2, . . .}.
The rest of the paper is organized in this way : Section 2 contains basic definitions and preliminary material from nonsmooth variational analysis. Section 3 addresses main results (optimality conditions). A conclusion is given Section 4.

2.
Preliminaries. In this section, we give some definitions, notations and results, which will be used in the sequel. For a subset D ⊆ X, cl D , co D, co D and cone D stand for the closure, the convex hull, the closure of the convex hull and the convex cone generated by D, repectively. The following definitions are crucials for our investigation.
Definition 2.1. [2] Let Ω ⊂ X be locally closed aroundx ∈ Ω. Then the Fréchet normal cone N (x; Ω) and the Mordukhovich normal cone N (x; Ω) to Ω atx are defined by Directly from (1) , it follows that for any locally closed set Ω and any x ∈ Ω The set Ω is called regular at x if (2) holds as equality. Remark that the set of regular sets includes all convex sets.
3. The singular subdifferential of ϕ at x is defined by Here, epi (ϕ) denotes the epigraph of ϕ defined by One clearly has Besides the classical cases of convex functions and those strictly differentiable at x (in particular, smooth functions), the latter class includes substantially broader collections of functions encountered in variational analysis and optimization; see [2,4,7]. Since X is Asplund, if ϕ is lower regular at x and locally Lipschitzian around this point, one gets ∂ϕ (x) = ∅; see [2, Corollary 2.25]. Remark that both the lower regularity and the locally Lipschitzity around x are automatic when ϕ is convex and continuous around x. The following result is a separation theorem, which can be considered as a generalization of the convex separation theorem to nonconvex sets. We will use it in our investigation of optimality conditions. Theorem 2.3. [8] Let X be an Asplund space and A, A 1 , ..., A n be nonempty closed (not necessarily convex) subsets of X such that A is compact and A∩ Then, for any ε ∈ ]0, +∞[ and ρ ∈ ]0, 1[ there exist a ∈ A, a i ∈ A i and a * i ∈ X * , i = 1, ..., n, such that the following statements hold: Here, γp (A1, ..., An, A) denotes the (p-weighted) non-intersect index of A1, ..., An, A defined by 3. Main result. Theorem 3.1 provides necessary optimality conditions for the optimization problem (P ) .
Theorem 3.1. Let x ∈ Ω be a strict local optimal solution of (P ) . Then, there exist sequences {v k } ⊂ Ω and {ω k } ⊂ Ω such that Since ϕ is lower semicontinuous and since Ω is compact, one deduces that A and A 1 are closed subsets of X×R, A is a compact subset of X×R and that ϕ (x) > −∞. Moreover, since (x, ϕ (x)) ∈ epi (ϕ) , one has • Let us prove that A ∩ A 1 = ∅.
By contrary, suppose that there exists x k ∈ Ω such that Then, and It follows from (3) that (v k , α k ) is not an interior point of epi (ϕ) ; consequently, Then, using (4) , one obtains the following inequalities : This implies that -Since w k ∈ Ω and since Ω is compact, taking a subsequence if necessary, we can assume that {w k } converges to a point w in Ω. Consequently, The continuity of the function ϕ allows us to deduce that ϕ (w) = ϕ (x) . Since x ∈ Ω is a strict optimal solution of of (P ) , one deduces that w = x.
Using the Lipschitz property of ϕ, we can find s * k ∈ ∂ϕ (v k ) and α * k ∈ N (Ω, w k ) such that where L is the Lipschitz constant of ϕ. Since X is Asplund and the sequence {s * k } is bounded, it is weak * sequentially compact. Letting k → ∞, taking a subsequence if necessary, we may assume that s * k w * → s * ∈ X * such that s * ∈ −N (Ω, x) , s * ≤ L and (s * , −1) ∈ N (gr (ϕ) , (x, ϕ (x))) .
By [4,Theorem 3.11], from (9) , we deduce that Since N (Ω, w k ) is a cone, one deduces that for every Finally, one gets Next we present a consequence of Theorem 3.1 in case g is convex continuous around x. Corollary 1. Let x ∈ Ω be a strict optimal solution of (P ) . Suppose that f (x) > 0 and that g is convex continuous around x. Then, there exist sequences {v k } ⊂ Ω and {ω k } ⊂ Ω such that and Proof. It is well known from convex analysis that the subdifferential of every convex function is nonempty at a point of continuity [5, Proposition 1.11].
4. Conclusion. In this paper, we are concerned with a fractional optimization problem (P ) . Assuming data locally Lipschitz, one investigates necessary optimality conditions. With the help of an exact separation theorem [8], one gives necessary optimality conditions for (P ) in terms of Fréchet subdifferentials.