����������� ������ 55 (2003), 21–30

UDK 512.644

���������� ������ ��� research paper

OPTIMALITY CONDITIONS AND TOLAND’S DUALITY FOR A NONCONVEX MINIMIZATION PROBLEM

M. Laghdir

Abstract. This paper studies necessary and sufficinet conditions and provides a duality theory for a wide class of problems arising in nonconvex optimization, such as minimizing a difference of two convex functions subject to a convex vector constraint taking values in an ordered topological vector space. These results are then used to study a problem of nondifferentiable optimization.

1. Introduction

Many of the nonconvex minimization problems arising in applied mathematics, operations research and mathematical programming can be formulated as the following nonconvex problem

(P) \{ \begin{matrix}inf f_{1}(x) − f_{2}(x) \\ h(x) ∈ −Y_{+} \\ x ∈ C, \end{matrix}

where X and Y are two real topological vector spaces and Y is equipped with a preorder induced by a convex cone Y_{+}, f_{1}, f_{2} : X ⟶ R ∪ {+∞} are two convex functions and h: X ⟶ Y ∪ {+∞} is a convex mapping and C is a nonempty convex subset of X. The problem (P) includes a wide family of DC-minimization problems subject to vector constraint taking values in finite or infinite dimensional spaces. The purpose of this paper is to present the optimality conditions related to the problem (P) and to formulate its Toland’s dual problem.

The main tools used to deal with this class of problems are the formulas of the subdifferential and Legendre-Fenchel conjugate function of the composition of a nondecreasing convex function with a convex mapping taking values in an ordered topological vector space (see [1] and [2]).

AMS Subject Classification: 49 K 27, 90 C 25

Keywords and phrases: DC-minimization, optimality conditions, Toland’s duality, fractional programmation, partial order.

This paper is organised as follows. In section 2 we give some notations and recall some definitions and results. Section 3 studies necessary and sufficient optimality conditions linked to problem (P). In Section 4, we deal with the dual problem of (P) in the sense of Toland’s duality. In Section 5, we give an applications to nondifferentiable fractional programming problem.

2. Notations and Preliminaries

All the vector spaces introduced in the sequel are real spaces. By X and Y we denote two topological vector spaces with respective topological duals X^{∗} and Y^{∗}. The canonical bilinear form on X^{∗} × X (resp. Y^{∗} × Y) is denoted by ⟨·, ·⟩. The space Y will be assumed to be equipped with a preorder ≤_{Y} induced by closed convex cone Y_{+} as follows

y_{1} ≤_{Y} y_{2} ⟺ y_{2} − y_{1} ∈ Y_{+},

and an abstract maximal element +∞ will be adjoined to Y. By Y_{+}^{∗} we denote the dual positive cone, i.e., the cone of positive linear functionals on Y given by

Y_{+}^{∗} := { y^{∗} ∈ Y^{∗} : ⟨y^{∗}, y⟩ ≥ 0, ∀y ∈ Y_{+} }.

Recall that an operator H: X ⟶ Y ∪ {+∞} is said to be Y_{+}-convex if

H(αx_{1} + (1 − α)x_{2}) ≤_{Y} αH(x_{1}) + (1 − α)H(x_{2})

for each x_{1}, x_{2} in X and for each α ∈ [0, 1]. By dom H := { x ∈ X : H(x) ∈ Y }, Epi H := { (x, y) ∈ X × Y : H(x) ≤_{Y} y } and Im H := H(dom H) we denote, respectively, the effective domain, the epigraph and the effective range of H. A function G: Y ⟶ R ∪ {+∞} is said to be Y_{+}-nondecreasing on Y if

y_{1} ≤_{Y} y_{2} ⟹ G(y_{1}) ≤ G(y_{2}).

Given F: X ⟶ R ∪ \{+∞\}, the subdifferential of F at \overline{x} ∈ dom F, denoted by ∂F (\overline{x}), is defined as

∂F (\overline{x}) := \{ x^{∗} ∈ X^{∗} : F (x) ≥ F (x̄) + ⟨x^{∗}, x − x̄⟩, ∀x ∈ X \}

and its conjuguate function F^{∗}: X^{∗} ⟶ R ∪ {+∞} is defined by

F^{∗}(x^{∗}) := sup_{x∈X}{ ⟨x^{∗}, x⟩ − F (x) }.

If C is a nonempty subset of X, then the cone that it generates is

R_{+}C := ⨆_{λ≥0} λC,

its indicator function is δ_{C}: X ⟶ R ∪ {+∞} defined for every x ∈ X by

δ_{C}(x) := \{ \begin{matrix}0, & if x ∈ C \\ +∞, & otherwise, \end{matrix}

and its support function δ_{C}^{∗} defined on the dual space X^{∗} is

δ_{C}^{∗}(x^{∗}) := sup_{x∈C}⟨x^{∗}, x⟩.

Also, we define the cone of normal directions to C at x_{0} ∈ C by

N_{C}(x̄) := { x^{∗} ∈ X^{∗} : ⟨x^{∗}, x − x̄⟩ ≤ 0, ∀x ∈ C }.

Throughout, we will adopt two conventions; the first is (+∞) − (+∞) = +∞ and the other is linked to the composite function, that is

x ∈ X ↦ (G∘H)(x) := \{ \begin{matrix}G(H(x)), & if x ∈ dom H \\ +∞, & otherwise. \end{matrix}

Now, let us recall some definitions and results that will be used throughout the paper.

Definition 2.1. [4]

(i) A function f: X ⟶ R ∪ {+∞} is called a DC-function if it can be represented as a difference of two convex functions defined on X.

(ii) f: X ⟶ R ∪ {+∞} is called a polyhedral convex function if

f(x) = max_{1≤i≤n}{ ⟨x_{i}^{∗}, x⟩ + b_{i} }, ∀x ∈ X

where x_{1}^{∗}, x_{2}^{∗},…, x_{n}^{∗} ∈ X^{∗} and b_{1}, b_{2},…, b_{n} ∈ R.

In order to establish our main results we shall need the following results due respectively to J. B. Hiriarty Urruty [4] and J. F. Toland [5]. These results established respectively the local optimality conditions and Toland’s dual problem related to an unconstrained DC-mathematical programming problem.

Proposition 2.1. [4] (i) Let us consider two convex functions f_{1} and f_{2} : X ⟶ R ∪ \{+∞\}. The condition ∂f_{2}(\overline{x}) ⊂ ∂f_{1}(\overline{x}) is necessary for \overline{x} being a local minimum of f = f_{1} − f_{2} on X.

(ii) If, furthermore, we assume that f_{2} is a polyhedral function, then the condition ∂f_{2}(\overline{x}) ⊂ ∂f_{1}(\overline{x}) is necessary and sufficient for x̄ being a local minimum of f = f_{1} − f_{2} on X.

Proposition 2.2. [5] Let f_{1} : X ⟶ R∪{+∞} be any function and f_{2} : X ⟶ R ∪ {+∞} be a convex and lower semicontinuous function. We have

inf_{x∈X}{f_{1}(x) − f_{2}(x)} = inf_{x^{∗}∈X^{∗}}{f_{2}^{∗}(x^{∗}) − f_{1}^{∗}(x^{∗})}.

We finish this section by recalling some formulas established in [1] (see also [2]) by C. Combari, M. Laghdir and L. Thibault, concerning the computation of the subdifferential and conjugate function of the composition of a nondecreasing convex function with a convex mapping taking values in a partially ordered topological vector space. For this, let us consider the following constraint qualification

(C.Q.A_{0}.B_{0}) \{ \begin{matrix}X and Y are Fréchet spaces, \\ G is convex, proper, lower semicontinuous and \\ Y_{+}-nondecreasing on Im h + Y_{+}, \\ H is Y_{+}-convex, proper with closed epigrah in X × Y, \\ R_{+}[dom G − Im(domF ∩ dom H)] is a closed vector subspace. \end{matrix}

Theorem 2.1. [1] If the condition (C.Q.A_{0}.B_{0}) holds then we have:

i) ∂(F + G ∘ H)(x̄) = ⨆_{y^{∗}∈∂G(H(x̄))} ∂(F + y^{∗} ∘ H)(x̄), for any x̄ ∈ X and

ii) (F + G ∘ H)^{∗}(x^{∗}) = min_{y^{∗}∈Y_{+}^{∗}}[G^{∗}(y^{∗}) + (F + y^{∗} ∘ H)^{∗}(x^{∗})], for any x^{∗} ∈ X^{∗}.

Remark 2.1. The above proposition holds also (see [1]), if we suppose, instead of the condition (C.Q.A_{0}.B_{0}.), that X and Y are locally convex topological real linear spaces and G is finite and continuous at some point ȳ = h(x̄) where x̄ ∈ C ∩ dom f_{1} ∩ dom h.

3. Optimality conditions

In order to establish the optimality conditions related to problem (P) we will need the following constraint qualification

(C.Q.A_{1}.B_{1})\{ \begin{matrix}X and Y are Fréchet spaces, \\ f_{1} and f_{2} are convex, proper and lower semicontinuous \\ h is convex and Epi h is a closed subset of X × Y \\ R_{+}[Y_{+} + h(dom f_{1} ∩ C ∩ dom h)] is a closed vector subspace. \end{matrix}

Now, we are ready to state the necessary optimality conditions.

Proposition 3.1. Let x̄ be a feasible point of (P), i.e., x̄ ∈ C ∩ h^{−1}(−Y_{+}). We suppose that the constraint qualification (C.Q.A_{1}.B_{1}) is satisfied. If x̄ is a local minimum for the problem (P) then for each x^{∗} ∈ ∂f_{2}(x̄) there exists some y^{∗} ∈ Y_{+}^{∗} satisfying ⟨y^{∗}, h(x̄)⟩ = 0 and x^{∗} ∈ ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄).

Proof. First, let us notice that the function y ↦ δ_{−Y_{+}}(y) is convex and Y_{+}- nondecreasing on the whole space (see also [1]) and with the convexity of mapping h we obtain that the composite function is also convex.

It is easy to see that a feasible point x̄ of (P) is a local minimum of (P) if and only if x̄ is a local minimum of the following unconstrained DC-minimization problem (Q)

(Q) inf_{x∈X}{f_{1}(x) + δ_{C}(x) + (δ_{−Y_{+}} ∘ h)(x) − f_{2}(x)}.

Hence, by applying Proposition 2.1, it follows that

∂f_{2}(x̄) ⊂ ∂(f_{1} + δ_{C} + δ_{−Y_{+}} ∘ h)(x̄),

and according to Theorem 2.1, the condition (C.Q.A_{1}.B_{1}) ensures that

∂f_{2}(x̄) ⊂ ⨆_{\begin{matrix}y^{∗}∈N(h(x̄),−Y_{+}) \\ ⟨y^{∗},h(x̄)⟩=0 \end{matrix}} ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄),

i.e., for each x^{∗} ∈ ∂f_{2}(x̄), there exists some y^{∗} ∈ Y_{+}^{∗} such that ⟨y^{∗}, h(x̄)⟩ = 0 and x^{∗} ∈ ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄).

By taking into account Remark 2.1 we obtain the same result as above in the setting of locally convex topological real vector spaces.

Proposition 3.2. Let X and Y be two locally convex topological real vector spaces, f_{1}, f_{2} : X ⟶ R ∪ {+∞} are two proper convex functions with f_{2} lower semicontinuous, h: X ⟶ Y ∪ {+∞} is a proper and Y_{+}-convex mapping and C is a closed convex subset of X. If there exists some x̄ ∈ C ∩ dom f_{1} ∩ dom h such that ȳ := h(x̄) ∈ − int Y_{+} where int Y_{+} denotes the topological interior of the cone Y_{+}, then for each x^{∗} ∈ ∂f_{2}(x̄) there exists some y^{∗} ∈ Y_{+}^{∗} such that ⟨y^{∗}, h(x̄)⟩ = 0 and x^{∗} ∈ ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄).

In order to establish the sufficient conditions linked to problem (P), let us denote by I(x̄) the subset of {1, 2,…, n} ⊂ N^{∗} given by

I(x̄) := { i ∈ {1, 2,…, n} : f_{2}(x̄) = ⟨x_{i}^{∗}, x̄⟩ + b_{i} }

where f_{2}(x) := max_{1≤i≤n}{⟨x_{i}^{∗}, x⟩ + b_{i}} is a polyhedral function with x_{1}^{∗}, x_{2}^{∗}, …, x_{n}^{∗} ∈ X^{∗} and b_{1}, b_{2},…, b_{n} ∈ R.

Proposition 3.3. If we assume that f_{2} is a polyhderal convex fuction, then under the same assumptions as in Proposition 3.1 or Proposition 3.2 we obtain that a feasible point x̄ of (P) is a local minimum for the problem (P) if and only if for each i ∈ I(x̄) there exists some y_{i}^{∗} ∈ Y_{+}^{∗} such that ⟨y_{i}^{∗}, h(x̄)⟩ = 0 and x_{i}^{∗} ∈ ∂(f_{1} + δ_{C} + y_{i}^{∗} ∘ h)(x̄).

Proof. By proceeding as in the proof of Proposition 3.1 and by combining assertion ii) of Proposition 2.1 and Proposition 2.2, we obtain that a feasible point x̄ ∈ X is a local minimum of the problem (P) if and only if

∂f_{2}(x̄) ⊂ ∂(f_{1} + δ_{C} + δ_{−Y_{+}} ∘ h)(x̄) = ⨆_{\begin{matrix}y^{∗}∈N(h(x̄),−Y_{+}) \\ ⟨y^{∗},h(x̄)⟩=0 \end{matrix}} ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄).

As f_{2} is the supremum of a finite family of affine functions, thus it is easy to check (or see [1]) that

∂f_{2}(x̄) = co{ x_{i}^{∗} : i ∈ I(x̄) }  (3.1)

where co stands for the convex hull. Since the subdifferential is a convex subset, we deduce that x̄ is a local minimum of (P) if and only if for each i ∈ I(x̄) there exists some y_{i}^{∗} ∈ Y_{+}^{∗} such that ⟨y_{i}^{∗}, h(x̄)⟩ = 0 and x_{i}^{∗} ∈ ∂(f_{1} + δ_{C} + y_{i}^{∗} ∘ h)(x̄).

Corollary 3.1. If, in addition to the assumptions of the above Proposition 3.3, we assume that f_{1} and h are finite and continuous at some point of C, then a feasible point x̄ is a local minimum of (P) if and only if for each i ∈ I(x̄) there exists some y_{i}^{∗} ∈ Y_{+}^{∗} such that ⟨y_{i}^{∗}h(x̄)⟩ = 0 and x_{i}^{∗} ∈ ∂f_{1}(x̄) + N(x̄, C)∂(y_{i}^{∗} ∘ h)(x̄).

4. Toland’s dual problem associated to (P)

Let us consider the following functions f_{1}, f_{2} : X ⟶ R∪{+∞}, g: Y ⟶ R∪ {+∞} and h: X ⟶ Y ∪ {+∞}. By (L) we shall mean the following minimization problem

(L) : inf_{x∈X}{(f_{1} + g ∘ h)(x) − f_{2}(x)}.

By virtue of Proposition 2.2, if we assume that f_{2} = f_{2}^{∗∗} then

inf_{x∈X}{(f_{1} + g ∘ h)(x) − f_{2}(x)} = inf_{x^{∗}∈X^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{1} + g ∘ h)^{∗}(x^{∗})}

where f_{1}, g and h are arbitrary functionals. Hence the dual problem (L^{∗}) associated to (L) takes the following form

(L^{∗}) : inf_{x^{∗}∈X^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{1} + g ∘ h)^{∗}(x^{∗})}.

Now, we can state the following duality result.

Proposition 4.1. Let us assume that f_{1} and f_{2} : X ⟶ R∪{+∞} are convex, proper and lower semicontinuous, g: Y ⟶ R∪{+∞} is Y_{+}-nondecreasing, proper, convex and lower semicontinuous and h: X ⟶ Y ∪{+∞} is Y_{+}-convex and proper with closed epigraph. If R_{+}[dom g −h(dom f_{1} ∩dom f_{2})] is a closed vector subspace of Y then we have inf(L) = inf(L^{∗}) where

inf(L^{∗}) = inf_{x^{∗}∈X^{∗}} max_{y^{∗}∈Y_{+}^{∗}}{f_{2}^{∗}(x^{∗}) − g^{∗}(y^{∗}) − (f_{+}y^{∗} ∘ h)^{∗}(x^{∗})}.

Proof. We have already pointed out that the following equality

inf_{x∈X}{(f_{1} + g ∘ h)(x) − f_{2}(x)} = inf_{x^{∗}∈X^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{1} + g ∘ h)^{∗}(x^{∗})}

holds under the assumption f_{2} = f_{2}^{∗∗} which is fulfilled since f_{2} is proper, convex and lower semicontinuous. By applying again Theorem 2.1, we get

(f_{1} + g ∘ h)^{∗}(x^{∗}) = min_{y^{∗}∈Y_{+}^{∗}}{g^{∗}(y^{∗}) + (f_{1} + y^{∗} ∘ h)^{∗}(x^{∗})},

and thus we may write that

inf_{x∈X}{(f_{1} + g ∘ h)(x) − f_{2}(x)}

= inf_{x^{∗}∈X^{∗}}{f_{2}^{∗}(x^{∗}) − min_{y^{∗}∈Y_{+}^{∗}}{g^{∗}(y^{∗}) + (f_{1} + y^{∗} ∘ h)^{∗}(x^{∗})}}

= inf_{x^{∗}∈X^{∗}} max_{y^{∗}∈Y_{+}^{∗}}{f_{2}^{∗}(x^{∗})g^{∗}(y^{∗}) − (f_{1} + y^{∗} ∘ h)^{∗}(x^{∗})}.

Now, let us go back to our main problem (P),

(P) \{ \begin{matrix}inf f_{1}(x) − f_{2}(x) \\ h(x) ∈ −Y_{+} \\ x ∈ C. \end{matrix}

From the previous results the Toland’s dual problem associated to (P) is defined by

(P^{∗}) inf_{x^{∗}∈X^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{1} + δ_{C} + δ_{−Y_{+}} ∘ h)^{∗}(x^{∗})}.

It was mentioned previously that when f_{2} is convex, proper and lower semicontinuous we have inf(P) = inf(P^{∗}).

Proposition 4.2. Let us consider two convex, proper and lower semicontinuous functions f_{1} and f_{2} : X ⟶ R ∪ {+∞}, a Y_{+}-convex and proper mapping h: X ⟶ Y ∪ {+∞} with closed epigraph and let C be a nonempty closed and convex subset of X. If R_{+}[Y_{+} + h(dom f_{1} ∩ C)] is a closed vector subspace of Y, then we have inf(P) = inf(P^{∗}) with

inf(P^{∗}) = inf_{x^{∗}∈X^{∗}} max_{y^{∗}∈Y_{+}^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{1} + δ_{C} + y^{∗} ∘ h)^{∗}(x^{∗})}.

Proof. It suffices to observe that (P) has the form of the problem (L), i.e., (P) is equivalent to

inf_{x∈X}{(f_{1} + δ_{C} + δ_{−Y_{+}} ∘ h)(x) − f_{2}(x)},

and that the function g: Y ⟶ R ∪ {+∞} given by g(y) = δ_{−Y_{+}}(y) is convex, proper, lower semicontinuous and Y_{+}-nondecreasing on the whole space Y (see [1]) and g^{∗} = δ_{−Y_{+}}^{∗} = δ_{Y_{+}^{∗}} and therefore by applying Proposition 4.1 we obtain there desired result.

Under the conditions of Remark 2.1 we obtain the above result in the setting of locally convex topological vector spaces.

Proposition 4.3. Let X and Y be two locally convex topological vector spaces, f_{1}, f_{2} : X ⟶ R ∪ {+∞} two convex proper functions, h: X ⟶ Y ∪ {+∞} is a Y_{+}-convex and proper mapping and C is a nonempty convex subset of X. If we assume that there exists some x̄ ∈ C ∩ dom f_{1} ∩ dom h such that ȳ:= h(x̄) ∈ int Y_{+}, then we have inf(P) = inf(P^{∗}) with

inf(P^{∗}) = inf_{x^{∗}∈X^{∗}} max_{y^{∗}∈Y_{+}^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{+}δ_{C} + y^{∗} ∘ h)^{∗}(x^{∗})}.  (4.1)

Corollary 4.1. If we assume, in addition to the assumptions of Proposition 4.2 or Proposition 4.3, that f_{1} and h are finite and continuous at some point of C then inf P = inf P∗ with

inf(P^{∗}) = inf_{x^{∗}∈X^{∗}} max_{(y^{∗},u^{∗},v^{∗})∈Y_{+}^{∗}×X^{∗}×X^{∗}}{f_{2}^{∗}(x^{∗})−f_{1}^{∗}(x^{∗}−u^{∗}−v^{∗})−δ_{C}^{∗}(u^{∗})−(y^{∗}∘h)(x^{∗})}.

Proof. Since h is finite and continuous at some point x̄ ∈ dom f_{1} ∩ C, we have according to [1] that

(f_{1} + δ_{C} + y^{∗} ∘ h)^{∗}(x^{∗}) = min_{u^{∗}∈X^{∗}}{(f_{1} + δ_{C})^{∗}(x^{∗} − u^{∗}) + (y^{∗} ∘ h)^{∗}(u^{∗})}.

Also, since f_{1} is finite and continuous at the same point x̄, we obtain

(f_{1}+δ_{C}+y^{∗}∘h)^{∗}(x^{∗}) = min_{u^{∗}∈X^{∗}} min_{v^{∗}∈X^{∗}}{f_{1}^{∗}(x^{∗}−u^{∗}−v^{∗})+δ_{C}^{∗}(v^{∗})+(y^{∗}∘h)^{∗}(u^{∗})}. (4.2)

Substituting (4.2) in (4.1) we get

inf(P^{∗}) = inf_{x^{∗}∈X^{∗}} max_{y^{∗}∈Y_{+}^{∗}}{f_{2}^{∗}(x^{∗}) − (f_{1} + δ_{C} + y^{∗} ∘ h)^{∗}(x^{∗})} =

inf_{x^{∗}∈X^{∗}} max_{(y^{∗},u^{∗},v^{∗})∈Y_{+}^{∗}×X^{∗}×X^{∗}}{f_{2}^{∗}(x^{∗}) − f_{1}^{∗}(x^{∗} − u^{∗} − v^{∗}) − δ_{C}^{∗}(v^{∗}) − (y^{∗} ∘ h)^{∗}(v^{∗})}.

5. Application to fractional mathematical programming problems

In this section, we illustrate the previous results by deriving optimality conditions for a scalar nondifferentiable fractional programming problem subject to a convex vector constraint. Let us consider the following optimization problem

(H) \{ \begin{matrix}inf & \frac{f_{1}(x)}{f_{2}(x)} \\ h(x) ∈ −Y_{+} \\ x ∈ C, \end{matrix}

where f_{1}, f_{2} : X ⟶ R ∪ {+∞} are two convex functions with f_{1}(x) ≥ 0 and f_{2}(x) > 0 for all x ∈ X, h: X ⟶ Y ∪ {+∞} is a Y_{+}-convex mapping and C is a nonempty convex closed subset of X. By (Hs̄) we denote the following minimization problem

(Hs̄) \{ \begin{matrix}inf\{f_{1}(x) − s̄f_{2}(x)\} \\ h(x) ∈ −Y_{+} \\ x ∈ C \end{matrix}

where s̄ is a real number.

Remark 5.1. Since f_{1} and f_{2} may take both the value +∞, we adopt throughout this section the convention (+∞) × 0 = 0.

The following result from [3] allows us to write equivalently the fractional programming problem as a DC-minimization problem.

Proposition 5.1. [3] The point x̄ is a local (resp. global) solution of the problem (H) if and only if x̄ is a local (resp. global) solution of the problem (Hs̄)

with s̄ = \frac{f_{1}(x̄)}{f_{2}(x̄)}.

Let us consider the following constraint qualification

(C.Q.A_{2}.B_{2}) \{ \begin{matrix}X and Y are Fréchet spaces, \\ f_{1} and f_{2} are convex, proper and lower semicontinuous, \\ Epi h is a closed subset of X × Y, \\ R_{+}[Y_{+} + h(dom f_{1} ∩ C ∩ dom h)] is a closed vector subspace. \end{matrix}

Now, we state the necessary optimality condition for the scalar fractional programming problem (H).

Proposition 5.2. Let x̄ be a feasible point of (H), i.e., x̄ ∈ C ∩ h^{−1}(−Y_{+}). If we assume that the condition (C.Q.A_{2}.B_{2}) is satisfied and x̄ is a local minimum for the problem (H), then we have: for any x^{∗} ∈ ∂f_{2}(x̄) there exists some y^{∗} ∈ Y_{+}^{∗} verifying ⟨y^{∗}, h(x̄)⟩ = 0 and s̄x^{∗} ∈ ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄).

Proof. For proving the necessity of optimality conditions we distinguish two cases. The first is when s̄ > 0. For this, from Proposition 5.1, a feasible point x̄ of (H) is a local minimum if and only if x̄ is a local minimum of (Hs̄). As ∂(s̄f_{2}) = s̄∂f_{2}(x̄) and by applying Proposition 3.1 we get under the condition (C.Q.A_{2}.B_{2}) that for each x^{∗} ∈ ∂f_{2}(x̄) there exists some y^{∗} ∈ Y_{+}^{∗} satisfying ⟨y^{∗}, h(x̄)⟩ = 0 and

s̄x^{∗} ∈ ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄).

The case when s̄ = 0 implies f_{1}(x̄) = 0 and by virtue of Remark 5.1 and the fact that x̄ is a local minimum of (H) we obtain

0 ∈ ∂(f_{1} + δ_{C} + δ_{−Y +} ∘ h)(x̄),

which ensures under the condition (C.Q.A_{2}.B_{2}) the existence of y^{∗} ∈ Y_{+}^{∗} satisfying ⟨y^{∗}, h(x̄)⟩ = 0 and0 ∈ ∂(f_{1} + δ_{C} + y^{∗} ∘ h)(x̄). This completes the proof.

Corolarry 5.1. If we assume, in addition to the assumptions of the above Proposition 5.2, that f_{1} and h are finite and continuous at some point of C, that the condition (C.Q.A_{2}.B_{2}) holds and x̄ is a local minimum of (H) then we obtain: for any x^{∗} ∈ ∂f_{2}(x̄) there exists some y^{∗} ∈ Y_{+}^{∗} satisfying ⟨y^{∗}, h(x̄)⟩ = 0 and

s̄x^{∗} ∈ ∂f_{2}(x̄) + N_{C}(x̄) + ∂(y^{∗} ∘ h)(x̄).

The related sufficient optimality conditions are given by

Proposition 5.3. If we suppose that f_{2} is a polyhedral convex function, then under the same assumptions in Proposition 5.2 we obtain that a feasible point x̄ of (H) is a local minimum for the problem (H) if and only if for each i ∈ I(x̄) there exists some y_{i}^{∗} ∈ Y_{+}^{∗} such that ⟨y_{i}^{∗}, h(x̄)⟩ = 0 and

s̄x_{i}^{∗} ∈ ∂(f_{1} + δ_{C} + y_{i}^{∗} ∘ h)(x̄).

Proof. We apply Proposition 3.3 and Proposition 5.2.

Corollary 5.2. If, under the assumptions of Propositions 5.3, f_{1} and h are assumed to be finite and continuous at some point of C, then x̄ is a local solution of (H) if and only if for each i ∈ I(x̄), there exists some y_{i}^{∗} ∈ Y_{+}^{∗} such that ⟨y_{i}^{∗}, h(x̄)⟩ = 0 and s̄x_{i}^{∈}∂f_{1}(x̄) + N_{C}(x̄) + ∂(y_{i}^{∗} ∘ h)(x̄).

Proof. We use the same arguments as in Corollary 3.1.

REFERENCES

[1] C. Combari, M. Laghdir and L. Thibault, Sous différentiel de fonctions convexes composées, Ann. Sci. Math. Quebec 18 (1994), 119–148.

[2] C. Combari, M. Laghdir and L. Thibault, A note on subdifferentials of convex composite functionals, Arch. Math., 67 (1996), 239–252.

[3] W. Dinkelbach, On Nonlinear Functional Programming, Management Sciences, 9 (1967), 492– 498.

[4] J. B. Hiriarty-Urruty, From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality, in: Nonsmooth Optimization and Related Topics, F. H. Clarke, V. F. Demyanov and F. Gianessi, eds., Plenum Press, New York, 1989, p.p. 219–239.

[5] J. F. Toland, Duality in nonconvex optimization, Journal. Math. Anal. Appl, 6 (1987), 399– 415.

(received 10.09.2001)

Département de Mathématiques, Faculté des Sciences, B.P. 20, El-Jadida, Maroc.