0 (real). Definition 6 The partially ordered space (Y, P ) is called lattice if each pair of points a and b in Y , admits a supremum and an infimum in the order induced by P. Definition 7 (Pareto Optimal) Let A be a subset of Y . A point a ∈ A is said to be P-Maximal (resp. P-minimal) if a P x (resp. x P a), for some x ∈ A, then x P a (resp. a P x). MaxP (A) (resp. MinP (A)) will denote the set of P-Maximal (resp. P-minimal) points of A. Remark 6 1. Maximal and minimal points are also called efficient points. Note that for A ⊂ Y maxP (A) ⊂ MaxP (A) (resp. minP (A) ⊂ MinP (A)). If, maxP (A) (resp. minP (A)) is nonempty then we can easily check that maxP (A) = MaxP (A) (resp. minP (A) = MinP (A)), (see also for example [24]), and whenever the cone P is pointed, at most maxP (A) (resp. minP (A)) reduces to a singleton. 2. If the cone P is pointed, then a point a ∈ A is P-Maximal (resp. P-minimal) point of A if, and only if (a + P) ∩ A = {a} (resp. (a − P) ∩ A = {a}). 3. Observe that, if maxP (A) = 0/ (resp. minP (A) = 0), / then a maximal (resp. minimal) point is not necessarily unique even if the cone is pointed. Maximal and minimal points may not exist even if Y is totally ordered. 4. MaxP (A) = Min−P (A) and MaxP (A) = −MinP (−A) for every A ⊂ Y. Definition 8 The ordering cone P is said to be Daniell, if the infimum of a decreasing net in Y exists and is its topological limit. Definition 9 A nonempty convex subset B of (the ordering cone) P is called a base for P if for each nonzero element x ∈ P has a unique representation of the form x = λ b with λ > 0 and b ∈ B. If the cone P has a bounded convex base B and 0Y ∈ / cl (B), then P is called well-based.
On the minima and maxima of directed free disposal subsets
7
Remark 7 – If B is a base of the (non-trivial convex) cone P, then 0Y ∈ / B. – If P has a compact base (and Y is locally convex), then P is Daniell. Definition 10 We say that (Y, P) is order-complete if every nonempty P-upper bounded ( resp. P-lower bounded) subset admits a P−supremum (resp. P−infimum). Lemma 2 The following assertions are satisfied: 1. P is order-complete and Int (P) 6= 0/ imply (Y, P ) is lattice. 2. Suppose (Y, P) is order-complete. Then, any full subset A ⊂ Y w.r.t P (i.e., A = [A]P ) is order-complete. Proof 1. Can be easily checked (see also the [23, Preliminaries Section]). 2. Let 0/ 6= B ⊂ A = [A]P such that B is P-upper bounded in A. Let a0 be a P-upper bound of B in A. supP (B) := b exists in Y. Then, b ∈ B +P ⊂ A +P. On the other hand, b P a0 , then b ∈ a0 −P ⊂ A −P. Thus, b ∈ (A + P) ∩ (A − P) = [A]P = A. Definition 11 The cone P is called normal (relatively to τ) if the origin 0 has a neighborhood base formed by full sets w.r.t P. Remark 8 One of the important properties of the normality of the cone are: 1. every order-bounded subset is bounded. The converse is true without normality provided that the cone has a nonempty interior; 2. if Y is Hausdorff space, then every normal cone in Y is pointed. Definition 12 If (Y, k.k) is a normed space, the convex cone P is said to be supernormal if, and only if, there exists y∗ ∈ Y ∗ such that P ⊂ {y ∈ Y | kyk ≤ hy, y∗ i} .
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Remark 9 1. If P is supernormal, then the linear functional in (4) satisfies y∗ ∈ qi(P+ ) (the quasi-interior of P+ ). 2. The supernormality of the cone is defined in general in a Hausdorff topological space where the topology is given by a family of semi-norms, say P, and (4) is obtained by replacing the norm of y by the value at y of an element p ∈ P. 3. If (Y, P) is a Hausdorff locally convex space, then P well-based ⇒ P supernormal ⇒ P normal. 4. If Y is a normed space then P supernormal ⇒ P well-based. Remark 10 Notice that Daniell cones are necessarily pointed if the space is Hausdorff. Definition 13 A subset A ⊂ Y is said to be asymptotically compact, a.c for short, if there exists γ > 0 and a neighborhood of 0Y such that ([0, γ]A) ∩U is relatively compact. The unit ball in every reflexive Banach space is a.c. In particular, every subset of any finite dimensional real linear space is a.c. Definition 14 The asymptotic cone of a nonempty subset A ⊂ Y is defined by: A∞ = {y ∈ Y : ∃(ti )i∈I ⊂ (0, ∞),ti ↓ 0, ∃(ai )i∈I ⊂ A : ti ai → y}. Remark 11 1. Note that if Y is a normed space, we can replace nets in A∞ by sequences. \ 2. If A is closed and convex then A∞ is given by the convex analysis formula: A∞ = t(A − a) for t>0 some a ∈ A. Definition 15 A subset A of Y is called upward-directed (resp.downward-directed) if for all x, y ∈ A, there exists z ∈ A such that x P z and y P z (resp. z P x and z P y). A subset will be said directed if it is directed both upwards and downwards. The space Y itself is directed if and only if Y = P − P i.e., P is reproducing (see [30]). If (Y, P ) is a lattice or int (P) is nonempty, then P is reproducing (see [10]).
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Definition 16 A subset A of Y is called upward-free disposal (resp. downward-free disposal) if A = A + P (resp. A = A − P). Remark 12 1. Sometimes in the literature, a downward-free disposal (resp. upward-free disposal) set is called simply a downward (resp. upward) set. 2. The free disposal property is due to Debreu [20], and it has been proved to play an important role in economic theory and many fields of mathematics (see for instance [44]). 3. Any downward/upward-free disposal subset of Y is full with respect to P. 3 New theorems on existence of supremum/infimum points
Now we present an existence Theorem in the more general setting of locally convex spaces provided that the quasi-interior of the dual cone is nonempty. Theorem 1 Assume that (H1 ) is satisfied i.e., Y is a locally convex Fr´echet vector space and P is closed and the quasi-interior qi(P+ ) of P+ is nonempty. Then the following assertions are satisfied: a) Let A be a nonempty P-upper bounded subset of Y such that for some ξ ∈ qi(P+ ), for every a ∈ cl (A), cl (A+ P (a)) + Ker(ξ ) is closed and cl (A) is upward-directed. Then A admits a P-supremum in its closure, i.e., supP (A) ∩ cl (A) 6= 0. / b) Let B be a nonempty P-lower bounded subset of Y such that for some ξ ∈ qi(P+ ), for every b ∈ cl (B), cl (B− P (b)) + Ker(ξ ) is closed and cl (B) is downward-directed. Then B admits a P-infimum in its closure i.e., infP (B) ∩ cl (B) 6= 0. / Remark 13 Before proving Theorem 1, let us make the following observations: 1. The closeness of cl (A+ P (a)) + Ker(ξ ) (in the assumptions of assertion a) in the theorem above) is in particular satisfied if the upper section A+ P (a) (of A) is compact. However, this compactness hypothesis is not needed in the proof below. 2. Theorem 1 extends and improves [5, Theorem 3.8] by dropping the normality of the cone, and replacing the non-emptiness of the interior of the latter by that of the quasi-interior of its dual cone. Also, the space Y is extended to locally convex setting. 3. If P is supernormal, then qi(P+ ) 6= 0. / 4. If Y is a locally convex space, then qi(P+ ) 6= 0/ if, and only if P has a base B with 0Y ∈ / cl (B), (see [26, Theorem 2.2.12]). 5. If Y is a locally convex space, then qi(P+ ) 6= 0/ ensures that P is pointed. The converse is true if Y is a finite-dimensional space, see [26]. 6. If Y is a Banach space, then qi(P+ ) 6= 0/ if, and only if, there exists another convex cone C with Int (C) 6= 0/ such that P\{0Y } ⊂ Int (C). Proof of Theorem 1. We prove only part a), part b) can be omitted. Let A be a subset satisfying the assumptions of assertion a). Let u be a P-upper bound of A and consider the family of closures of upper + sections {cl (A+ P (a)}a∈cl (A) of A. Set, for simplicity, Fa = cl (AP (a)) for each a ∈ cl (A). We see that Fa is nothing else but cl ({y ∈ A| a P y}). Remark that the family (Fa )a∈cl (A) is decreasing i.e., if a, b ∈ cl (A) such that a P b then Fb ⊂ Fa . Therefore, given that cl (A) is upward-directed, the family (Fa )a∈cl (A) has the finite intersection property i.e., for every {a1 , ..., an } ⊂ cl (A), n \
Fai 6= 0. /
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i=1
Now, by the assumption, there exists ξ ∈ qi(P+ ) (so ξ ∈ P+ ) such that Fa + Ker(ξ ) is closed for every a ∈ cl (A). On the other hand, ξ being in qi(P+ ), a fortiori ξ 6= 0. Hence, ξ is surjective, which implies that ξ (Y )(= R) is closed. Therefore, from [49, Corollary 1.3.15], we obtain that ξ (Fa ) is closed for every a ∈ cl (A). In addition, it follows from (5) that for every {a1 , ..., an } ⊂ cl (A),
n \ i=1
ξ (Fai ) 6= 0. / Moreover, since
ξ ∈ P+ , for every a ∈ cl (A), ξ (Fa ) is upper bounded by ξ (u) and lower bounded by ξ (a) (in the real line). Thus, the family ξ (Fa )a∈cl (A) has the finite intersection property and all of its elements are closed of the compact [ξ (a), ξ (u)] (in the real line). Accordingly, the well known compactness lemma (see [15,
On the minima and maxima of directed free disposal subsets
\
Lemma 3.1]) implies that
9
ξ (Fa ) 6= 0. / Let y0 ∈
\
ξ (Fa ) 6= 0. / We claim therefore that there
a∈cl (A)
a∈cl (A)
exists a unique x0 ∈
\
Fa such that y0 = ξ (x0 ), otherwise there are some a1 , a2 ∈ cl (A), there are
a∈cl (A)
some x1 ∈ Fa1 and x2 ∈ Fa2 such that y0 = ξ (x1 ) = ξ (x2 )
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with x1 6= x2 . First case: If x1 − x2 ∈ P i.e., x2 P x1 , then x1 6= x2 =⇒ ξ (x1 − x2 ) > 0 (since ξ ∈ qi(P+ )), a contradiction with (6). Second case: If x1 − x2 ∈ / P, then from directness of cl (A) the pair {x1 , x2 } admits a P-upper bound c in cl (A). Of course, y0 ∈ ξ (Fc ), then there exists xc ∈ Fc such that y0 = ξ (xc ). It follows that ξ (xc − x1 ) = 0.
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But x1 P c P xc . Remark that xc 6= x1 (otherwise x2 P x1 , contradicting the assumption of the case: x1 − x2 ∈ / P). Therefore, ξ (xc − x1 ) > 0 (since ξ ∈ qi(P+ )), a contradiction with (7). Accordingly, \ \ x0 ∈ Fa ⊂ Fa . a∈cl (A)
a∈A
Thus, by the closeness of P we obtain that x0 ∈ maxP (cl(A)) and A P x0 . If y is a P-upper bound of A, then y is also a P-upper bound of cl (A), implying that x0 P y. Hence, x0 ∈ supP (A). Which finishes the proof. Remark 14 Let us stress that 1. The family (Fa )a∈cl (A) has the finite intersection property, and has a nonempty intersection without compacity of any element of it. Then, the technique in the above proof can be considered as a kind of scalarization. 2. Also, the P-supremum of A is unique thanks to the pointedness of P (according to 5) of Remark 13 and 1) of Remark 4). The following descends directly from Theorem 1: Theorem 2 Assume that (H1 ) is satisfied. Then i) any nonempty P-upper bounded, upward-directed and closed subset of Y admits a unique P-maximum; ii) any nonempty P-lower bounded, downward-directed and closed subset of Y admits a unique Pminimum. If the quasi-interior of P is empty, the following result presents an alternative in which the subset A verifies additional properties. Theorem 3 Assume that (H2 ) is satisfied i.e., Y is a Hausdorff locally convex space such that (Y, P) is order-complete, and either Y is lattice or Int (P) 6= 0. / Then a) any nonempty P-upper bounded, convex, downward-free disposal and upward-directed subset A in Y admits its P-supremums in its closure i.e., supP (A) ⊂ cl (A) 6= 0/ ; b) any nonempty P-lower bounded, convex, upward-free disposal and downward-directed subset B in Y admits its P-infimums in its closure i.e., minP (B) ⊂ cl (B) 6= 0. / Proof As before, we only prove part a), part b) descends immediately form a). Since (Y, P) is ordercomplete, supP (A) is nonempty. Let a0 ∈ supP (A). Assume for contradiction that a0 ∈ / cl (A). Thus, using ∗ the standard separation theorem ([26, Theorem 2.2.8]), there exists ξ ∈ Y and a real number α such that sup ξ (A) ≤ sup ξ (cl (A)) ≤ α < ξ (a0 ).
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Since A = A − P, for each p ∈ P and each a ∈ A, we have ξ (a − p) ≤ sup(ξ (A)) < +∞ (since A is P-upper bounded). Then, ξ (a) ≤ sup(ξ (A)) + ξ (p) for all a ∈ A and all p ∈ P, concluding that ξ (p) ≥ 0. Thus ξ ∈ P+ .
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On the other hand, Y is a lattice (in the case where Int (P) 6= 0, / order-completeness ensures that Y is a lattice, see Lemma 2). Then, A being P-upper bounded and upward-directed and given that ξ ∈ P+ we obtain sup ξ (A) = ξ (a0 ) (9) Indeed, every linear functional in the positive cone of P preserves the P−supremum of P-upper bounded and upward-directed subsets, see for instance [46] . Clearly, (9) contradicts (8), concluding that a0 ∈ cl (A), which completes the proof. Remark 15 Theorem 3 puts in a general setting the result of Lemma 6.7 of [3]. A particular and useful case of this Theorem 3 is the following Theorem 4 Assume that (H2 ) is satisfied. If, in addition, the cone P is pointed, then a) any nonempty P-upper bounded, closed, convex, downward-free disposal and upward-directed subset A in Y admits a unique P-maximum ; b) any nonempty P-lower bounded, closed, convex, upward-free disposal and downward-directed subset A in Y admits a unique P-minimum. 4 Cone-completeness, maximality and minimality
Definition 17 A net {xi | i ∈ I} in Y is said to be decreasing (resp. increasing) with respect to the ordering cone P if xi P x j and xi 6= x j (resp. x j P xi and xi 6= x j ) for each i, j ∈ I, j i, where (I, ) is a directed set. Definition 18 ([24]) A subset A is called strongly P-complete (resp. P-complete) if it has no covers of the form {(xi − P)c | i ∈ I} (resp. {(xi − cl (P))c | i ∈ I}), where (xi )i∈I is a decreasing net in A. Remark 16 1. If a subset is strongly P-complete, then it is P complete and whenever P is closed, strong P-completeness and P-completeness coincide. 2. A subset A is strongly P-complete if, and only if, every decreasing net (xi )i∈I in A has a P-lower bound in A. Thus, every P-lower bounded subset A with a lower bound in A is trivially strongly P-complete. The space Y itself is not strongly P-complete (see Remark 19 in the next section). 3. (strong) P-completeness is intimately related to existence of minimal points. Then, dually we can speak about (strong)−P-completeness i.e., (strong) completeness w.r.t the cone −P. Then a subset B is called strongly −P-complete (resp.−P-complete) if it has no covers of the form {(xi + P)c | i ∈ I} (resp. {(xi + cl (P))c | i ∈ I}), where (xi )i∈I is an increasing net in A. Equivalently, B is strongly −Pcomplete if, and only if, every increasing net (xi )i∈I in B has a P-upper bound in B. Lemma 3 Let A ⊂ Y and x ∈ Y. Then + 1. MinP (A− (x)) ⊂ Min (A) resp. Max (A (x)) ⊂ Max (A) . P P P P P − + + 2. MinP (A− (x)) = Min (A) ∩ A (x) resp. Max (A (x)) = Max (A) ∩ A (x) . P P P P P P P − Proof 1. Let a ∈ MinP (A− P (x)). Of course a ∈ AP (x) = A ∩ (x − P), thus a ∈ A and a P x. Now let y ∈ A such that y P a. We see that y P x. Therefore, y ∈ A ∩ (x − P) = A− P (x). Hence, y P a =⇒ a P y (as a ∈ MinP (A− (x))), concluding that a ∈ Min (A). P P 2. Straightforward.
Definition 19 A subset A ⊂ Y satisfies the domination property, (DP)min for short, w.r.t P if and only if for each a ∈ A, MinP (A− P (a)) is nonempty. Lemma 4 Let 0/ 6= A ⊂ Y. The following assertions are equivalent: i) A has the (DP)min property; ii) For each x ∈ A, there exists a ∈ MinP (A) such that a P x; iii) A ⊂ MinP (A) + P.
On the minima and maxima of directed free disposal subsets
11
Proof i) =⇒ ii) Suppose i) is true. Then for each x ∈ A, MinP (A− / which means that for each P (x)) 6= 0, − x ∈ A, ∃a ∈ A− (x) and a ∈ Min (A (x)) ⊂ Min (A) (thanks to Lemma 3), implying that a P x and P P P P a ∈ MinP (A). ii) is then satisfied. ii) =⇒ iii) immediate. iii) =⇒ i) Let a ∈ A. Then there exists x ∈ MinP (A) such that a ∈ x+P. Hence, x ∈ A− P (a)∩MinP (A) = − MinP (AP (a)) (by using again Lemma 3), completing the proof. Remark 17 Notice that 1. Assertion ii) of Lemma 4 is adopted as a definition of domination property by many authors as in Luc [24]. 2. In [26], the authors adopted the following definition: a subset A ⊂ Y satisfies the domination property w.r.t P if and only if for each a ∈ A, MaxP (A+ P (a)) is nonempty. 3. According to previous point of the present remark, in the sequel we write (DP)max for the domination property in the sense of [26]. 4. Clearly (DP)min w.r.t P ⇐⇒ (DP)max w.r.t −P. Similarly we have the following: Lemma 5 Let 0/ 6= A ⊂ Y. The following assertions are equivalent: i) A has (DP)max w.r.t P; ii) For each x ∈ A, there exists a ∈ MaxP (A) such that x P a; iii) A ⊂ MaxP (A) − P. Definition 20 A subset A ⊂ Y satisfies the domination property, (DP) for short, w.r.t P if, and only if, A fulfills both of (DP)max and (DP)min . Lemma 6 The following assertions hold true. i) Every P-interval is full w.r.t P and fulfills (DP). ii) Conversely, if P is pointed then every full subset of Y , say A, w.r.t P such that maxP (P) and minP (A) are nonempty is a P-order interval. Proof i) is clearly satisfied. ii) Let A ⊂ Y satisfying conditions of ii). Since P is pointed, there exist a, a¯ ∈ A such that ¯ minP (A) = MinP (A) = {a} and maxP (A) = MaxP (A) = {a}. Obviously, we have A ⊂ [a, a] ¯ P . Now let x ∈ [a, a]. ¯ Then, x ∈ (a + P) ∩ (a¯ − P) ⊂ (A + P) ∩ (A − P) = [A]P = A (since A is full w.r.tP), concluding that A = [a, a] ¯ P. Lemma 7 Assume that P is pointed. Let A be a full and upward-directed subset of (Y, P). Then, every + upper section A+ ¯ P (a) of A having a P-Maximal point a¯ is an order-interval w.r.t Pi.e., AP (a) = [a, a]. Proof Since A is upward-directed and P is pointed, according to Remark 6, we can easily check that + MaxP (A+ ¯ On the other hand, a + P is full w.r.t P, then A+ P (a)) = maxP (A) = maxP (AP (a)) = {a}. P (a) is full w.r.t P as an intersection of two full subsets. Clearly, {a} = minP (A+ (a)), then the result is a P consequence of Lemma 6. In the rest of this section we provide the conditions under which a nonempty closed, convex, downwardfree disposal subset and upward-directed, say A ∈ CCD f Ud (Y ), admits an efficient point. Unless additional hypotheses are specified, the cone P is a non-trivial convex cone. We begin with the following Proposition 1 Every subset A in D f (Y ) has no strongly P-complete lower section. In particular, MinP (A) = 0. / Proof Let 0/ 6= A 6= Y be a (proper) subset of Y such that A ∈ D f (Y ) i.e., A − P = A. Clearly, MinP (A) = 0. / Then, by [24, Theorem 3.4], A has no strongly P-complete lower section.
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Proposition 2 Let 0/ 6= A ⊂ Y. If A ∈ D f (Y ), then for all x ∈ Y the lower section A− P (x) ∈ D f (Y ). In − particular, MinP (AP (x)) = 0. / Proof Let 0/ 6= A ⊂ Y such that A = A − P and let A− P (x) = A ∩ (x − P) be a lower section of A for some − x ∈ Y. Obviously, A− (x) ⊂ A (x) − P (0 ∈ P). Let y ∈ A− P P P (x) − P ⊂ A − P = A. Then, y ∈ A and y = a − c − with a ∈ AP (x) and c ∈ P. Therefore, a ∈ A and x − a ∈ P. Since y − x = a − c − x = −c − (x − a) ∈ −P − P ⊂ −P, − − we see that y ∈ (x − P) ∩ A = A− P (x). Hence, AP (x) − P = AP (x).
From Proposition 2 we obtain the following Proposition 3 Let 0/ 6= B ⊂ Y. If B ∈ U f (Y ), then for all x ∈ Y, the upper section B+ P (x) ∈ U f (Y ). In particular, MaxP (B+ (x)) = 0. / P Proposition 4 Let 0/ 6= A ⊂ Y. Assume that (Y, P) is a lattice. If A ∈ D f (Y ), then for all x ∈ A, the lower − section A− P (x) is upward-directed i.e., AP (x) ∈ Ud (Y ). Proof Let 0/ 6= A ⊂ Y. and let x ∈ A. (Y, P) being lattice, for every a, b ∈ A− P (x), supP {a, b} exists in Y, set x0 = supP {a, b}. As x is a P-upper bound of {a, b}, we clearly have x0 P x. Then, x0 ∈ x−P ⊂ A−P = A, completing the proof. Similarly, we also have: Proposition 5 Let 0/ 6= B ⊂ Y. Assume that (Y, P) is a lattice. If B ∈ U f (Y ), then for all x ∈ B, the lower + section B+ P (x) is downward-directed i.e., BP (x) ∈ Dd (Y ). Remark 18 As opposed to lower sections, upper sections have a P-minimum and then a P-minimal point. Remark also that any subset of D f (Y ) doesn’t satisfy the (DP)min . We continue with the following result in which we prove that Y contains a proper downward (resp. upward) free disposal subset if, and only if, Y is not strongly P-complete (resp −P-complete). Theorem 5 The following assertions are satisfied. a) D f (Y )\{0,Y / } is nonempty ⇐⇒ Y is not strongly P-complete. b) U f (Y )\{0,Y / } is nonempty ⇐⇒ Y is not strongly −P-complete. Proof We only justify the first statement, the second one is similar. (=⇒) Let A ⊂ Y be a proper downward-free disposal subset. Assume for contradiction that Y is strongly P-complete. The subset A is strongly P-complete otherwise it has cover of the form {(xi − P)c | i ∈ I}, where (xi )i∈I being a decreasing net in A. Since A = A − P, it follows that xi − P ⊂ A for all i ∈ I. Then Ac ⊂ (xi − P)c for all i ∈ I. Thus, Y has cover of the form {(xi − P)c | i ∈ I}, (xi )i∈I being a decreasing net in A (and then in Y ), contradicting the strong P-completeness of Y. Now, let A− P (x) − be a lower section of A. From Proposition 2 we have A− (x) ∈ D (Y ). The same arguments, for A f P P (x) − instead of A, shows that AP (x) is strongly P-complete. Therefore, using [24, Theorem 3.4] we obtain that MinP (A) 6= 0, / contradicting that MinP (A) = 0/ (obtained again from Proposition 2). Hence, Y is not strongly P-complete. (⇐=) It suffices to take A = x − P for some x ∈ Y, which is a proper downward-free disposal since Y 6= P, the cone P being non-trivial. Remark 19 1. Observe that Theorem 5 a) can be equivalently expressed as follows: Y is strongly P-complete ⇐⇒ D f (Y )\{0,Y / } = 0. / 2. Since the cone P is a non-trivial, the space Y itself is not strongly P-complete. 3. If P is a trivial cone, then Y is strongly P-complete in the case where P = Y while Y is not strongly P-complete in the case P = {0Y }. Theses two cases are to be excluded in the sequel as P is considered to be a non-trivial cone. 4. A subset \ A is not strongly P-complete if, and only if, there exists a decreasing net (xi )i∈I in A such that (xi − P) = 0. / i∈I
On the minima and maxima of directed free disposal subsets
13
Proposition 6 Let A be a nonempty closed and convex subset in Y and let A∞ be the asymptotic cone of A. Assume that A is P-upper bounded. Then, A∞ ∩ P ⊂ P ∩ (−cl (P)). In particular, if P is closed then A∞ ∩ P ⊂ l(P), where l(P) = P ∩ (−P). \
Proof Since A is closed and convex, A∞ is given by A∞ = t(A − a) for some a ∈ A. A being P-upper t>0 bounded, there exist u ∈ Y such that A ⊂ u − P. Now, let x ∈ A∞ ∩ P. Then, x ∈ P and x ∈ t(A − a) for every t > 0. Hence, for all t > 0, x ∈ tA − ta. Equivalently, ∀t > 0, t −1 x + a ∈ A ⊂ u − P. Accordingly, for all t > 0, x + ta ∈ tu − P. In particular, for tn −→ 0, we obtain x + tn (a − u) ∈ −P. Passing to limit we get x ∈ −cl (P). Thus, x ∈ P ∩ (−cl (P)), concluding that A∞ ∩ P ⊂ P ∩ (−cl (P)). As a direct consequence of Proposition 6, we state the following Theorem 6 Assume that (H3 ) is satisfied i.e., Y is a reflexive Banach space and P is closed and pointed. Let A be a nonempty closed and convex subset in Y. If A is P-upper bounded, then, A∞ ∩ P = {0Y }, where A∞ is the asymptotic cone of A. The following results provide (DP)max/min property and, consequently, efficient (P-Maximal/Minimal) point for any nonempty closed, convex and P-upper/lower bounded subset A of Y. Theorem 7 Assume that (H3 ) is satisfied. Let A be a nonempty closed and convex subset in Y. Then a) If A is P-upper bounded, then A has the (DP)max property. Moreover, every upper section of A is w-compact. In particular, MaxP (A) 6= 0. / b) If A is P-lower bounded, then A has the (DP)min property. Moreover, every lower section of A is w-compact. In particular, MinP (A) 6= 0. / Proof Let us prove part a), part b) can be deduced from a) by inverting the order. Let A satisfying the conditions of the theorem. Y being reflexive, its closed unit ball B is w-compact. Thus, for some γ > 0, [0, γ]A ∩ B is w-compact, which means that A is asymptotically compact. Now, by Theorem 6, we have A∞ ∩ P = {0Y }. Hence, from [26, Proposition 3.2.26] we obtain that every upper section, say A+ P (x) with x ∈ Y, of A is w-compact. [26, Corollary 3.2.27] allows us to conclude that A has the (DP)max property. In particular, MaxP (A) 6= 0. / Remark 20 As we will see subsequently in Corollary 1, if in Theorem 7 the subset A is in addition upward (resp. downward) directed, then its P-maximal (resp. minimal) points become P-maximums (resp. minimums) ones. Then, Theorem 7 improves [5, Theorem 3.8] by dropping both the normality and the non-emptiness of the interior of the ordering cone. Instead, we require that the cone be pointed. A similar result to Theorem 7, whenever the cone P is normal, is obtained in the following Theorem 8 Assume that (H4 ) is satisfied i.e., Y is a reflexive Banach space and P is closed and normal. Let A be a nonempty closed and convex subset in Y. a) If A is P-upper bounded, then A has the (DP)max property. Moreover, every upper section of A is w-compact. In particular, MaxP (A) 6= 0. / b) If A is P-lower bounded, then A has the (DP)min property. Moreover, every lower section of A is w-compact. In particular, MinP (A) 6= 0. / Proof As in the previous theorem, it suffices to prove a). Let u be a P-upper bound of A. Any upper section A+ P (a) of A is closed, convex and P-bounded (having u as a P-upper bound and a as P-lower bound. Then, the normality of P implies that A+ P (a) is bounded (thanks to Remark 8), therefore it is w-compact. The result follows therefore from [26, Proposition 3.2.20]. Remark 21 As in Remark 20, Theorem 8 improves [5, Theorem 3.8] by dropping the non-emptiness of the interior of the cone in the case of directed subsets. Next, we obtain similar existence results of maximal/minimal points for the same category of subsets in normed spaces but the cone will be required to have a compact base. Theorem 9 Assume that (H5 ) is satisfied i.e., Y is a normed space and P has a compact base. Let A be a nonempty closed and convex subset in Y.
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a) If A is a P-upper bounded subset, then A has the (DP)max property. Moreover, every upper section of A is compact. In particular, MaxP (A) 6= 0. / a) If A is P-lower bounded subset, then A has the (DP)min property. Moreover, every lower section of A is compact. In particular, MinP (A) 6= 0. / Proof Since P has a compact base, it is pointed and closed. Let A satisfying the conditions of the theorem. All the hypotheses of Theorem 6 are fulfilled, then A∞ ∩ P = {0Y }. Hence, the conclusion is a direct from [26, Corollary 3.2.27 ]. Part b) is similar to a). Now, reverting to the case where Y is a Banach space, we can improve the existence of maximal points in Theorem 9 by weakening the condition on the cone P. Also the subset A is not needed to be convex. Precisely, we have the following Theorem 10 Assume that (H6 ) is satisfied i.e., Y is a Banach space and P is closed and supernormal (i.e., P verifies the angle property (4)). Let A be a nonempty closed subset in Y . a) If A is P-upper bounded, then A has the (DP)max property. In particular, MaxP (A) 6= 0. / b) If A is P-lower bounded, then A has the (DP)min property. In particular, MinP (A) 6= 0. / Proof Let y∗ ∈ Y ∗ satisfying the angle property (4). Then, since A is P-upper bounded, y∗ (A) is bounded above (in the real line). Therefore, [9, Theorem 2.5 ] allows us to conclude that A satisfies the (DP)max property is. In particular, MaxP (A) 6= 0. / Part b) is similar. Remark 22 1. If Y is Hausdorff locally convex space, then P has compact base ⇒ P well-based ⇒ P supernormal ⇒ P normal, and P has compact base ⇒ P is Daniell. 2. If Y is a normed space, then P supernormal ⇒ P well-based. 3. Every non-trivial normed space (X, k · k) has a convex well-based cone with nonempty interior (see for example [26] for more details). Corollary 1 Under the assumptions of one of Theorems 7, 8, 9 or 10, if, in addition, P is pointed and A is upward (resp. downward) directed, then A admits a unique P-maximum (resp. P-minimum). A similar conclusion is obtained under (H7 ) as follows. Theorem 11 Assume that (H7 ) is satisfied i.e., Y is Hausdorff and P is closed and Daniell. a) Then every P-upper/lower bounded and closed subset A in Y has a P-maximal/minimal point. b) If in addition, P is pointed and A is upward/downward directed, then A admits a unique P-maximum/minimum point. Proof By [39, Theorem 3.3], A admits a P-maximal/minimal point. Part b) is guaranteed by the pointedness of P and directness of A. To present our unified results on the existence of maximal/minimal points, we complete our treatment by the finite dimensional case as follows. Theorem 12 Assume that (H8 ) is satisfied i.e., Y is a finite dimensional normed space and P is pointed. Then, any upward (resp. downward)-directed subset A admits a P-maximal (resp. P-minimal) in its closure. If, in addition, A is closed and upward (resp. downward) directed, then A admits a unique Pmaximum (resp. P-minimum). Proof Since Y is a finite dimensional normed space, it is reflexive and complete. Now, given that P is pointed, by [26, corollaries 2.1.23 and 2.2.11], it is normal. Then we are in the conditions of Theorem 8 to conclude the first part of the theorem. The last part of the conclusion is easy to check. Remark 23 Theorem 12 improves [5, Corollary 3.10] since here we don’t need the non-emptiness of the cone.
On the minima and maxima of directed free disposal subsets
15
5 Unified existence results of efficient points
To formulate our unified theorems, we need to fix the following notations which will also be used in the remaining of the paper: • • • • • •
D f (Y ) := the set of downward-free disposal subsets of Y. C(Y ) := the set of closed subsets of Y. CC(Y ) := the set of closed and convex subsets of Y. CCD f (Y ) := the set of closed, convex and downward-free disposal subsets of Y. Ud (Y ) := the set of upward-directed subsets of Y. CCD f Ud (Y ) := the set of closed, convex, downward-free disposal and upward-directed subsets of Y. Our dual treatment requires equally to consider the following notations:
• • • •
U f (Y ) := the set of upward-free disposal subsets of Y. CCU f (Y ) := the set of closed, convex and upward-free disposal subsets of Y. Dd (Y ) := the set of downward-directed subsets of Y. CCU f Dd (Y ) := the set of closed, convex, upward-free disposal and downward-directed subsets of Y.
Remark 24 The empty set and the space itself (0/ and Y ) are elements of D f (Y ), U f (Y ), C(Y ), CC(Y ), CCD f (Y ), CCU f (Y ). If in addition the cone P is reproducing, then they are also elements of Ud (Y ), Dd (Y ), CCD f Ud (Y ) and CCU f Dd (Y ). In the next section we focus our attention on existence of efficient points for subsets in CCD f Ud (Y ). Dually, we shall deduce efficient points for subsets in CCU f Dd (Y ). From Theorems 2, 4, 7, 8, 9, 11 and 12 we have the following: Theorem 13 Let A (resp. B) be a nonempty P-upper (resp. P-lower) bounded subset of Y. Then, MaxP (A) 6= 0/ (resp. MinP (B) 6= 0/ ) in one of the following situations: • (Hi ) holds for some i ∈ {1, 6, 7, 8} and A ∈ C(Y ) (resp. B ∈ C(Y )); • (Hi ) holds for some i ∈ {3, 4, 5} and A ∈ CC(Y ) (resp. B ∈ CC(Y )); • (H2 ) holds and A ∈ CCD f (Y ) (resp. B ∈ CCU f (Y )). If, in addition, the underlying subset is in addition upward/downward-directed, then taking into account Corollary 1, we obtain the uniqueness of the P-maximum/minimum element as follows: Theorem 14 Let A (resp. B) be a nonempty P-upper (resp. P-lower) bounded subset of Y. Then, A (resp. B) admits a unique P− maximum (resp. minimum)i.e., ∃!a ∈ A s.t maxP (A) = {a} (resp. ∃!b ∈ B s.t minP (B) = {b}) in one of the following situations: • (Hi ) for some i ∈ {1, 6, 7, 8} and A ∈ CUd (Y ) (resp. B ∈ CDd (Y )) • (Hi ) for some i ∈ {3, 4, 5} and A ∈ CCUd (Y ) (resp. B ∈ CCDd (Y )). • (H2 ), P is pointed and A ∈ CCD f Ud (Y ) (resp. B ∈ CCU f Dd (Y )). Remark 25 1. Observe that (H4 ) =⇒ (H3 ) (since Y is Hausdorff, then P normal=⇒ P pointed). 2. If Y is a Banach space, (H5 ) =⇒ (H6 ). 3. If Y is a reflexive Banach space, (H5 ) or (H6 ) =⇒ (H4 ). 4. If Y is a normed space, (H5 ) =⇒ (H7 ). 5. If Y is a Banach space, (H6 ) =⇒ (H1 ). 6. If P has a compact base, then it is pointed and closed. 6 Application to hypo/epi-graphical level sets and related Extended regularizations
The aim of this section is to apply the results of the previous section to obtain P-maximum/minimum points for hypo/epi-graphical level sets Axf and Bxf defined in the introduction section for a given mapping f : X → Y • , where Y is as before (i.e., a real linear topological space ordered by a convex cone P) and X is a real vector normed space. Since the use of these level sets is easier in their sequential formulation1 , for simplicity, when we involve (H2 ) or (H7 ) we assume that Y is in addition metrizable, for the cases of (Hi ), i = 1, 3, 4, 5, 6, 8 this metrizability assumption is automatically satisfied. Remark 26 Notice that 1
See [3, Proposition 4.2]. Some examples of these sets can be found in [3].
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1. the upper epi-graphical level Bxf can be obtained from the lower hypo-graphical Axf by inverting the orders P and −P. 2. given that Axf = lim inf H (z) and Bxf = lim inf E (z), Axf and Bxf are closed subsets (since the lower z→x
z→x
limit of given set-valued map is always closed). Proposition 7 Let f : X → Y • and x ∈ Dom f . Then, Axf (resp. Bxf ) is: i) P-upper (resp. lower) bounded by f (x); ii) Convex; iii) upward (downward) directed if Y is a lattice space; iv) downward (resp. upward) free disposal. Proof The proofs of these elementary properties given in [4, Proposition 2.4] or else in [3] are still true in the generality (independently of the partial order of Y ). The directness in iii) requires that Y be, in addition, a lattice space. In the perspective to a global set-valued analysis we are conducted to consider the set-valued hypo/epi profile mappings A f : X → 2Y , x 7→ Axf , and B f : X → 2Y , x 7→ Bxf respectively. Thus, from Proposition 7, we obtain the following: Proposition 8 Let f : X → Y • . Then, for all x ∈ Dom ( f ), A f (x) = Axf ∈ CCD f (Y ) and B f (x) = Bxf ∈ CCU f (Y ). If, in addition, Y is a lattice space, then A f (x) = Axf ∈ Ud (Y ) and B f (x) = Bxf ∈ Dd (Y ). According to Proposition 8 and Theorem 5, we have to suppose in the remaining of the paper that Y is neither strongly P-complete nor strongly −P-complete to ensure that Y contains proper downward and upward free disposal subsets i.e., D f (Y )\{0,Y / } 6= 0/ and U f (Y )\{0,Y / } 6= 0. / The following result collects further properties on the set-valued profiles: Proposition 9 Assume that X is a real linear normed space and Y is a metrizable linear topological space partially ordered by a non-trivial convex cone P. Let f : X → Y • . Then the assertions below are satisfied. If f is positively homogenous, then so are A f and B f in their domains. If Int (P) 6= 0, / then A f and B f are lower semicontinuous in their domains. If Int (P) 6= 0, / then both A f and B f admit a continuous selection. A f and B f are order-convex valued that is the values are full w.r.t P. T If the cone P is pointed, T then set-valued upper (lower) sections defined by A f (.) = A f (.) (. + P) (resp. B f (.) = B f (.) (. − P)) is order-interval valued. f) A f and B f order-complete-valued if (Y, P) is order-complete. T g) If Y is a lattice space, T then set-valued lower (resp. upper) sections defined by A f (.) = A f (.) (. − P) (resp. B f (.) = B f (.) (.+P)) range in the upward (resp. downward) directed subsets set Ud (Y ) (resp. Dd (Y )). T h) If Y is a lattice space, T then set-valued lower (resp. upper) sections defined by A f (.) = A f (.) (. − P) (resp. B f (.) = B f (.) (. + P)) range in the downward (upward) free disposal subsets set D f (Y ) (resp. U f (Y )).
a) b) c) d) e)
Proof Part a) (resp. b)) is satisfied by the same arguments of [2, Lemma 3] (resp., [2, Lemma 2]). The assertion c) is a consequence of b) and the well known Michael selection Theorem. The claim in d) descends from iv) of Proposition 7 and and 3) of Remark 12. Part e) comes from d) and Lemma 7. Now, part f ) steams immediately from d) and Lemma 2. On the other hand, the assertion g) is a quick combination of assertion of Proposition 8 and Propositions 4 (resp. 5). Finally, Propositions 2 and 3 imply part h), ending the proof. Lemma 8 Let f : X → Y • and x ∈ Dom f . Consider the following: a) Axf (resp. Bxf ) is nonempty; b) f is P-lower (resp. upper) bounded around x.
On the minima and maxima of directed free disposal subsets
17
Then i) b) =⇒ a). ii) If, in addition, Axf (resp. Bxf ) admits a P-maximum (resp. minimum) point, then b) ⇐⇒ a). Proof The assertion i) is immediate while ii) comes from [4, b) of Theorem 5.4]. Now, as an immediate application of Theorem 14, we deduce the following Theorem 15 Let X be a real linear normed space, Y be a linear topological space partially ordered by a non-trivial convex cone P and f : X → Y • a vector-valued map. Assume that (Y, P) is a lattice space and one of the following assumptions hold: (Hi ) for some i ∈ {1, . . . , 8}\{2, 7}; (H2 ), Y is metrizable and P is pointed. (H7 ) and Y is metrizable. Let x ∈ Dom ( f ) such that f is P-bounded around x. Then a) Axf is nonempty and admits a unique P-maximum i.e., there exists a unique ax ∈ Axf such that maxP (Axf ) = {ax }; b) Bxf is nonempty and admits a unique P-minimum i.e., there exists a unique bx ∈ Bxf such that minP (Bxf ) = {bx }. Proof Since f is P-bounded around x, it follows from i) of Lemma 8 that Axf (resp. Bxf ) is nonempty. Moreover, from Proposition 8 it results that Axf ∈ CCD f Ud (Y ) (resp. Bxf ∈ CCU f Dd (Y )). Then, the conclusion of the theorem descends immediately from Theorem 14. Remark 27 Note that 1. In the above theorem it is not needed to assume that Y is a lattice space in the case of (H2 ) (it is automatically satisfied In view of Lemma 2). 2. Theorem 15 extends [3, Lemma 6.7] to the setting where Y is a locally convex order-complete spaces (the case of (H2 )). 3. Theorem 15 contains [4, Proposition 4.3] (the case of (H7 )). 4. Theorem 15 improves [4, Proposition 4.4] by dropping the normality of the (pointed) cone (the case of (H3 )). 5. Theorem 15 brings several further answers of the question treated in [5] concerning directed sets admitting a supremum in their closure. Corollary 2 Let X be a real linear normed space, Y be a linear topological space partially ordered by a non-trivial convex cone P and f : X → Y • a vector-valued map. Assume that (Y, P) is a lattice space and one of the following assumptions hold: (Hi ) for some i ∈ {1, . . . , 8}\{2, 7}; (H2 ), Y is metrizable and P is pointed. (H7 ) and Y is metrizable. Then, for all x ∈ Dom ( f ). Axf (resp. Bxf ) is nonempty if, and only if, it admits a unique P-maximum (resp. minimum). Proof This result is a quick combination of Lemma 8 and Theorem 15.
Now, thanks to Theorem 15, we are in a position to define the related regularizations in an unified framework. To do that, we need the following notation: for any f : X −→ Y • , we will write: D( f ) = {x ∈ Dom ( f ) and f is P − lower bounded around x}; and D( f ) = {x ∈ Dom ( f ) and f is P − upper bounded around x}. Definition 21 Under the assumptions of Corollary 2, we define the following:
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M. Ait Mansour and H. Riahi
The lower regularization of f denoted by f : maxP (Ax¯f ) f (x) ¯ := −∞ +∞
if x¯ ∈ D( f ) if x¯ ∈ Dom ( f ) ∩ (D( f ))c if x¯ ∈ / Dom ( f ).
The upper regularization of f denoted by f : minP (Bx¯f ) f (x) ¯ := +∞ +∞
if x¯ ∈ D( f ) if x¯ ∈ Dom ( f ) ∩ (D( f ))c if x¯ ∈ / Dom ( f ).
Remark 28 The notation f (x) ¯ (resp. f (x)) ¯ coincide with that adopted in [3] when x¯ ∈ D( f ) (resp. x¯ ∈ D( f )) i.e., f (x) ¯ = v − lim inf f (x) and f (x) ¯ = v − lim sup f (x). The letter v in v − lim inf and v − lim sup x→x¯
x→x¯
refers to the vector-values of f . Thanks to their compatibility with usual lower and upper limits in the finite-dimensional case (see [3, Section 7]), f (resp. f ) will also be denoted by v − lim inf f (resp. v − lim sup f ). Remark 29 Notice that the regularizations of Definition 21 • • • •
extend (correspondingly to (H2 )) to locally Hausdorff spaces those introduced in [3]; coincide (correspondingly to (H4 )) with that of [2]; improve (correspondingly to (H3 )) that of [2] and [4] by dropping the normality of the cone; leads to new (extended) definitions in the settings of (Hi ), i ∈ {1, . . . , 7}\{2, 3, 4}.
Moreover, these regularizations enjoy the semicontinuity property in the sense of Penot and Thera [40] (see also [3] or [18]). Precisely, we state the following: Proposition 10 Let X be a real linear normed space, Y be a linear topological space partially ordered by a non-trivial convex cone P and f : X → Y • a vector-valued map. Assume that (Y, P) is a lattice space and one of the following assumptions hold: (Hi ) for some i ∈ {1, . . . , 8}\{2, 7}; (H2 ), Y is metrizable and P is pointed. (H7 ) and Y is metrizable. If, in addition, P has a nonempty interior, then a) b) c) d)
f f f f
is P-lower semicontinuous; is P-lower semicontinuous at a point x if, and only if, f (x) = f (x); is P-upper semicontinuous; is P-upper semicontinuous at a point x if, and only if, f (x) = f (x).
In addition, f (resp. f ) is the greatest (resp. least) P-lower (resp. P-upper) semicontinuous map minorizing (resp. majorizing) f . Proof The same proof of [4, Theorem 5.4] still is valid by remarking that the P-semicontinuity of these regularizations depend only on the non-emptiness of the interior of the cone. 7 From semi-continuous regularizations to extended Radial epi-derivatives and global vectorial optimization 7.1 Extended Radial epi-derivatives
Radial epi-derivatives of real/vector/set-valued maps together with (vector/set-valued) optimization have attracted the attention of many optimizers during the last four decades, see [37, 38, 47, 48, 7, 8, 17, 31, 27,22,33]. In [23], under the order completeness hypothesis on (Y, P), the author investigates the class of extended vector-valued maps by considering the lower and upper radial epi-derivatives defined by means of the standard regularizations missing the semicontinuity property. This reason has motivated the authors of [2] to consider the more recent P-lower/upper semicontinuous regularizations defined by the above maps f and f for the case of reflexives Banach spaces ordered by a normal cone with a nonempty interior. The aim of this subsection is to provide the possible extensions through various assumptions on the ordering cone and topology of the target space. For a given extended vector-valued map f : X → Y , we first fix the following notations:
On the minima and maxima of directed free disposal subsets
19
¯ • the first order difference quotient of f at a point x : (∆t f )x¯ (.) := t −1 ( f (x¯ + t.) − f (x));
t > 0;
¯ u) = sup(∆t f )x¯ (u). ¯ u) = inf(∆t f )x¯ (u) and f+0 (x; • f−0 (x; t>0
t>0
In the following definition, X is supposed to be real linear normed space, Y a linear topological space partially ordered by a non-trivial convex cone P and f : X → Y • a vector-valued map. Moreover, to apply the regularization (of Section 6), the space (Y, P) is assumed to be a lattice space such that one of the following assumptions hold (the same assumptions of Definition 21): (Hi ) for some i ∈ {1, . . . , 8}\{2, 7}; (H2 ), Y is metrizable and P is pointed. (H7 ) and Y is metrizable. Definition 22 For x¯ ∈ Dom( f ), we define the • Lower radial epiderivative of f by DRe f (x; ¯ u) := v − lim inf f−0 (x; ¯ u0 ) ∈ Y , for every u ∈ X; 0 u →u
• Upper radial epiderivative of f by R
De f (x; ¯ u) := v − lim sup f+0 (x; ¯ u0 ) ∈ Y • , for every u ∈ X. u0 →u
¯ .)(u) and f+0 (x; With the following notations f−0 (x; ¯ u) := f−0 (x; ¯ u) := f+0 (x; ¯ .)(u), we are able to give the following complete definition: 0 if u ∈ D( f−0 (x; ¯ .)) ¯ u) f− (x; R ¯ u) = −∞ De f (x; if u ∈ Dom ( f−0 (x; ¯ .)) ∩ (D( f−0 (x; ¯ .)))c 0 +∞ if u ∈ / Dom ( f− (x; ¯ .)), and
where and
f+0 (x; ¯ u) R De f (x; ¯ u) = +∞ +∞
if u ∈ D( f+0 (x; ¯ .)) if u ∈ Dom ( f+0 (x; ¯ .)) ∩ (D( f−0 (x; ¯ .)))c 0 if u ∈ / Dom ( f+ (x; ¯ .)).
¯ .)) = u ∈ Dom ( f−0 (x; ¯ .)) : f−0 (x; ¯ .) is P-lower bounded around u D( f−0 (x; D( f+0 (x; ¯ .)) = u ∈ Dom ( f+0 (x; ¯ .)) : f+0 (x; ¯ .) is P-lower bounded around u .
In other words, for any x¯ ∈ Dom( f ), f 0 (x;.) ¯
DRe f (x; ¯ u) = maxP (Au−
f 0 (x;.) ¯
R
) and De f (x; ¯ u) = minP (Bu+
).
Remark 30 Definition 22 extends the radial epiderivatives of [23, 2]. Lemma 9 For all x ∈ Dom( f ), for all u ∈ X, we have R
DRe f (x; ¯ u) P• f−0 (x; ¯ u) P• f+0 (x; ¯ u) P• De f (x; ¯ u). Proof Straightforward from [3, Proposition 5.1] and the definition of the radial epi-derivatives.
Remark 31 By remarking that the proofs presented in [2] depend only on the hypo/epi-graphical sets of the considered function, the following properties of [2, 23] are extended to the setting of Definition 22: 1. [2, Theorem 5] or [23, Theorem 3. 13]; 2. [2, Theorem 6, Theorem 8] or [23, Theorem 3. 2]; 3. [2, Theorem 7] or [23, Proposition 3. 7] which can also be proved directly with the help of a) of R Proposition 9 i.e., DRe f (x; ¯ .) and De f (x; ¯ .) are positively homogenous; 4. [2, Theorem 9] or [23, Theorem 3. 3, Proposition 3.1]; 5. [2, Theorem 10] or [23, Proposition 3.1]; 6. [2, Theorem 12].
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7.2 Global maximality conditions for nonconvex vector-valued functions
Since the introduction of dynamic programming by Bellman (1953) and the maximum principle by Pontryagin (1958), global optimization problems are subjects of intensive studies that find the motivation in many fields of applications: production planning, design of water and gaz, transportation, power distribution systems and so forth. So, finding a global optimum of a function or even a feasible point nearest to the (theoretical) global one in such domains of applications is of great interest. For convex or quasi-convex differentiable extended real-valued functions, global optimization results are very known in the literature. In this section, we present new conditions in the nonconvex nondifferentiable extended vector-valued case. Precisely, we consider the following vector maximization problem: (P) Maximize f (x) subject to x ∈ D where f : X −→ Y • is an extended vector-valued mapping, X being a Banach space, Y a Banach space ordered by a closed, convex, pointed and proper cone P (these assumptions correspond to the setting of (H3 )) such that (Y, P) be latticial, and D ⊂ X a nonempty convex (constraint) set. A point x¯ ∈ Dom ( f ) is a (global) solution to (P) if x¯ ∈ D and f (x) P f (x) ¯ for all x ∈ D. We first present a necessary optimality condition in the following. Theorem 16 Let x be a solution to problem (P) and Ec ( f ) = {x ∈ D : f (x) = c}, for c ∈ Y . Then DRe f (y; x − y) P 0, ∀y ∈ E f (x) ( f ), ∀x ∈ D. Proof Let y ∈ E f (x) ( f ) and x ∈ D. For all t ∈ (0, 1), y + t(x − y) ∈ D and by the assumption we have f (y + t(x − y)) P f (x) ¯ = f (y). Thus, using Lemma 9, we are able to write DRe f (y; x − y) P f−0 (y; x − y) = inf(∆t f )y (x − y) P (∆t f )y (x − y) t>0
= t −1 [ f (y + t(x − y)) − f (y)] P 0,
which completes the proof. Remark 32 From the proof of Theorem 16, ∀y ∈ E f (x) ( f ), ∀x ∈ D, we have DRe f (y; x − y) P t −1 [ f (y + t(x − y)) − f (y)], ∀t > 0.
(10)
Then, a sufficient condition for a point x to be a global minimum of f over D is 0 P DRe f (y; x − y) for all x ∈ D and all y ∈ E f (x) ( f ). From [23, c) of Corollary 3.5], it can be easily seen that (10) is also necessary for x to be global minimum R of f over D. The condition 0 P De f (y; x − y) for all x ∈ D and all y ∈ E f (x) ( f ) is equally a necessary p condition for x to be global minimum of f but not sufficient as shows the counter-example of |x| at x = 1 (see [23, Remark 3.6]). We present now a sufficient optimality condition for the maximization problem (P). Theorem 17 Let x ∈ D and Ec ( f ) = {x ∈ X : f (x) = c}, for c ∈ Y . R If, for every y ∈ E f (x) ( f ) and for every x ∈ D, De f (y; x − y) P 0, then x is a solution to (P). Proof Let x ∈ D and y ∈ E f (x) ( f ). Using Lemma 9 we obtain R
sup(∆t f )y (x − y) = f+0 (y; x − y) P De f (y; x − y) P 0. t>0
Accordingly, (∆t f )y (x − y) P 0, ∀t > 0, and hence, t −1 [ f (y + t(x − y)) − f (y)] P 0. In particular with t = 1, we derive f (x) P f (y) = f (x), achieving the proof.
On the minima and maxima of directed free disposal subsets
21
Remark 33 From the proof of Theorem 17, we see that R
t −1 [ f (y + t(x − y)) − f (y)] P De f (y; x − y), ∀t > 0, ∀y ∈ E f (x) ( f ), ∀x ∈ D. R
Then, a necessary condition for a point x to be a global minimum of f over D is 0 P De f (y; x − y) for all x ∈ D and all y ∈ E f (x) ( f ). Next, we provide the link between the radial epiderivatives and the gradient of a real-valued differentiable function defined on X. Proposition 11 Let f : X −→ R be a differentiable function, and f 0 denotes the gradient of f . Then, for every x, y ∈ X, R
DRe f (y; x − y) ≤ h f 0 (y), x − yi ≤ De f (y; x − y).
(11)
Proof Let t > 0 and x, y ∈ X. Of course, by the differentiability of f , for t close to 0 we can write o(tkx − yk) 0 , f (y + t(x − y)) − f (y) = t h f (y), x − yi + t o(tkx − yk) converges to 0 when t goes to 0. Therefore, for a nonnegative real number t close to where t zero, it results that (∆t f )y (x − y) = (h f 0 (y), x − yi +
o(tkx − yk) ). t
(12)
Clearly, lim(∆t f )y (x − y) = h f 0 (y), x − yi. t→0
On the other hand, by Lemma 9, for all t > 0, we have R
DRe f (y; x − y) ≤ f−0 (y; x − y) ≤ (∆t f )y (x − y) ≤ f+0 (y; x − y) ≤ De f (y; x − y), which in turn implies that R
DRe f (y; x − y) ≤ (∆t f )y (x − y) ≤ De f (y; x − y).
(13)
Passing to limit in (13) when t goes to 0 we end at R
DRe f (y; x − y) ≤ h f 0 (y), x − yi ≤ De f (y; x − y),
(14)
completing the proof. Remark 34 Proposition 11 is true for both Gˆateaux and Fr´echet differentiability of f .
Let us now recall the notion of quasiconvex real functions and compare the classic optimality conditions of this class of functions with those obtained in Theorems 16 and 17. Definition 23 Let K be a convex subset in X. A function f : K −→ R is said to be quasiconvex if for all x, y ∈ K f (tx + (1 − t)y) ≤ max{ f (x), f (y)}, ∀t ∈ [0, 1]. If f : K −→ R is quasiconvex differentiable function, then: • a necessary optimality condition for a point x to be a global maximum of f over K is: h f 0 (y), x − yi ≤ 0, ∀y ∈ E f (x) ( f ), ∀x ∈ K.
(15)
• a sufficient optimality condition for a point x to be a global maximum of f over K is: f 0 (y) 6= 0 and h f 0 (y), x − yi ≤ 0, ∀y ∈ E f (x) ( f ), ∀x ∈ K.
(16)
Then, thanks to Proposition 11, the necessary condition of Theorem 16 gives more information than the condition (16) while the sufficient one in Theorem 17 implies the second part of (16). Notice that (15) and (16) are very known in the literature, they are always derived from the characterization of quasiconvexity due to Arrow and Enthoven [6] (see also [25, Theorem 1]). For further discussions on quasi-convex programming using the normal operator, subdifferential or generalized differential we refer for example to [11] and [34, 35].
22
M. Ait Mansour and H. Riahi
8 Conclusion and further research questions
In this paper we have presented new and unified existence theorems of optimum (cone maximum/minimum points) of directed convex free disposal subsets in a partially ordered space. This class of subsets has been proved to be of a particular relevance since it covers the (recent) so-called hypo/epi-graphical level sets for general extended vector-valued maps. Those remarkable points for these level sets are the key idea in defining adequate regularizations and consequently radial epi-derivatives for vector-valued maps that express the optimality conditions of strong Pareto optima for vector-valued functions. Notice that from c) of Proposition 9, we can even define continuous regularizations rather than semicontinuous ones. Also, a further question is to find conditions under which these regularizations preserve the convexity or quasi-convexity of vector-valued maps. (Note that in the case where the cone P is Daniell the above regularizations have been already proved to preserve convexity see [4]). Another point that deserves a forthcoming attention is to characterize the weak Pareto optima for vector-valued functions by the use of the above extended radial epi-derivatives as in [23]. Acknowledgements The authors are grateful to an anonymous referee for his useful remarks and suggestions, leading to the improvement of the presentation of this paper.
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