The set-valued optimization problem minC F(x), G(x)\(-K) 6= ; is considered, where F : Rn Rm and G : Rn Rp are set-valued functions, and C Rm and K Rp are closed convex cones. Two type of solutions, called w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers), are treated. In terms of the Dini set-valued directional derivative first-order necessary conditions for a point to be a w-minimizer, and first-order sufficient conditions for a point to be an i-minimizer are established, both in primal and dual form. Key words: Set-valued optimization, First-order optimality conditions, Dini derivatives.

Ginchev Ivan

Rocca Matteo

English

Research Papers in Economics

UNIVERSITÀ DELL'INSUBRIA
FACOLTÀ DI ECONOMIA
http://eco.uninsubria.it
Ivan Ginchev, Matteo Rocca
On constrained set-valued
optimization
2007/10© Copyright Ivan Ginchev, Matteo Rocca
Printed in Italy in Ottobre 2007
Università degli Studi dell'Insubria
Via Monte Generoso, 71, 21100 Varese, Italy
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Authors only, and not necessarily the ones of the Economics
Faculty of the University of Insubria.On constrained set-valued optimization∗
Ivan Ginchev† Matteo Rocca‡
Abstract
The set-valued optimization problem minC F(x), G(x)∩(−K) 6= ∅, is considered,
where F : Rn   Rm and G : Rn   Rp are set-valued functions, and C ⊂ Rm
and K ⊂ Rp are closed convex cones. Two type of solutions, called w-minimizers
(weakly eﬃcient points) and i-minimizers (isolated minimizers), are treated. In
terms of the Dini set-valued directional derivative ﬁrst-order necessary conditions
for a point to be a w-minimizer, and ﬁrst-order suﬃcient conditions for a point to
be an i-minimizer are established, both in primal and dual form.
Key words: Set-valued optimization, First-order optimality conditions, Dini
derivatives.
Math. Subject Classiﬁcation: 49J53, 49J52, 90C29, 90C30, 90C46.
1 Introduction
The constrained set-valued optimization problem (svp)
minCF(x), G(x) ∩ (−K) 6= ∅, (1)
is considered, where F : Rn   Rm and G : Rn   Rp are set-valued functions (svf)
with non-empty values, and C ⊂ Rm and K ⊂ Rp are closed convex cones. First order
optimality conditions in terms of the Dini set-valued directional derivative are derived.
The obtained results generalize those of [3] from vector to set-valued problem, and of
[1] from unconstrained to constrained problem. Recently optimality conditions for svp
are studied mainly by means of epiderivatives, e. g. in [4], [5] and [2]. We consider the
optimality conditions based on directional derivatives as certain alternative of those based
on epiderivatives. Some comparison of the two methods is done in [1].
∗Communicated at: New Trends in Mathematics and Informatics, Jubilee International Conference
60 years Institute of Mathematics and Informatics Bulgarian Academy of Sciences, 6–8 July, 2007, Soﬁa
(Bulgaria).
†University of Insubria, Department of Economics, Via Monte Generoso 71, 21100 Varese, Italy,
iginchev@eco.uninsubria.it.
‡University of Insubria, Department of Economics, Via Monte Generoso 71, 21100 Varese, Italy,
mrocca@eco.uninsubria.it.
12 Preliminaries
The space Rk is considered with the usual topology. It is generated by an arbitrary
norm. Since all the norms in Rk are equivalent, we make use of this sometimes choosing a
special norm. The dual pairing in Rk is a denoted h·,·i. It is a mapping Rk ×(Rk)∗ → R,
where (Rk)∗ stands for the dual of Rk. In fact (Rk)∗ can be identiﬁed with Rk as linear
and topological spaces, but when Rk is considered with a norm, the dual space (Rk)∗ is
supplied with the dual norm. When Rk is supplied with an Euclidean norm, then (Rk)∗
can be identiﬁed with Rk also as a norm space. The notations Bk and ¯ Bk are used for
the open and closed unit balls, and Bk(x0) and ¯ Bk(x0) for the open and closed unit balls
with center x0.
For a given closed convex cone M ⊂ Rk its positive polar cone is deﬁned by M0 = {ξ ∈
(Rk)∗ | hξ, xi ≥ 0 for all y ∈ M}. When x0 ∈ M we put M0[x0] = {ξ ∈ M0 | hξ,x0i = 0}
and M[x0] = (M0[x0]))0. It holds M ⊂ M[x0].
When Rk is considered with a concrete norm, the distance from a point x ∈ Rk to a set
A ⊂ Rk is given by d(x,A) = inf{kx − yk | a ∈ A}. The oriented distance from x to A
is deﬁned by D(x,A) = d(x,A) − d(x,Rk \ A). When M ⊂ Rk is a proper closed convex
cone, then D(x,−M) = sup{hξ, xi | ξ ∈ M0, kξk = 1}. We deﬁne the oriented distance
D(P,A) from a set P ⊂ Rk to the set A ⊂ Rk putting D(P,A) = inf{D(x,A) | x ∈ P}.
Using the oriented distance we introduce the following notation. Let M ⊂ Rk be a
cone and let a be a real number. Then we put M(a) = {x ∈ Rk | D(x, M) ≤ akxk}.
The weakly eﬃcient frontier (w-frontier) w-MinMA and the properly eﬃcient frontier (p-
frontier) p-MinMA of A are deﬁned respectively by w-MinMA = {x ∈ A | A∩(x−intM) =
∅} and p-MinMA = {x ∈ A | ∃a ∈ (0, 1) : A ∩ (x − M(a)) = {x}}.
The set of the feasible points of svp (1) is deﬁned by G = {x ∈ Rn | G(x) ∩ (−K) 6=
∅}. Further N(x0) denotes the family of the neighbourhoods of x0. We deal with local
solutions of (1), which in any case are pairs (x0,y0), y0 ∈ F(x0), with x0 feasible. Here
we use the following concepts of solutions for problem (1). The pair (x0,y0), x0 ∈ Rn,
y0 ∈ F(x0), is said a w-minimizer (weakly eﬃcient point) if there exists U ∈ N(x0) such
that x ∈ U ∩ G implies F(x) ∩ (y0 − intC) = ∅ (then necessary y0 ∈ w-MinCF(x0)).
The pair (x0,y0) is said an i-minimizer (isolated minimizer) if there exists U ∈ N(x0)
and a constant A > 0 such that D(F(x) − y0,−C) ≥ Akx − x0k and y0 ∈ p-MinCF(x0)
for x ∈ U ∩ G (the concept of i-minimizer is norm-independent, since all norms in ﬁnite-
dimensional spaces are equivalent).
The svf Φ : Rn   Rk is said locally Lipschitz at x0 ∈ Rn, if there exists U ∈ N(x0) and
a constant L > 0, such that for x1, x2 ∈ U it holds Φ(x2) ⊂ Φ(x1) + Lkx2 − x1k ¯ Bk. The
svf Φ is said locally Lipschitz, if it is locally Lipschitz at each x0 ∈ Rn. Given a cone
M ⊂ Rk, we say that Φ is locally M-Lipschitz at x0 if the svf x   Φ(x) + M is locally
Lipschitz at x0. The svf Φ is said locally M-Lipschitz, if it is locally M-Lipschitz at each
point x0.
The cone M ⊂ Rk is said pointed if (−M)∩M = {0}, and M is contained in a half-space
of Rk. If Rk is supplied with an Euclidean norm, then the cone M is said non-obtuse, if
hx1,x2i ≥ 0 for all x1,x2 ∈ M. Each non-obtuse cone is pointed. The following result
converts in some sense this statement.
Lemma 1 ([1]) If M ⊂ Rk is a pointed closed convex cone, then there exists an Euclidean
norm in Rk with respect to which M is a non-obtuse cone.
2The next lemma is essential for the proof of the optimality conditions for (1).
Lemma 2 ([1]) Let M ⊂ Rk be a non-obtuse closed convex cone and M \ {0} 6= ∅. Let
the svf Φ : Rn   Rk be C-Lipschitz with constant L in U ∈ N(x0) and y0 ∈ Φ(x0).
Suppose that for some σ ∈ (0, 1/2) it holds Φ(x0) ∩ (y0 − M(2σ)) = {y0}. Then for each
x ∈ U and each y ∈ Φ(x) ∩ (y0 − M(σ)) it holds
ky − y
0k ≤
L(1 + σ)
σ
kx − x
0k.
Our aim is to obtain optimality conditions for svp (1) in terms of Dini derivatives. For
the svf Φ : Rn   Rk the Dini derivative of Φ at (x0,y0), y0 ∈ Φ(x0), in direction u ∈ Rn
is deﬁned as the upper limit
Φ
0(x
0,y
0;u) = Limsup
t → 0+
1
t
 
Φ(x
0 + tu) − y
0
.
3 First-order optimality conditions
Theorem 1 (Necessary Conditions, w-minimizers) Consider svp (1) with C ⊂ Rm
and K ⊂ Rp closed convex cones, and F : Rn   Rm and G : Rn   Rp svf. Let the pair
(x0,y0), x0 ∈ Rn, y0 ∈ F(x0), be a w-minimizer of svp (1), and let z0 ∈ G(x0) ∩ (−K).
Then
∀u ∈ R
m : (F × G)
0(x
0,(y
0,z
0);u) ∩ (−(intC × intK[−z
0]) = ∅. (2)
Proof. Suppose the contrary, that there exists (¯ y0, ¯ z0) ∈ (F × G)0(x0,(y0,z0);u0), such
that ¯ y0 ∈ −intC, ¯ z0 ∈ −intK[−z0]. Let ¯ y0 = limk(1/tk)(yk−y0), ¯ z0 = limk(1/tk)(zk−z0),
where yk ∈ F(x0 + tku0) and zk ∈ G(x0 + tku0) for some tk → 0+ and u0 ∈ Rn. These
equalities imply that yk → y0 and zk → z0, and the boundedness of the sequences {yk}
and {zk}.
Let ¯ η ∈ K0, k¯ ηk = 1. We show that there exists a positive integer k(¯ η) and a neighbour-
hood V (¯ η) of ¯ η, such that hη,zki < 0 for k > k(¯ η) and η ∈ V (¯ η). For this purpose we
consider the following cases:
10. ¯ η ∈ K0[−z0]. Since ¯ z0 ∈ −intK[−z0], we have
lim
k
1
tk
h¯ η,z
k − z
0i = h¯ η, ¯ z
0i < 0.
Therefore there exists k(¯ η), such that for all k > k(¯ η) it holds h¯ η,zki < h¯ η,z0i = 0. Now
the boundedness of the sequence {zk} implies the existence of a neighbourhood V (¯ η) of
¯ η, such that h¯ η,zki < 0 for all k > k(¯ η).
20. ¯ η ∈ K0 \ K0[−z0]. We have h¯ η,z0i < 0, whence h¯ η,zki < 0 for all k > k(¯ η) with
suitable k(¯ η). This implies as above hη,zki < 0 for all k > k(¯ η) and η ∈ V (¯ η) with
suitable neighbourhood V (¯ η) of ¯ η.
The set Γ = {η ∈ K0 | kηk = 1} is compact, whence Γ ⊂ V (¯ η1) ∪ ... ∪ V (¯ ηs) for some
¯ η1,..., ¯ ηs ∈ Γ. Let k0 = max(k(¯ η1),...,k(¯ ηs)). Take k > k0. Then hη,zki < 0 for all
η ∈ Γ, and hence for all η ∈ K0. Therefore zk ∈= intK ⊂ −K. In other words, the points
x0 + tku0 are feasible.
3According to the made assumption ¯ y0 = limk(1/tk)(yk−y0) ∈ −intC. Therefore yk−y0 ∈
−intC for all suﬃciently large k, a contradiction to the hypothesis that (x0,y0) is a w-
minimizer of (1). 2
Remark 1 Condition (2), which can be referred to as primal form condition, can be
substituted by the equivalent dual form condition
∀u ∈ Rn \ {0} : ∀(¯ y0, ¯ z0) ∈ (F × G)0(x0,(y0,z0);u) :
∃(ξ,η) ∈ C0 × K0[−z0],(ξ,η) 6= (0, 0) : hξ, ¯ y0i + hη, ¯ z0i ≥ 0. (3)
Theorem 2 (Suﬃcient Conditions, i-minimizers) Consider svp (1) with C ⊂ Rm
pointed closed convex cone, K ⊂ Rp closed convex cone, F : Rn   Rm locally C-Lipschitz
svf, and G : Rn   Rp locally Lipschitz svf. Suppose that the pair (x0,y0), x0 ∈ Rn,
y0 ∈ F(x0), is such that y0 ∈ p-MinCF(x0), and there exists z0 ∈ G(x0) for which
∀u ∈ R
n \ {0} : (F × G)
0(x
0,(y
0,z
0);u) ∩ (−(C × K[−z
0])) = ∅. (4)
Suppose also that the svf G satisﬁes the following condition:
G(x0,z0) :
∃U ∈ N(x0) : ∃` > 0 : ∀x ∈ U :
G(x) ∩ (−K) 6= ∅ ⇒ G(x) ∩ `kx − x0k ¯ Bp(z0) ∩ (−K) 6= ∅
Then (x0,y0) is an i-minimizer of svp (1).
Proof. We can assume without loss of generality that Rm (the image space of F) is
supplied with an Euclidean norm, with respect to which the cone C is non-obtuse. We
may assume that F is C-Lipschitz with constant L > 0 on ¯ Bn(x0). Suppose that (x0,y0)
is not an i-minimizer. Fix a sequence εk → 0+. According to the assumption, there exist
sequences tk → 0+, and uk ∈ Rn, kukk = 1, such that:
10. G(x0 + tkuk) ∩ (−K) 6= ∅,
20. D(F(x0 + tkuk) − y0,−C) < εktk.
Passing to a subsequence, we may assume that uk → u0, and 0 < tk < r.
By the C-Lipschitz property of F we have
D(
1
tk
 
F(x
0 + tku
0) − y
0
,−C) < εk + Lku
k − u
0k.
Let yk ∈ F(x0+tku0) be such that D(¯ yk,−C) < εk+Lkuk−u0k, where ¯ yk = (1/tk)(yk−y0).
The sequence {¯ yk} is bounded, which follows from the following reasoning. Since y0 ∈
p-MinCF(x0), there exists σ, 0 < σ < 1/2, such that F(x0) ∩ (y0 − C(2σ)) = {y0}.
Let k be such that εk + Lkuk − u0k < L, whence D(yk − y0,−C) < Ltk. Then it
holds k¯ ykk ≤ L(1 + 1/σ). Indeed, assume on the contrary, that k¯ ykk > L(1 + 1/σ), or
equivalently kyk − y0k > L(1 + 1/σ)tk. We have
D(y
k − y
0,−C) < Ltk
σ
Ltk(1 + σ)
ky
k − y
0k < σ ky
k − y
0k.
This inequality shows that yk − y0 ∈ −C(σ), whence, from Lemma 2 we get
ky
k − y
0k ≤
L(1 + σ)
σ
k(x
0 + tku
0) − x
0k = L

1 +
1
σ

tk ,
4a contradiction.
We proved that the sequence {¯ yk} is bounded and k¯ ykk ≤ L(1 + 1/σ) for all suﬃciently
large k. Passing to a subsequence, we may assume that ¯ yk → ¯ y0, whence k¯ y0k ≤ L(1 +
1/σ) and ¯ y0 ∈ F 0(x0,y0;u0). Taking a limit in the inequality D(¯ yk,−C) < εk+Lkuk−u0k
we get D(¯ y0,−C) ≤ 0. Since C is closed, this inequality gives ¯ y0 ∈ −C.
The hypothesis G(x0,z0) together with the condition G(x0 + tkuk) ∩ (−K) 6= ∅ give
that G0(x0 + tkuk) ∩ (−K) 6= ∅, where G0(x) = G(x) ∩ `kx − x0k ¯ Bp(z0). The local
Lipschitz property of G gives that there exists a point zk ∈ G(x0 + tku0) such that
D(zk,G0(x0 + tkuk)) ≤ Ltk kuk − u0k (here we suppose that G is locally Lipschitz with
constant L on r ¯ Bn). From the triangle inequality we get kzk −z0k ≤ (`+Lkuk −u0k)tk.
Putting ¯ zk = (1/tk)(zk − z0), we have k¯ zkk ≤ (` + Lkuk − u0k) ≤ ` + 2L. Therefore the
sequence ¯ zk is bounded. Passing to a subsequence we may assume that ¯ zk → ¯ z0.
The construction of zk yields the existence of ˜ zk ∈ G0(x0 + tkuk) ∩ (−K), such that
zk ∈ ˜ zk + Ltk kuk − u0k ¯ Bp, whence for arbitrary η ∈ K0[−z0], kηk = 1, we have
hη, ¯ z
ki =
1
tk
hη,z
ki ≤
1
tk
hη, ˜ z
ki + Lku
k − u
0k ≤ Lku
k − u
0k.
Here we have used hη, ˜ zki ≤ 0, a consequence of ˜ zk ∈ −K. Taking the limit in the above
inequality, we get hη, ¯ z0i ≤ 0, whence
D(¯ z
0,−K[−z
0]) = sup{hη, ¯ z
0i | η ∈ K
0[−z
0], kηk = 1} ≤ 0.
Regarding that K0[−z0] is closed, this gives ¯ z0 ∈ −K[−z0].
We have used the same sequence tk → 0+ to construct both ¯ y0 and ¯ z0, hence we have
(¯ y0, ¯ z0) ∈ (F ×G)0(x0,(y0,z0);u0). So far we have proved that (¯ y0, ¯ z0) ∈ −(C ×K[−z0]).
On the other hand condition (4) gives (¯ y0, ¯ z0) / ∈ −(C × K[−z0]), a contradiction. 2
Remark 2 Condition (4), which can be referred to as primal form condition, can be
substituted by the equivalent dual form condition
∀u ∈ Rn \ {0} : ∀,(¯ y0, ¯ z0) ∈ (F × G)0(x0,(y0,z0);u) :
∃(ξ,η) ∈ C0 × K0[−z0],(ξ,η) 6= (0, 0) : hξ, ¯ y0i + hη, ¯ z0i > 0. (5)
The next example shows that without condition G(x0,x0) Theorem 2 is not true.
Example 1 Consider problem (1) with n = 1, m = 1, p = 2, C = R+, K = R2
+, F :
R → R arbitrary single-valued diﬀerentiable function, and G : R   R2 given by G(x) =
[(|x|,−1),(−|x|,0)]. Let x0 = 0, y0 = F(x0), z0 = (0,−1). All conditions of Theorem
2, with exception of G(x0,z0), are satisﬁed, independently on the concrete function F.
In particular K[−z0] = R+ × R and (F × G)0(x0,(y0,z0);u) = (F 0(0)u,G0(x0,z0;u)),
where G0(x0,z0;u) = {|u|} × R+, which veriﬁes condition (4). Since any point x ∈ R is
feasible, problem (1) is equivalent to the optimization problem minF(x), x ∈ R. But, if
for instance F(x) = −x2, the point x0 is not an i-minimizer.
The following theorem is a modiﬁcation of Theorem 2 and is proved similarly.
5Theorem 3 Consider svp (1) with C ⊂ Rm and K ⊂ Rp pointed closed convex cones, and
F : Rn   Rm and G : Rn   Rp respectively locally C-Lipschitz and locally K-Lipschitz
svf. Suppose that the pair (x0,y0), x0 ∈ Rn, y0 ∈ F(x0), is such that y0 ∈ p-MinCF(x0),
and there exists z0 ∈ G(x0) for which z0 ∈ p-MinKG(x0) and condition (4) holds. Suppose
also that the svf G satisﬁes condition G(x0,z0). Then (x0,y0) is an i-minimizer of svp
(1).
When the functions F and G are single-valued, then problem (1) transforms into the vector
optimization problem minCF(x), G(x) ∈ −K, and Theorems 1 and 2 reduce to those
proved in [3]. Then the conditions y0 ∈ p-MinCF(x0) and G(x0,z0) are automatically
satisﬁed.
Though condition G(x0,z0) does not appear in Theorem 1, the interesting applications of
this theorem could be those in which G(x)∩(−K) possess points near z0. Indeed, suppose
that ∀` > 0 : ∃U ∈ N(x0) : ∀x ∈ U : G(x) ∩ (−K) 6= ∅ ⇒ G(x) ∩ `kx − x0k ¯ Bp(x0) ∩
(−K) = ∅. Then (F ×G)0(x0,(y0,z0);u) = ∅ for all u ∈ Rn, and condition (2) is satisﬁed
for arbitrary svf F.
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6

A multiplier rule in set-valued optimisation.

Contingent epiderivatives and set-valued optimization.

First and second order optimality conditions for set-valued optimization problems.

First order optimality conditions in set-valued optimization.

First-order conditions for C0,1 constrained vector optimization.

On constrained set-valued optimization

http://eco.uninsubria.it/dipeco/Quaderni/files/QF2007_10.pdf

On constrained set-valued optimization

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