Applied Analysis and Optimization

Yokohama Publishers

ISSN 2189-1664 Online Journal

C Copyright 2018

28

S. SUZUKI AND D. KUROIWA

of uncertain problem whenever the uncertainty aspects the vector-valued objectivefunction.
Hence, in this paper, we study quasiconvex vector optimization with data uncertainty via robust optimization. To the purpose, we introduce surrogate duality forrobust vector optimization by using scalarization. We investigate surrogate minmax duality for quasiconvex vector optimization with uncertain objective and/or
constraints. For the problem with uncertain objective, we introduce its robust
counterpart as a set-valued optimization problem.
The remainder of the paper is organized as follows. In Section 2, we introduce
some preliminaries and previous results. In Section 3, we investigate surrogate
min-max duality for quasiconvex vector optimization with uncertain constraints. In
Section 4, we investigate surrogate min-max duality for quasiconvex vector optimization with uncertain objective and constraints. In Section 5, we discuss about
our results.
2. Preliminaries
Let X be a locally convex Hausdorﬀ topological vector space, X ∗ the continuous
dual space of X, and ⟨x∗ , x⟩ the value of a functional x∗ ∈ X ∗ at x ∈ X. Given a
set A∗ ⊂ X ∗ , we denote the w∗ -closure, the boundary, the interior, the convex hull,and the conical hull generated by A∗ , by cl A∗ , bd A∗ , int A∗ , convA∗ , and cone A∗ ,respectively. By convention, we deﬁne cone ∅ = {0}. The indicator function δA of
A ⊂ X is deﬁned by{
0, x ∈ A,
δA (x) :=∞, otherwise.
Let f be a function from X to R := [−∞, +∞]. We denote the domain of f by
domf := {x ∈ X | f (x) < ∞}. The epigraph of f is epif := {(x, r) ∈ X × R |f (x) ≤ r}, and f is said to be convex if epif is convex. The Fenchel conjugate off , f ∗ : X ∗ → R, is deﬁned as f ∗ (u) := supx∈domf {⟨u, x⟩ − f (x)}. Deﬁne the levelsets of f with respect to a binary relation ⋄ on R as
L(f, ⋄, β) := {x ∈ X | f (x) ⋄ β}
for any β ∈ R. A function f is said to be quasiconvex if for each β ∈ R, L(f, ≤, β) isconvex. Any convex function is quasiconvex, but the opposite is not generally true.
Let Y be a locally convex Hausdorﬀ topological vector space, partially ordered
by a nonempty, closed and convex cone K ⊂ Y , that is, for y, z ∈ Y , the notationy ≤K z will mean z − y ∈ K. A cone K is said to be solid if int K is nonempty.
Let Y ∗ be the continuous dual space of Y , and g a function from X to Y . The
positive polar cone of K is K + := {λ ∈ Y ∗ | ∀y ∈ K, ⟨λ, y⟩ ≥ 0}. A function g issaid to be K-convex if for all x1 , x2 ∈ X, and α ∈ [0, 1], (1 − α)g(x1 ) + αg(x2 ) ∈g((1 − α)x1 + αx2 ) + K. It is well known that g is K-convex if and only if λ ◦ g isconvex for all λ ∈ K + . A function g is said to be K-quasiconvex if for all y ∈ Y , x1 ,x2 ∈ X, and α ∈ [0, 1] with y ∈ (g(x1 )+K)∩(g(x2 )+K), y ∈ g((1−α)x1 +αx2 )+K.
Also, g is said to be proper quasi K-concave if for each x, y ∈ X and α ∈ (0, 1),g((1 − α)x + αy) ∈ (g(x) + K) ∪ (g(y) + K), in detail, see [4, 28].

SURROGATE DUALITY FOR ROBUST VECTOR OPTIMIZATION

29

The following set containment characterizations are well known in convex optimization. There are some similar results in convex and quasiconvex optimization,for example see, [3, 8, 9, 19, 22].
Theorem 2.1 ([3]). Let X be a locally convex Hausdorﬀ topological vector space,
I an arbitrary set, gi a proper lower semicontinuous convex function from X to R
for each i ∈ I, A a closed convex subset of X, {x ∈ A | ∀i ∈ I, gi (x) ≤ 0} ̸= ∅,x∗ ∈ X ∗ , and α ∈ R. Then, the following statements are equivalent:∗(i) {x ∈ A | ∀i ∈ I, gi (x) (≤ 0} ⊂ {x ∈ X | ⟨x
) , x⟩ ≤ α},∪∗epigi∗ + epiδA.
(ii) (x∗ , α) ∈ cl cone conv
i∈I

Theorem 2.2 ([8]). Let X and Y be locally convex Hausdorﬀ topological vector
spaces, K ⊂ Y a nonempty, closed and convex cone, A a closed convex subset of X,
Y partially ordered by K, and g a continuous and K-convex function. Assume thatS = {x ∈ A | g(x) ∈ −K} is nonempty. Then,∪∗epiδS∗ = cl 
epi(λ ◦ g)∗ + epiδA.
λ∈K +3. Quasiconvex vector optimization with uncertain constraints
In this section, let X, Y and Z be locally convex Hausdorﬀ topological vector
spaces, K ⊂ Y and C ⊂ Z nonempty, solid, pointed, closed and convex cones,
A a closed convex subset of X, Y partially ordered by K, Z partially ordered by
C, f a continuous C-quasiconvex function from X to Z, V a set, g a function
from X × V to Y such that g(·, v) is continuous and K-convex for each v ∈ V,
F(v,λ) = {x ∈ A | λ ◦ g(x, v) ≤ 0} for each (v, λ) ∈ V × K + , and F = {x ∈ A | ∀v ∈V, g(x, v) ∈ −K} ̸= ∅.
We investigate the following vector optimization problem with data uncertainty(UP):
Minimize f (x),subject to x ∈ A, g(x, v) ∈ −K.
In this problem, v ∈ V indicates data uncertainty. Because of the complexity ofreal-world optimization problems, measurement errors, and the other uncertainty,
it is diﬃcult to determine constraint (or objective) functions clearly. In (UP), wecannot determine v clearly, however, we know that v is an element of the uncertainty
set V. In order to solve such a problem robustly, robust optimization have been
investigated. In robust optimization, we consider the following robust counterpart(RC):
Minimize f (x),subject to x ∈ A, ∀v ∈ V, g(x, v) ∈ −K.
In (RC), the constraint set is the intersection of the constraint sets of (UP). Hence,a feasible solution of (RC) is also a feasible solution of (UP) for each v. It is clearthat val(RC) ≥ val(U P ), where val(RC) (val(U P )) is the minimum value of (RC)((UP), respectively). Because of these properties, robust optimization is called

30

S. SUZUKI AND D. KUROIWA‘worst-case approach’. Since (RC) is a semi-inﬁnite programming problem, we investigate (RC) by the similar way in previous results via semi-inﬁnite programming.
We deﬁne a solution of (RC) as follows.
Definition 3.1. An element x ∈ F is said to be a weakly optimal solution of (RC)if f (F ) ∩ (f (x) − intC) is empty.
Let e ∈ intC and z0 ∈ Z. We denote by φe,z0 the scalarizing function from Z toR, that is,
φe,z0 (z) = inf{t ∈ R | z ∈ z0 + te − C}.
It is well known that the following statements hold, in detail, see [6, 15].
Theorem 3.2 ([6, 15]). Let e ∈ intC and z0 ∈ Z. Then, the following statements
hold:(i) φe,z0 is a real-valued continuous function,(ii) L(φe,z0 , ≤, t) = z0 + te − C,(iii) L(φe,z0 , <, t) = z0 + te − intC,(iv) L(φe,z0 , =, t) = z0 + te − bdC,(v) if f is C-quasiconvex, then φe,z0 ◦ f is quasiconvex,(vi) if f is proper quasi C-concave, then φe,z0 ◦ f is quasiconcave,(vii) x0 is a weakly optimal solution of (RC) if and only if for each e ∈ intC,
there exists z0 ∈ Z such that φe,z0 ◦ f (x0 ) = minx∈F φe,z0 ◦ f (x).
We need the following proposition.
Proposition 3.3. Let e ∈ intC and z0 ∈ Z Then, for each µ ∈ R,
φe,z0 (µe) = µ + φe,z0 (0).
Proof. By the deﬁnition of φe,z0 , for each ε > 0, there exists tε < φe,z0 (0) + ε suchthat 0 ∈ z0 + tε e − C. Hence,µe ∈ µe + z0 + tε e − C = z0 + (µ + tε )e − C.
This shows that
φe,z0 (µe) ≤ µ + tε < µ + φe,z0 (0) + ε.
Therefore, φe,z0 (µe) ≤ µ + φe,z0 (0).
Assume that φe,z0 (µe) < µ + φe,z0 (0). Then, there exists t ∈ R such that
φe,z0 (µe) < t < µ + φe,z0 (0) and µe ∈ z0 + te − C. Hence, 0 ∈ z0 + (t − µ)e − C.
This shows that φe,z0 (0) ≤ t − µ. This is a contradiction.

We show the following characterizations for robust vector optimization.
Theorem 3.4. The following conditions hold:∗(i) epiδF∗ (v,λ) = cl {cone epi (λ ◦ g(·, v))∗ + epiδA},∪∗(ii) epiδF∗ = cl conv
cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.
v∈V,λ∈K +SURROGATE DUALITY FOR ROBUST VECTOR OPTIMIZATION

Proof. By Theorem 2.2,
epiδF∗ (v,λ)

= cl


∪


∗
epi (tλ ◦ g(·, v))∗ + epiδA

t≥0

31





∗
= cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.

This shows that the condition (i) holds.
(ii) Since K is a closed convex cone,F= {x ∈ A | ∀v ∈ V, ∀λ ∈ K + , λ ◦ g(x, v) ≤ 0}∩=F(v,λ)
v∈V,λ∈K +∩=L(δF(v,λ) , ≤, 0).

v∈V,λ∈K +Since epiδF∗ (v,λ) is a convex cone, by Theorem 2.1 and the condition (i) of thistheorem,∪∗ 
epiδF∗ = cl cone conv 
epiδF∗+ epiδX(v,λ)

v∈V,λ∈K +∪= cl cone conv

epiδF∗ (v,λ)

v∈V,λ∈K +∪= cl conv

epiδF∗ (v,λ)

v∈V,λ∈K +∪= cl conv∗
cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.

v∈V,λ∈K +
Remark 3.5. It is clear thatcl conv∪∗
cl {cone epi (λ ◦ g(·, v))∗ + epiδA}v∈V,λ∈K += cl conv∪∗{cone epi (λ ◦ g(·, v))∗ + epiδA}.

v∈V,λ∈K +However, the operator ‘cl’ in the union is necessary and important to prove surrogate
duality and its constraint qualiﬁcation.
Remark 3.6. By Theorem 3.4, we show that the following set containment characterization. Let x∗ ∈ X ∗ , and α ∈ R. Then, the following statements are equivalent:(i) F ⊂ {x ∈ X | ⟨x∗ , x⟩∪≤ α},∗(ii) (x∗ , α) ∈ cl conv
cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.
v∈V,λ∈K +32

S. SUZUKI AND D. KUROIWA

By Theorem 3.4, the following robust characteristic cone∪∗}cl {cone epi (λ ◦ g(·, v))∗ + epiδAv∈V,λ∈K +is closed and convex if and only if∪∗
epiδF∗ ⊂cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.
v∈V,λ∈K +In the following theorem, we investigate surrogate min-max duality for quasiconvex vector optimization with uncertain constraints.
Theorem 3.7. The following statements are equivalent:(i) the following robust characteristic cone is closed and convex:∪∗
cl {cone epi (λ ◦ g(·, v))∗ + epiδA},v∈V,λ∈K +(ii) for each x0 ∈ F and continuous C-quasiconvex function f , x0 is a weakly
optimal solution of (RC) if and only if for each e ∈ intC, there exists z0 ∈ Zsuch that
φe,z0 ◦ f (x0 ) =max

inf {φe,z0 ◦ f (x) | λ ◦ g(x, v) ≤ 0}.

v∈V,λ∈K + x∈A

Proof. Assume that (i) holds. Let x0 ∈ F and f a continuous C-quasiconvex function. By Theorem 3.2 (vii), x0 is a weakly optimal solution of (RC) if and only if
for each e ∈ intC, there exists z0 ∈ Z such that φe,z0 ◦ f (x0 ) = minx∈F φe,z0 ◦ f (x).
We show that
(3.1)

inf φe,z0 ◦ f (x) =

x∈F

max

inf {φe,z0 ◦ f (x) | λ ◦ g(x, v) ≤ 0}.

v∈V,λ∈K + x∈A

It is clear that the weak duality holds, that is,
inf φe,z0 ◦ f (x) ≥

x∈F

sup

inf {φe,z0 ◦ f (x) | λ ◦ g(x, v) ≤ 0}.

v∈V,λ∈K + x∈A

Let m = inf x∈F φe,z0 ◦ f (x). If L(φe,z0 ◦ f, <, m) = ∅, then for each v ∈ V and λ = 0,
the equation (3.1) holds. If L(φe,z0 ◦ f, <, m) ̸= ∅, there exists (x∗ , α) ∈ X ∗ × Rsuch that for all x ∈ F and y ∈ L(φe,z0 ◦ f, <, m),⟨x∗ , x⟩ ≤ α < ⟨x∗ , y⟩ ,since L(φe,z0 ◦ f, <, m) ∩ F = ∅ and L(φe,z0 ◦ f, <, m) is a nonempty, open andconvex set. Because of the condition (i) and Theorem 3.4,(x∗ , α) ∈ epi δF∗∪⊂∗
cl {cone epi (λ ◦ g(·, v))∗ + epiδA}v∈V,λ∈K +=∪v∈V,λ∈K +epi δF∗ (v,λ) .

SURROGATE DUALITY FOR ROBUST VECTOR OPTIMIZATION

33

Therefore, there exists (v̄, λ̄) ∈ V × K + such that (x∗ , α) ∈ epi δF∗. For each(v̄,λ̄)x ∈ F(v̄,λ̄) ,⟨x∗ , x⟩ ≤ α =⇒ x ∈/ L(φe,z0 ◦ f, <, m) ⇐⇒ φe,z0 ◦ f (x) ≥ m,by the above separation inequality. This shows that inf x∈A {φe,z0 ◦f (x) | λ̄◦g(x, v̄) ≤0} ≥ m. Hence, the equation (3.1) holds. This shows that (i) implies (ii).
Next, we prove that (ii) implies (i). We only show that∪∗
epiδF∗ ⊂cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.
v∈V,λ∈K +Let (x∗ , α) ∈ epi δF∗ . We deﬁne a function f from X to Z as follows:{− ⟨x∗ , x⟩ e, ⟨x∗ , x⟩ > δF∗ (x∗ ),f (x) =
−δF∗ (x∗ )e, ⟨x∗ , x⟩ ≤ δF∗ (x∗ ),where e ∈ intC. Then, f is continuous C-quasiconvex and for each x ∈ F , x is aweakly optimal solution of (RC). Let x0 ∈ F . By the statement (ii), there existz0 ∈ Z, v ∈ V, and λ ∈ K + such that
φe,z0 ◦ f (x0 ) = inf {φe,z0 ◦ f (x) | λ ◦ g(x, v) ≤ 0}.
x∈A

By Proposition 3.3,
φe,z0 ◦ f (x) ={− ⟨x∗ , x⟩ + φe,z0 (0), ⟨x∗ , x⟩ > δF∗ (x∗ ),−δF∗ (x∗ ) + φe,z0 (0), ⟨x∗ , x⟩ ≤ δF∗ (x∗ ).

Hence, for each x ∈ F(v,λ) ,φe,z0 ◦ f (x) ≥ φe,z0 ◦ f (x0 )⇐⇒⇐⇒

φe,z0 ◦ f (x) ≥ −δF∗ (x∗ ) + φe,z0 (0)⟨x∗ , x⟩ ≤ δF∗ (x∗ ).

This implies that δF∗ (v,λ) (x∗ ) ≤ δF∗ (x∗ ) ≤ α. By Theorem 3.4.
∗(x∗ , α) ∈ epi δF∗ (v,λ) = cl {cone epi (λ ◦ g(·, v))∗ + epiδA}.

This shows that (i) holds.



4. Quasiconvex vector optimization with uncertain objective andconstraints
In this section, let X and Y be locally convex Hausdorﬀ topological vector spaces,
K ⊂ Y a nonempty, solid, pointed, closed and convex cone, A a closed convex
subset of X, Y partially ordered by K, U a compact set, f a continuous function
from X × U to Rn , Rn++ = {x ∈ Rn | ∀i ∈ {1, . . . , n}, xi > 0}, V a set, g afunction from X × V to Y such that g(·, v) is continuous K-convex for each v ∈ V,
F(v,λ) = {x ∈ A | λ ◦ g(x, v) ≤ 0} for each (v, λ) ∈ V × K + , and F = {x ∈ A | ∀v ∈V, g(x, v) ∈ −K} ̸= ∅.
We study vector optimization with uncertain objective and constraints. Consider
the following uncertain problem:
Minimize f (x, u),subject to x ∈ A, g(x, v) ∈ −K.
In this paper, we introduce the following robust counterpart (RC):34

S. SUZUKI AND D. KUROIWA

Minimize f (x, U),subject to x ∈ A, ∀v ∈ V, g(x, v) ∈ −K,
where f (x, U) = {f (x, u) | u ∈ U }. Hence, we regard (RC) as a set-valued optimization problem with vector-valued constraints. For each i ∈ {1, . . . , n}, there exists
fi : X × U → R, i ∈ {1, . . . , n} such thatf (x, u) = (f1 (x, u), . . . , fn (x, u)).
Since f is continuous and U is compact, fi (x, ·) attains its maximum on U for each
i and x. Let fˆ be a function from X to Rn as follows:fˆ(x) = (max f1 (x, u), . . . , max fn (x, u)).
u∈U

u∈U

In [14], Kuroiwa and Lee introduce the function, and study a similar robust counterpart. Now we deﬁne a solution of (RC).
Definition 4.1. An element x0 ∈ F is said to be a solution of (RC) if f (x, U ) ̸⊂(fˆ(x0 ) − intC) for each x ∈ F .
We compare a solution of (RC) with a solution of set optimization in Section 5.
Theorem 4.2. Let x0 ∈ F , e ∈ Rn++ and z0 ∈ Rn . Then,φe,z0 (fˆ(x0 )) = sup φe,z0 ◦ f (x0 , u).
u∈U

Proof. It is clear that f (x0 , U) ⊂ fˆ(x0 ) − Rn+ . Hence,
φe,z0 (fˆ(x0 )) ≥ sup φe,z0 ◦ f (x0 , u).
u∈U

By Theorem 3.2, there exists z̄ ∈ bdRn+ such that fˆ(x0 ) = z0 + φe,z0 (fˆ(x0 ))e − z̄.
Since z̄ ∈ bdRn+ , there exists i ∈ {1, . . . , n} such that z̄i = 0. Let ȳ = fˆ(x0 ) −f (x0 , ui ) where ui ∈ U such that fi (x, ui ) = maxu∈U fi (x, u). Then, ȳ ∈ Rn+ andȳi = 0. This implies that z̄ + ȳ ∈ bdRn+ . Hence,f (x0 , ui ) = fˆ(x0 ) − ȳ
= z0 + φe,z0 (fˆ(x0 ))e − z̄ − ȳ
∈ z0 + φe,z (fˆ(x0 ))e − bdRn .
0+This shows that φe,z0 (f (x, ui )) = φe,z0 (fˆ(x0 )). This completes the proof.
In the following theorem, we show a characterization of the solution of (RC).
Theorem 4.3. Let x0 ∈ F . The following statements are equivalent:(i) x0 is a solution of (RC),(ii) for each e ∈ Rn++ , there exists z0 ∈ Rn such thatsup φe,z0 ◦ f (x0 , u) = min sup φe,z0 ◦ f (x, u).

u∈Ux∈F u∈USURROGATE DUALITY FOR ROBUST VECTOR OPTIMIZATION

35

Proof. We show that (i) implies (ii). Let e ∈ Rn++ and z0 = fˆ(x0 ). Then, by
Theorem 4.2, supu∈U φe,z0 ◦ f (x0 , u) = φe,z0 (fˆ(x0 )) = 0. By the statement (i),for each x ∈ F , there exists u ∈ U such that f (x, u) ∈/ fˆ(x0 ) − Rn++ . Henceφe,z0 ◦ f (x, u) ≥ φe,z0 (fˆ(x0 )). This shows that (ii) holds.
Conversely, assume that (ii) holds and there exists x̄ ∈ F such that f (x̄, U) ⊂fˆ(x0 ) − Rn++ . Let e ∈ Rn++ . By the assumption and Theorem 4.2, there exists z0 ∈Rn such that φe,z0 (fˆ(x0 )) = supu∈U φe,z0 ◦ f (x0 , u) = minx∈F supu∈U φe,z0 ◦ f (x, u).
By monotonicity of the scalarizing function, φe,z0 (fˆ(x0 )) > φe,z0 ◦ f (x̄, u) for eachu ∈ U. Since U is compact and f is continuous, φe,z0 (fˆ(x0 )) > maxu∈U φe,z0 ◦f (x̄, u).
This is a contradiction.
Let F be the following set of functions;
f : continuous,






U
:
compact
convex,n
F = f :X ×U →R.
f (·, u) : Rn+ -quasiconvex ∀u ∈ U,






f (x, ·) : proper quasi Rn+ -concave ∀x ∈ X.
In the following theorem, we investigate surrogate min-max duality for quasiconvex
vector optimization with uncertain objective and constraints.
Theorem 4.4. The following statements are equivalent:(i) the following robust characteristic cone is closed and convex:∪∗
cl {cone epi (λ ◦ g(·, v))∗ + epiδA},v∈V,λ∈K +(ii) for each x0 ∈ F and f ∈ F, x0 is a solution of (RC) if and only if for eache ∈ Rn++ , there exists z0 ∈ Rn such thatsup φe,z0 ◦ f (x0 , u) =

u∈U

max

inf {φe,z0 ◦ f (x, u) | λ ◦ g(x, v) ≤ 0}.

v∈V,λ∈K + ,u∈U x∈A

Proof. By Theorem 3.2, φe,z0 ◦f (·, u) is continuous quasiconvex for each u ∈ U. Since
U is compact, and f is continuous, supu∈U φe,z0 ◦ f (·, u) is continuous quasiconvex.
Hence, we can prove the following equation by the similar way of the proof ofequation (3.1),
inf sup φe,z0 ◦ f (x, u) =x∈F u∈Umax

inf {sup φe,z0 ◦ f (x, u) | λ ◦ g(x, v) ≤ 0}.

v∈V,λ∈K + x∈A u∈U

Since f is continuous and f (x, ·) is proper quasi Rn+ -concave, by Theorem 3.2,
φe,z0 ◦ f (x, ·) is continuous quasiconcave function. By Sion’s min-max theorem,
inf {sup φe,z0 ◦ f (x, u) | λ ◦ g(x, v) ≤ 0}x∈A u∈U= max inf {φe,z0 ◦ f (x, u) | λ ◦ g(x, v) ≤ 0}.
u∈U x∈A

This shows that (i) implies (ii). The proof of converse implication is similar to theproof of Theorem 3.7.


36

S. SUZUKI AND D. KUROIWA

Remark 4.5. In the deﬁnition of F, we assume that U is compact convex. Exactly
speaking, we assume that U is a compact convex subset of a topological vector
space. It is not necessary that U is a subset of given topological vector spaces, X,
Y , or Rn .
5. Discussion
In this section, we discuss about our results in this paper. Especially, we investigate a necessary and suﬃcient constraint qualiﬁcation for surrogate duality
via robust optimization. Also, we compare a solution of (RC) in Section 4 with asolution of set optimization.
Recently, many types of necessary and suﬃcient constraint qualiﬁcations for duality theorems have been investigated, see [8, 10, 18–26]. In Theorem 3.7 and Theorem 4.4, we show surrogate min-max duality theorems for robust quasiconvex vector
optimization with its necessary and suﬃcient constraint qualiﬁcation. In the following theorem, we investigate surrogate duality theorem for real-valued optimization
with uncertain objective and constraints. The proof is similar to Theorem 3.7 andTheorem 4.4, and will be omitted.
Theorem 5.1. The following statements are equivalent:(i) the following characteristic cone is closed and convex:∪∗
cl {cone epi (λ ◦ g(·, v))∗ + epiδA},v∈V,λ∈K +(ii) for each upper semicontinuous quasiconvex function f from X to R,
inf f (x) =x∈F

inf {f (x) | λ ◦ g(x, v) ≤ 0},maxv∈V,λ∈K + x∈A(iii) for each continuous function f from X × U to R with U is compact convex,
f (·, u) is quasiconvex for each u ∈ U, and f (x, ·) is quasiconcave for eachx ∈ X,
inf sup f (x, u) =x∈F u∈Umax

inf {f (x, u) | λ ◦ g(x, v) ≤ 0},v∈V,λ∈K + ,u∈U x∈A(iv) for each x0 ∈ F and continuous C-quasiconvex function f , x0 is a weakly
optimal solution of (RC) if and only if for each e ∈ intC, there exists z0 ∈ Zsuch that
φe,z0 ◦ f (x0 ) =max

inf {φe,z0 ◦ f (x) | λ ◦ g(x, v) ≤ 0},v∈V,λ∈K + x∈A(v) for each x0 ∈ F and f ∈ F, x0 is a solution of (RC) if and only if for eache ∈ Rn++ , there exists z0 ∈ Rn such thatsup φe,z0 ◦ f (x0 , u) =

u∈U

max

inf {φe,z0 ◦ f (x, u) | λ ◦ g(x, v) ≤ 0}.

v∈V,λ∈K + ,u∈U x∈A

In the present paper, we introduce a robust counterpart as a set-valued optimization problem. Unfortunately, it is not so evident which is the suitable robust counterpart of uncertain problem whenever the uncertainty aspects the vector-valued
objective function. Many researchers introduce various types of robust approachfor uncertain vector optimization problem. For example, in [5], Georgiev, Luc andSURROGATE DUALITY FOR ROBUST VECTOR OPTIMIZATION

37

Pardalos deﬁne the eﬃcient solutions of robust counterpart as those robust feasible solutions which are eﬃcient for any possible value of u ∈ U . In [7], Goberna,
Jeyakumar, Li, and López investigate uncertain problems by transferring the uncertainty from the objective function to the constraints. This approach is a very
elegant and important element for convex optimization problems. However, it is
not always valid for quasiconvex optimization problems, see the following example.
Example 5.2. We consider the following counterpart:
minimize t,
subject to x ∈ F , f (x, u) − t ∈ −C, ∀u ∈ U.
If f (·, u) is C-convex for each u ∈ U, then this counterpart is convex optimization
problem. However, even if f (·, u) is C-quasiconvex, this counterpart is not always
quasiconvex optimization problem.
Let f be the following real-valued quasiconvex function on R:{
f (x) :=

0, x ≥ 0,2x, otherwise.

Let f¯(x, t) := f (x) − t, z1 = (−1, 1) and z2 = (1, 3), then
(

1
1
f¯
z1 + z222)= −2 > −3 = max{f¯(z1 ), f¯(z2 )}This shows that f¯ is not a quasiconvex function on R2 .
In [14], Kuroiwa and Lee introduce a robust counterpart by a similar way in
Section 4 of this paper. We consider this robust counterpart is one of the natural
pessimistic approach to uncertain vector optimization problems.
In set-valued optimization, there are some deﬁnitions of solutions. Especially,
Kuroiwa investigate set optimization approach, see [12, 13]. Next, we compare asolution of (RC) in Section 4 with a solution of set optimization. In set optimization,
the following relation have been investigated. Let A and B be subsets of Z. The
notation A ≤u B will mean A ⊂ B − C. Also, A <u B will mean A ⊂ B − intC.
An element x0 ∈ F is said to be a u-type weak minimal solution if there does notexist x ∈ F such that f (x, U) <u f (x0 , U). We can check easily that if x0 ∈ F is asolution of (RC), then x0 is a u-type weak minimal solution. However, the converse
is not generally true, see the following example.
Example 5.3. Let x0 , x ∈ F , A = f (x0 , U) = {a ∈ R2 | ∥a∥ ≤ 1}, and B =f (x, U) = {a ∈ R2 | ∥a − ( 12 , 21 )∥ ≤ 38 }. Then, f (x, U) ̸⊂ f (x0 , U) − R2++ , that is
f (x, U) ̸<u f (x0 , U). If f (y, U) ̸<u f (x0 , U) for each y ∈ F , then x0 is a u-type weakminimal solution. However, fˆ(x0 ) = (max(x1 ,x2 )∈A x1 , max(x1 ,x2 )∈A x2 ) = (1, 1) andf (x, U) ⊂ fˆ(x0 ) − R2++ . Hence, x0 is not a solution of (RC). This indicates theexistence of a u-type weak minimal solution which is not a solution of (RC).

38

S. SUZUKI AND D. KUROIWAfˆ(x0 )B
A

Fig. 1In Theorem 4.3, we investigate a characterization of a solution of (RC) by thescalarizing function. We can check easily that if for each e ∈ Rn++ , there exists
z0 ∈ Rn such thatsup φe,z0 ◦ f (x0 , u) = min sup φe,z0 ◦ f (x, u),u∈Ux∈F u∈Uthen x0 ∈ F is a u-type weak minimal solution. However, the converse is not
generally true by Theorem 4.2 and Example 5.3. This means that a u-type weakminimal solution is not characterized by this type of scalarization. By using thenotion of the solution of (RC) in this paper and the scalarizing function, we show
surrogate min-max duality via robust quasiconvex vector optimization.
6. Conclusion
In this paper, we investigate surrogate duality for robust quasiconvex vector optimization. In Section 3 and Section 4, we investigate robust vector optimizationby using scalarization. We investigate surrogate min-max duality for quasiconvex
vector optimization with uncertain constraints and introduce a necessary and sufﬁcient constraint qualiﬁcation for surrogate duality. Also, we investigate vector
optimization with uncertain objective and constraints. Because of the uncertaintyof the objective function, the robust counterpart is a set-valued optimization problem. We introduce a notion of solutions of the robust counterpart, and we investigate surrogate min-max duality with its constraint qualiﬁcation. In Section 5, we
discuss about our results. We investigate surrogate duality for real-valued robust
optimization with its constraint qualiﬁcation. Also, we compare a solution of (RC)in Section 4 with a u-type weak minimal solution in set optimization.
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Manuscript received October 15 2017
revised January 2 2018

S. Suzuki
Department of Mathematics, Shimane University, Shimane, Japan
E-mail address: suzuki@riko.shimane-u.ac.jp
D. Kuroiwa
Department of Mathematics, Shimane University, Shimane, Japan
E-mail address: kuroiwa@riko.shimane-u.ac.jp