# Strong Second-Order Karush–Kuhn–Tucker Optimality Conditions for Vector
Optimization

NGUYEN QUANG HUY  Department of Mathematics, Hanoi Pedagogical University 2,
Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam<EMAIL_ADDRESS>,  DO SANG KIM†
Department of Applied Mathematics, Pukyong National University, Busan 48513,
Korea<EMAIL_ADDRESS>and  NGUYEN VAN TUYEN‡ Department of Mathematics,
Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the present paper, we focus on the vector optimization problems with
inequality constraints, where objective functions and constrained functions
are vector-valued functions with $C^{1,1}$ components defined on
$\mathbb{R}^{n}$. By using the second-order symmetric subdifferential and the
second-order tangent set, we propose two types of second-order regularity
conditions in the sense of Abadie. Then we establish some strong second-order
Karush–Kuhn–Tucker necessary optimality conditions for Geoffrion properly
efficient solutions of the considered problem. Examples are given to
illustrate the obtained results.

###### Key words and phrases:

Symmetric second-order subdifferential $\cdot$ Abadie second-order regularity
condition $\cdot$ Geoffrion properly efficient point $\cdot$ Second-order
Karush–Kuhn–Tucker optimality condition

###### 1991 Mathematics Subject Classification:

49K30 $\cdot$ 49J52 $\cdot$ 49J53 $\cdot$ 90C29$\cdot$ 90C46

†The research of the second author was supported by the National Research
Foundation of Korea Grant funded by the Korean Government
(NRF-2016R1A2B4011589)

‡The research of the third author was supported by the National Foundation for
Science and Technology Development (NAFOSTED), Vietnam grant 101.01-2014.39

##  1\. Introduction

In this paper, we are interested in second-order optimality conditions for the
following constrained vector optimization problem

$\displaystyle\text{min}\,f(x)$ (VP) $\displaystyle\text{s. t.}\ \ x\in
Q_{0}:=\\{x\in\mathbb{R}^{n}\,:\,g(x)\leqq 0\\},$

where $f:=(f_{i})$, $i\in I:=\\{1,\ldots,l\\}$, and $g:=(g_{j})$, $j\in
J:=\\{1,\ldots,m\\}$ are vector-valued functions with $C^{1,1}$ components
defined on $\mathbb{R}^{n}$. Recall that a real-valued function $\varphi$ is
said to be a $C^{1,1}$ function if it is Fréchet differentiable with a locally
Lipschitz gradient $\nabla\varphi(\,\cdot\,)$ on $\mathbb{R}^{n}$; see [1] for
more details.

It is well-known that if $x^{0}\in Q_{0}$ is an efficient solution to (VP),
then there exist Lagrange multipliers
$(\lambda,\mu)\in\mathbb{R}^{l}\times\mathbb{R}^{m}$ satisfying

$\displaystyle\sum_{i=1}^{l}\lambda_{i}\nabla
f_{i}(x^{0})+\sum_{j=1}^{m}\mu_{j}\nabla g_{j}(x^{0})=0,$ (1)
$\displaystyle\mu=(\mu_{1},\ldots,\mu_{m})\geqq 0,\mu_{j}g_{j}(x^{0})=0,$ (2)
$\displaystyle\lambda=(\lambda_{1},\ldots,\lambda_{l})\geqq
0,(\lambda,\mu)\neq 0;$ (3)

see [2, Theorem 7.4]. The conditions (1)–(3) are called the first-order
F.-John necessary optimality conditions. If $\lambda$ is nonzero, then this
conditions are called the Karush–Kuhn–Tucker $(KKT)$ optimality conditions.
$(KKT)$ optimality conditions are one of the most important results in
optimization theory. In vector optimization problems, there are two kinds of
$(KKT)$ optimality conditions. When all Lagrange multipliers corresponding to
the objective functions are positive, we say that strong first-order
Karush–Kuhn–Tucker $(SFKKT)$ conditions hold. On the other hand, when at least
one of Lagrange multipliers, corresponding to the objective functions, is
positive, we say that weak first-order Karush–Kuhn–Tucker $(WFKKT)$ conditions
hold for the problem. In this case, some Lagrange multipliers corresponding to
the components of the vector objective function may be zero. This means that
some components of the vector-valued objective function have no role in the
necessary conditions. To avoid this situation and to obtain positive Lagrange
multipliers associated with each of the objective functions, the problem has
to fulfill some assumptions. These assumptions are called constraint
qualifications $(CQ)$ when they have to be fulfilled by the constraints of the
problem, and they are called regularity conditions $(RC)$ when they have to be
fulfilled by both the objectives and the constraints of the problem.

The optimality conditions for vector problems, which use similar $(CQ)$’s as
those used for single-objective problems do not ensure $(SFKKT)$ conditions;
see [3, 4]. In 1994, Maeda [5] was the first to introduce a Generalized
Guignard regularity condition and established $(SFKKT)$ necessary conditions
for differentiable problems. Later on, Preda and Chiţescu [6] derived
$(SFKKT)$ necessary conditions for efficient solutions of semidifferentiable
vector optimization problems. In the recent years, there are many works
dealing with $(SFKKT)$ necessary optimality conditions for smooth and
nonsmooth vector optimization problems; see, for example, [9, 7, 8, 10].

One of the first investigations to obtain second-order (KKT) optimality
conditions for smooth vector optimization problems was carried out by Wang
[11]. Then, Bigi and Castellani [12, 13] obtained some weak second-order (KKT)
optimality conditions by introducing some types of the second-order regularity
conditions. On the line of their work, many authors have derived weak second-
order (KKT) necessary conditions for efficiency in vector optimization
problems both for smooth and nonsmooth cases; see, for example, [19, 18, 17,
20, 16, 21, 14, 15]. However, to the best of our knowledge, there are only a
few works considering strong second-order Karush–Kuhn–Tucker $(SSKKT)$
necessary optimality conditions. In [22], Maeda was the first to propose a
Abadie second-order regularity condition $(ASORC)$ and established $(SSKKT)$
necessary conditions in terms of generalized second-order directional
derivatives for $C^{1,1}$ vector optimization problems. Recently, Kim and
Tuyen [23] obtained some $(SSKKT)$ necessary optimality conditions for
Geoffrion properly efficient solutions of $C^{2}$ vector optimization problems
under the so-called generalized Abadie second-order regularity condition
$(GASORC)$. The $(GASORC)$ was first introduced by Rizvi and Nasser in [24].
As shown in [24], this condition is weaker than the condition $(ASORC)$.

Our aim is to extend [23, Theorem 3.2] to a larger class of vector
optimization problems. By using the second-order symmetric subdifferential,
which was introduced in [25], we propose two types of second-order regularity
conditions in the sense of Abadie for (VP). These regularity conditions
generalize corresponding regularity conditions in [24] to $C^{1,1}$ vector
optimization problems. Then we establish some $(SSKKT)$ necessary optimality
conditions in terms of second-order symmetric subdifferentials for Geoffrion
properly efficient solutions of (VP). As shown in [25], the second-order
symmetric subdifferential may be strictly smaller than the Clarke
subdifferential, and has some nice properties. In particular, every $C^{1,1}$
function has Taylor expansion in terms of its second-order symmetric
subdifferential. This property plays an important role in our paper.

The rest of the paper is organized as follows. In Section 2, we recall some
basic definitions and preliminaries from variational analysis, which are
widely used in the sequel. Section 3 is devoted to investigate second-order
regularity conditions in the sense of Abadie for vector optimization problems.
In Section 4, we establish some $(SSKKT)$ optimality conditions for Geoffrion
properly efficient solutions of (VP). Section 5 draws some conclusions.

##  2\. Preliminaries

In this section, we recall some basic definitions and preliminaries from
variational analysis, which are widely used in the sequel. Let $\Omega$ be a
subset in $\mathbb{R}^{n}$. The closure, convex hull and conic hull of
$\Omega$ are denoted, respectively, by $\mbox{cl}\,\Omega$, $\mbox{\rm
conv}\,\,\Omega$ and $\mbox{cone}\,\Omega$.

######  Definition 2.1.

Given $\bar{x}\in\mbox{cl}\Omega$. The limiting normal cone or the
Mordukhovich normal cone of $\Omega$ at $\bar{x}$ is the set

$N(\bar{x};\Omega):=\\{z^{*}\in\mathbb{R}^{n}:\exists
x^{k}\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\bar{x},\epsilon_{k}\downarrow
0,z^{*}_{k}\to
z^{*},z^{*}_{k}\in{\widehat{N}_{\epsilon_{k}}}(x^{k};\Omega),\,\,\forall
k\in\mathbb{N}\\},$

where

$\hat{N}_{\epsilon}(x;\Omega):=\left\\{{z^{*}\in{\mathbb{R}^{n}}\;:\;\limsup_{u\overset{\Omega}{\rightarrow}x}\frac{{\langle
z^{*},u-x\rangle}}{{\parallel u-x\parallel}}\leq\epsilon}\right\\}$

is the set of $\epsilon$-normals of $\Omega$ at $x$ and the notation
$u\xrightarrow{{\Omega}}x$ means that $u\rightarrow x$ and $u\in\Omega$.

######  Definition 2.2.

Let $C$ be a nonempty subset of $\mathbb{R}^{n}$, $x^{0}\in C$ and
$u\in\mathbb{R}^{n}$.

  1. (i)

The tangent cone to $C$ at $x^{0}\in C$ is the set defined by

$T(C;x^{0}):=\\{d\in\mathbb{R}^{n}\,:\,\exists t_{k}\,\downarrow 0,\exists
d^{k}\to d,x^{0}+t_{k}d^{k}\in C,\ \ \forall k\in\mathbb{N}\\}.$

  2. (ii)

The second-order tangent set to $C$ at $x^{0}$ with respect to the direction
$u$ is the set defined by

$T^{2}(C;x^{0},u):=\left\\{v\in\mathbb{R}^{n}:\exists t_{k}\,\downarrow
0,v^{k}\to v,x^{0}+t_{k}u+\frac{1}{2}t_{k}^{2}v^{k}\in C,\,\forall
k\in\mathbb{N}\right\\}.$

From the definition, we have $T^{2}(C;x^{0},0)=T(C;x^{0})$. Clearly,
$T(\cdot\,;x^{0})$ and $T^{2}(\cdot\,;x^{0},u)$ are isotone, i.e., if
$C^{1}\subset C^{2}$, then

$\displaystyle T(C^{1};x^{0})$ $\displaystyle\subset T(C^{2};x^{0}),$
$\displaystyle T^{2}(C^{1};x^{0},u)$ $\displaystyle\subset
T^{2}(C^{2};x^{0},u).$

It is well-known that $T(C;x^{0})$ is a nonempty closed cone. For each
$u\in\mathbb{R}^{n}$, the set $T^{2}(C;x^{0},u)$ is closed, but may be empty.
However, we see that the set $T^{2}(C;x^{0},0)=T(C;x^{0})$ is always nonempty.
If $C$ is convex, then

$T(C;x^{0})=\mbox{cl}\,\left\\{d\,:\,d=\beta(x-x^{0}),x\in C,\beta\geq
0\right\\},$

and for each $u\in T(C;x^{0})$ one has

$T^{2}(C;x^{0},u)\subset\mbox{cl cone}[\mbox{ cone}(C-x^{0})-u].$

Moreover, if $C$ is a polyhedral convex set, then
$T^{2}(C;x^{0},u)=T({T(C;x^{0})};u).$

Let $\varphi\colon\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be an extended-real-
valued function. The epigraph, hypergraph and domain of $\varphi$ are denoted,
respectively, by

$\displaystyle\mbox{epi }\varphi$
$\displaystyle:=\\{(x,\alpha)\in\mathbb{R}^{n}\times\mathbb{R}\,:\,\alpha\geq\varphi(x)\\},$
$\displaystyle\mbox{hypo }\varphi$
$\displaystyle:=\\{(x,\alpha)\in\mathbb{R}^{n}\times\mathbb{R}\,:\,\alpha\leq\varphi(x)\\},$
$\displaystyle\mbox{dom }\varphi$
$\displaystyle:=\\{x\in\mathbb{R}^{n}\,:\,|\varphi(x)|<+\infty\\}.$

######  Definition 2.3.

Let $\bar{x}\in\mbox{dom }\varphi$. The set

$\partial\varphi(\bar{x}):=\\{x^{*}\in\mathbb{R}^{n}\,:\,(x^{*},-1)\in
N((\bar{x},\varphi(\bar{x}));\mbox{epi }\varphi)\\}$

is called the limiting subdifferential, or the Mordukhovich subdifferential,
of $\varphi$ at $\bar{x}$. If $\bar{x}\notin\mbox{dom }\varphi$, then we put
$\partial\varphi(\bar{x})=\emptyset$.

######  Definition 2.4.

(see [26, p. 84]) Given $\bar{x}\in\mbox{dom }\varphi.$ The sets

$\displaystyle\partial^{+}\varphi(\bar{x})$
$\displaystyle:=\\{x^{*}\in\mathbb{R}^{n}\,:\,(-x^{*},1)\in
N((\bar{x},\varphi(\bar{x}));\mbox{hypo }\varphi)\\},$
$\displaystyle\partial_{S}\varphi(\bar{x})$
$\displaystyle:=\partial\varphi(\bar{x})\cup\partial^{+}\varphi(\bar{x}),$
$\displaystyle\partial_{C}\varphi(\bar{x})$ $\displaystyle:=\mbox{\rm
cl}\,\mbox{\rm conv}\,\partial_{S}\varphi(\bar{x})$

are called the upper subdifferential, the symmetric subdifferential and the
Clarke subdifferential of $\varphi$ at $\bar{x}$, respectively.

We note here that

$\partial\varphi(\bar{x})\subseteq\partial_{S}\varphi(\bar{x})\subseteq\partial_{C}\varphi(\bar{x}),$

and both inclusions may be strict, see [26, pp. 92–93].

Let $D$ be an open subset of $\mathbb{R}^{n}$. We denote by $C^{1,1}(D)$ the
class of all real-valued functions $\varphi$, which are Fréchet differentiable
on $D$, and whose gradient mapping $\nabla\varphi(\cdot)$ is locally Lipschitz
on $D$.

######  Definition 2.5.

(see [25, Definition 2.6]) Let $\varphi\in C^{1,1}(D)$ and $\bar{x}\in D$. The
second-order symmetric subdifferential of $\varphi$ at $\bar{x}$ is a
multifunction

$\partial_{S}^{2}\varphi(\bar{x})\colon\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$

defined by

$\partial_{S}^{2}\varphi(\bar{x})(v):=\partial_{S}\langle
v,\nabla\varphi\rangle(\bar{x})=\partial\langle
v,\nabla\varphi\rangle(\bar{x})\cup\partial^{+}\langle
v,\nabla\varphi\rangle(\bar{x}),\ \ \forall v\in\mathbb{R}^{n}.$

We now summarize some properties of the second-order symmetric subdifferential
that will be needed in this paper.

######  Proposition 2.1.

(see [25, Proposition 2.3]) Let $\varphi\in C^{1,1}(D)$ and $\bar{x}\in D$.
The following assertions hold:

  1. (i)

For any $\lambda\in\mathbb{R}$ and $v\in\mathbb{R}^{n}$, we have

$\partial_{S}^{2}\varphi(\bar{x})(\lambda
v)=\partial_{S}^{2}(\lambda\varphi)(\bar{x})(v)=\lambda\partial_{S}^{2}\varphi(\bar{x})(v).$

  2. (ii)

For any $v\in\mathbb{R}^{n}$, $\partial_{S}^{2}\varphi(\bar{x})(v)$ is a
nonempty compact set in $\mathbb{R}^{n}$.

  3. (iii)

For any $v\in\mathbb{R}^{n}$ the mapping
$x\mapsto\partial_{S}^{2}\varphi(x)(v)$ is locally bounded. Moreover, if
$x_{k}\to\bar{x}$, $x^{*}_{k}\to x^{*}$ and
$x^{*}_{k}\in\partial_{S}^{2}\varphi(x_{k})(v)$ for all $k\in\mathbb{N}$, then
$x^{*}\in\partial_{S}^{2}\varphi(\bar{x})(v)$.

######  Theorem 2.1 (Taylor’s formula).

(see [25, Corollary 2.1]) Let $\varphi\in C^{1,1}(\mathbb{R}^{n})$. Then, for
every $a,b\in\mathbb{R}^{n}$ there exists
$z^{*}\in\partial\varphi_{S}^{2}(\xi)(b-a)$, where $\xi\in\,(a,b)$, such that

$\varphi(b)-\varphi(a)-\langle\nabla\varphi(a),b-a\rangle=\dfrac{1}{2}\langle
z^{*},b-a\rangle.$

##  3\. Second-Order Abadie Regularity Conditions

In this section, we propose some types of second-order regularity conditions
in the sense of Abadie for vector optimization problems, investigate some
relations with the regularity conditions in [22, 24], and give some conditions
which assure that these regularity conditions hold true.

We first recall notations and definitions which will be used in the sequel.
Let $\mathbb{R}^{l}$ be the $l$-dimensional Euclidean space. For
$a,b\in\mathbb{R}^{l}$, by $a\leqq b$, we mean $a_{i}\leq b_{i}$ for all
$i=1,\ldots,l$; by $a\leq b$, we mean $a\leqq b$ and $a\neq b$; and by $a<b$,
we mean $a_{i}<b_{i}$ for all $i=1,\ldots,l$. For any two vectors
$a=(a_{1},a_{2})$ and $b=(b_{1},b_{2})$ in $\mathbb{R}^{2}$, we denote the
lexicographic order by

$\displaystyle a$ $\displaystyle\leqq_{\rm lex}b,\ \ {\rm iff}\ \ a_{1}<b_{1}\
\ {\rm or}\ \ a_{1}=b_{1}\ \ {\rm and}\ \ a_{2}\leq b_{2},$ $\displaystyle a$
$\displaystyle<_{\rm lex}b,\ \ {\rm iff}\ \ a_{1}<b_{1}\ \ {\rm or}\ \
a_{1}=b_{1}\ \ {\rm and}\ \ a_{2}<b_{2}.$

Fix $x^{0}\in Q_{0}$, the active index set at $x^{0}$ is defined by

$J(x^{0}):=\\{j\in J\,:\,g_{j}(x^{0})=0\\}.$

For each $u\in\mathbb{R}^{n}$, put $J(x^{0};u):=\\{j\in
J(x^{0})\,:\,\langle\nabla g_{j}(x^{0}),u\rangle=0\\}.$ We say that $u$ is a
critical direction of problem (VP) at $x^{0}\in Q_{0}$ iff

$\displaystyle\langle\nabla f_{i}(x^{0}),u\rangle$ $\displaystyle=0,\ \ \
\forall i\in I,$ $\displaystyle\langle\nabla g_{j}(x^{0}),u\rangle$
$\displaystyle\leq 0,\ \ \ \forall j\in J(x^{0}).$

The set of all critical direction of the problem (VP) at $x^{0}$ is denoted by
$K(x^{0})$. The following sets were introduced by Maeda [22]:

$\displaystyle Q^{i}$
$\displaystyle:=Q_{0}\cap\\{x\in\mathbb{R}^{n}\,:\,f_{k}(x)\leq
f_{k}(x^{0}),k\in I\setminus\\{i\\}\\},\ \ i\in I,$ $\displaystyle Q$
$\displaystyle:=Q_{0}\cap\\{x\in\mathbb{R}^{n}\,:\,f_{k}(x)\leq
f_{k}(x^{0}),k\in I\\}.$

If $I=\\{1\\}$, we set $Q^{i}:=Q_{0}$. The sets were proposed by Rizvi and
Nasser [24].

$M^{i}:=Q_{0}\cap\\{x\in\mathbb{R}^{n}\,:\,f_{i}(x)\leq f_{i}(x^{0})\\},\ \ \
i\in I.$

Clearly, $\displaystyle Q^{i}=\mathop{\bigcap\limits_{k\in I}}\limits_{k\neq
i}M^{k}$ for all $i\in I$ and $\displaystyle Q=\bigcap_{i\in
I}Q^{i}=\bigcap_{i\in I}M^{i}.$

######  Remark 3.1.

By the isotony of $T(\cdot\,;x^{0})$ and $T^{2}(\cdot\,;x^{0},u)$, we have

$T(Q^{i};x^{0})\subset T(M^{k};x^{0}),T^{2}(Q^{i};x^{0},u)\subset
T^{2}(M^{k};x^{0},u),\ \ \ \forall k\in I\setminus\\{i\\}.$

Thus

$\bigcap_{i\in I}T(Q^{i};x^{0})\subset\bigcap_{i\in I}T(M^{i};x^{0})\ \
\mbox{and}\ \ \bigcap_{i\in I}T^{2}(Q^{i};x^{0},u)\subset\bigcap_{i\in
I}T^{2}(M^{i};x^{0},u).$

The following example shows that the above inclusions may be strictly proper.

######  Example 3.1.

Let $f\colon\mathbb{R}^{2}\to\mathbb{R}^{3}$ and
$g\colon\mathbb{R}^{2}\to\mathbb{R}$ be two maps defined by

$\displaystyle f(x)$
$\displaystyle:=(f_{1}(x),f_{2}(x),f_{3}(x))=(x_{2},x_{1}+x_{2}^{2},-x_{1}-x_{1}|x_{1}|+x_{2}^{2})$
$\displaystyle g(x)$ $\displaystyle:=0,\ \ \forall
x=(x_{1},x_{2})\in\mathbb{R}^{2}.$

Clearly, $Q_{0}=\mathbb{R}^{2}$ and $x^{0}=(0,0)$ is a feasible point to
problem (VP). We have

$\displaystyle M^{1}=\\{(x_{1},x_{2})\,:\,x_{2}\leq
0\\},M^{2}=\\{(x_{1},x_{2})\,:\,x_{1}+x_{2}^{2}\leq 0\\},$ $\displaystyle
M^{3}=\\{(x_{1},x_{2})\,:\,-x_{1}-x_{1}|x_{1}|+x_{2}^{2}\leq 0\\},$
$\displaystyle Q^{1}=\\{x^{0}\\},Q^{2}=M^{1}\cap M^{3},Q^{3}=M^{1}\cap M^{2}.$

An easy computation shows that

$\displaystyle T(M^{1};x^{0})=M^{1},T(M^{2};x^{0})=\\{(u_{1},u_{2}):u_{1}\leq
0\\},$ $\displaystyle T(M^{3};x^{0})=\\{(u_{1},u_{2})\,:\,u_{1}\geq
0\\},\bigcap_{i=1}^{3}T(M^{i};x^{0})=\\{(u_{1},u_{2})\in\mathbb{R}^{2}:u_{1}=0,u_{2}\leq
0\\}.$

Clearly, $T(Q^{1};x^{0})=\\{x^{0}\\}$. Consequently,
$\displaystyle\bigcap_{i=1}^{3}T(Q^{i};x^{0})=\\{x^{0}\\}.$ Thus

$\bigcap_{i=1}^{3}T(Q^{i};x^{0})\subsetneqq\bigcap_{i=1}^{3}T(M^{i};x^{0}).$

From this and the fact that

$T^{2}(M^{i};x^{0},0)=T(M^{i};x^{0}),T^{2}(Q^{i};x^{0},0)=T(Q^{i};x^{0}),$

for all $i=1,2,3$, we have

$\bigcap_{i=1}^{3}T^{2}(Q^{i};x^{0},0)\subsetneqq\bigcap_{i=1}^{3}T^{2}(M^{i};x^{0},0).$

Now we define the first-order and the second-order linear approximation sets
to the set $Q$. Fix $u\in\mathbb{R}^{n}$. By the compactness of
$\partial_{S}^{2}f_{i}(x^{0})(u)$ and $\partial_{S}^{2}g_{j}(x^{0})(u)$, we
always denote hereafter that $\xi^{*i}$ and $\zeta^{*j}$ are elements in
$\partial_{S}^{2}f_{i}(x^{0})(u)$ and $\partial_{S}^{2}g_{j}(x^{0})(u)$,
respectively, such that

$\displaystyle\langle\xi^{*i},u\rangle$
$\displaystyle:=\max\left\\{\langle\xi^{i},u\rangle\,:\,\xi^{i}\in\partial_{S}^{2}f_{i}(x^{0})(u)\right\\},\
\ i\in I,$ $\displaystyle\langle\zeta^{*j},u\rangle$
$\displaystyle:=\max\left\\{\langle\zeta^{j},u\rangle\,:\,\zeta^{j}\in\partial_{S}^{2}g_{j}(x^{0})(u)\right\\},\
\ j\in J.$

For each $(u,v)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$, put

$\displaystyle F^{2}_{i}(x^{0};u,v)$ $\displaystyle:=\left(\langle\nabla
f_{i}(x^{0}),u\rangle,\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle\right),\ \ i\in I,$
$\displaystyle G^{2}_{j}(x^{0};u,v)$ $\displaystyle:=\left(\langle\nabla
g_{j}(x^{0}),u\rangle,\langle\nabla
g_{j}(x^{0}),v\rangle+\langle\zeta^{*j},u\rangle\right),\ \ j\in J.$

######  Definition 3.1.

Let $x^{0}\in Q_{0}$ and $u\in\mathbb{R}^{n}$.

  1. (i)

The linearizing cone to $Q$ at $x^{0}\in Q_{0}$ is defined by

$L(Q;x^{0}):=\\{u\in\mathbb{R}^{n}\,:\,\langle\nabla f_{i}(x^{0}),u\rangle\leq
0,\langle\nabla g_{j}(x^{0}),u\rangle\leq 0,i\in I,j\in J(x^{0})\\}.$

  2. (ii)

The second-order linearizing set of $Q$ at $x^{0}\in Q_{0}$ in the direction
$u$ is defined by

$\displaystyle L^{2}(Q;x^{0},u)$
$\displaystyle:=\\{v\in\mathbb{R}^{n}\,\,:\,F^{2}_{i}(x^{0};u,v)\leqq_{\rm
lex}(0,0),\ \ i\in I,$ $\displaystyle\qquad\qquad\text{and}\ \
G^{2}_{j}(x^{0};u,v)\leqq_{\rm lex}(0,0),\ \ j\in J(x^{0})\\}.$

Then, we introduce two types of second-order regularity conditions in the
sense of Abadie as follows.

######  Definition 3.2.

Let $x^{0}\in Q_{0}$ and $u\in\mathbb{R}^{n}$. We say that:

  1. (i)

The Abadie second-oder regularity condition holds at $x^{0}$ for the direction
$u$ iff

$L^{2}(Q;x^{0},u)\subset\bigcap_{i=1}^{l}T^{2}(Q^{i};x^{0},u).$

  2. (ii)

The generalized Abadie second-oder regularity condition holds at $x^{0}$ for
the direction $u$ iff

$L^{2}(Q;x^{0},u)\subset\bigcap_{i=1}^{l}T^{2}(M^{i};x^{0},u).$

######  Definition 3.3.

Let $x^{0}\in Q_{0}$. We say that the Abadie first-order regularity condition
$(AFORC)$ (resp., generalized Abadie first-oder regularity condition
$(GAFORC)$) holds at $x^{0}$ iff ((i)) (resp., ((ii))) holds at $x^{0}$ for
the direction $0$.

######  Remark 3.2.

  1. (i)

From the isotony of second-order tangent sets, if the ((i)) holds at $x^{0}$
for the direction $u$, then so does the ((ii)). The reverse does not hold in
general; see Example 4.2 in Sect. 4.

  2. (ii)

When $f$ and $g$ are $C^{2}$ functions, then the ((ii)) coincides with the
generalized Abadie second-oder regularity condition in the sense of Rizvi and
Nasser [24].

  3. (iii)

Let $\varphi\in C^{1,1}$. By [27, Proposition 2.1.2], we have

$\varphi^{\circ\circ}(x^{0};u,u)=\max\,\\{\langle\xi,u\rangle\,:\,\xi\in\partial_{C}\langle\nabla\varphi(\cdot),u\rangle(x^{0})\\}=\langle\xi^{*},u\rangle,$

where

$\displaystyle\varphi^{\circ\circ}(x^{0};u,u)$
$\displaystyle:=\limsup\limits_{\mathop{x\to x^{0}}\limits_{t\downarrow
0}}\frac{\langle\nabla\varphi(x+tu),u\rangle-\langle\nabla\varphi(x),u\rangle}{t},$
$\displaystyle\langle\xi^{*},u\rangle$
$\displaystyle:=\max\left\\{\langle\xi,u\rangle\,:\,\xi\in\partial_{S}^{2}\varphi(x^{0})(u)\right\\}.$

Thus, the ((i)) coincides with the Abadie second-oder regularity condition in
the sense of Maeda [22].

We now introduce a condition, which ensures that the ((ii)) holds at $x^{0}$
for the direction $u$.

######  Proposition 3.1.

Let $x^{0}\in Q_{0}$ and $u\in\mathbb{R}^{n}$. Suppose that, for each $i\in
I$, the following system (in the unknown $v$)

$\displaystyle\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle$
$\displaystyle<0,$ (4) $\displaystyle\langle\nabla
g_{j}(x^{0}),v\rangle+\langle\zeta^{*j},u\rangle$ $\displaystyle<0,$
$\displaystyle\ \ \ j\in J(x^{0};u),$ (5)

has at least one solution, say $v^{i}\in\mathbb{R}^{n}$. Then, the ((ii))
holds at $x^{0}$ for the direction $u$.

###### Proof.

Let $v$ be an arbitrary element in $L^{2}(Q;x^{0},u)$. Fix $i\in I$ and let
$v^{i}$ be a solution of the system (4)-(5). Let $\\{s_{h}\\}$ and
$\\{t_{p}\\}$ be any positive sequences converging to 0. For each
$h\in\mathbb{N}$, put $w^{h}:=s_{h}v^{i}+(1-s_{h})v$. Clearly,
$\displaystyle\lim_{h\to\infty}w^{h}=v$. Since $v\in L^{2}(Q;x^{0},u)$, we
have

$\begin{cases}F^{2}_{k}(x^{0};u,v)&=\left(\langle\nabla
f_{k}(x^{0}),u\rangle,\langle\nabla
f_{k}(x^{0}),v\rangle+\langle\xi^{*k},u\rangle\right)\leqq_{\rm
lex}(0,0),\,\,k\in I,\\\ G^{2}_{j}(x^{0};u,v)&=\left(\langle\nabla
g_{j}(x^{0}),u\rangle,\langle\nabla
g_{j}(x^{0}),v\rangle+\langle\zeta^{*j},u\rangle\right)\leqq_{\rm
lex}(0,0),\,\,j\in J(x^{0}).\end{cases}$ (6)

This implies that

$\begin{cases}\langle\nabla f_{k}(x^{0}),u\rangle\leq 0,\ \ \ \forall k\in
I,\\\ \langle\nabla g_{j}(x^{0}),u\rangle\leq 0,\ \ \ \forall j\in
J(x^{0}).\end{cases}$

For $h=1$, we have $w^{1}=s_{1}v^{i}+(1-s_{1})v$. We consider two cases of $i$
as follows.
Case 1. $\langle\nabla f_{i}(x^{0}),u\rangle=0$. By (6), we have
$\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle\leq 0.$ Since
$v^{i}$ is a solution of the system (4)-(5), we have

$\displaystyle\langle\nabla
f_{i}(x^{0}),w^{1}\rangle+\langle\xi^{*i},u\rangle=$ $\displaystyle
s_{1}[\langle\nabla f_{i}(x^{0}),v^{i}\rangle+\langle\xi^{*i},u\rangle]$
$\displaystyle+(1-s_{1})[\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle]<0.$ (7)

For each $p\in\mathbb{N}$, put
$x^{p}:=x^{0}+t_{p}u+\frac{1}{2}t^{2}_{p}w^{1}$. Then we have

$f_{i}(x^{p})-f_{i}(x^{0})=[f_{i}(x^{p})-f_{i}(x^{0}+t_{p}u)]+[f_{i}(x^{0}+t_{p}u)-f_{i}(x^{0})-t_{p}\langle\nabla
f_{i}(x^{0}),u\rangle].$

From the mean value theorem for differentiable functions, we have

$f_{i}(x^{p})-f_{i}(x^{0}+t_{p}u)=\langle\nabla
f_{i}(\gamma^{s}),\frac{1}{2}t_{p}^{2}w^{1}\rangle=\frac{1}{2}t_{p}^{2}\langle\nabla
f_{i}(\gamma^{p}),w^{1}\rangle$ (8)

for some $\gamma^{p}\in(x^{0}+t_{p}u,x^{p})$. By Taylor’s formula, there exist
$\eta^{p}\in(x^{0},x^{0}+t_{p}u)$ and
$\zeta^{s}\in\partial_{S}^{2}f_{i}(\eta^{p})(t_{p}u)$ such that

$f_{i}(x^{0}+t_{p}u)-f_{i}(x^{0})-t_{p}\langle\nabla
f_{i}(x^{0}),u\rangle=\frac{1}{2}\langle\zeta^{p},t_{p}u\rangle=\frac{1}{2}t_{p}\langle\zeta^{p},u\rangle.$

Furthermore, since
$\partial_{S}^{2}f_{i}(\eta^{p})(t_{p}u)=t_{p}\partial_{S}^{2}f_{i}(\eta^{p})(u)$,
there is $\xi^{p}\in\partial_{S}^{2}f_{i}(\eta^{p})(u)$ such that
$\zeta^{p}=t_{p}\xi^{p}$. Thus

$f_{i}(x^{0}+t_{p}u)-f_{i}(x^{0})-t_{p}\langle\nabla
f_{i}(x^{0}),u\rangle=\frac{1}{2}t^{2}_{p}\langle\xi^{p},u\rangle.$

From this and (8), one has

$\dfrac{f_{i}(x^{p})-f_{i}(x^{0})}{\frac{1}{2}t^{2}_{p}}=\langle\nabla
f_{i}(\gamma^{p}),w^{1}\rangle+\langle\xi^{p},u\rangle.$ (9)

Since $\partial_{S}^{2}f_{i}(\cdot)(u)$ is locally bounded at $x^{0}$ and
$\displaystyle\lim_{p\to\infty}\eta^{p}=x^{0}$, it follows that the sequence
$\\{\xi^{p}\\}$ is bounded. By the boundedness of $\\{\xi^{p}\\}$ and
Proposition 2.1(iii), without loss of any generality, we may assume that
$\\{\xi^{p}\\}$ converges to $\xi^{i}\in\partial_{S}^{2}f_{i}(x^{0})$. It is
easily seen that

$\lim_{p\to\infty}\langle\nabla f_{i}(\gamma^{p}),w^{1}\rangle=\langle\nabla
f_{i}(x^{0}),w^{1}\rangle.$

Letting $p\to\infty$ in (9) we have

$\lim_{p\to\infty}\dfrac{f_{i}(x^{p})-f_{i}(x^{0})}{\frac{1}{2}t^{2}_{p}}=\langle\nabla
f_{i}(x^{0}),w^{1}\rangle+\langle\xi^{i},u\rangle.$ (10)

Since (7), one has

$\langle\nabla
f_{i}(x^{0}),w^{1}\rangle+\langle\xi^{i},u\rangle\leq\langle\nabla
f_{i}(x^{0}),w^{1}\rangle+\langle\xi^{*i},u\rangle<0.$ (11)

By (10) and (11), there exists $N_{1}\in\mathbb{N}$ such that
$f_{i}(x^{p})-f_{i}(x^{0})<0$ for all $p\geq N_{1}$, or, equivalently,

$f_{i}(x^{p})<f_{i}(x^{0}),\ \ \ \forall p\geq N_{1}.$ (12)

Case 2. $\langle\nabla f_{i}(x^{0}),u\rangle<0$. Since

$\lim_{p\to\infty}\dfrac{f_{i}(x^{p})-f_{i}(x^{0})}{t_{p}}=\langle\nabla
f_{i}(x^{0}),u\rangle,$

there exists $N_{2}\in\mathbb{N}$ such that $f_{i}(x^{p})<f_{i}(x^{0})$ for
all $p\geq N_{2}.$

We now claim that $g_{j}(x^{p})<0$ for all $j\in J$ and $p$ large enough.
Indeed, we consider the following cases:

  1. (a)

$j\in J\setminus J(x^{0})$. Then $g_{j}(x^{0})<0$. Hence, by
${\displaystyle\lim_{p\to\infty}}x^{p}=x^{0}$ and the continuity of $g_{j}$,
there exists $N_{3}\in\mathbb{N}$ such that $g_{j}(x^{p})<0$ for all $p\geq
N_{3}$.

  2. (b)

$j\in J(x^{0})\setminus J(x^{0};u)$. Then $g_{j}(x^{0})=0$ and $\langle\nabla
g_{j}(x^{0}),u\rangle<0$. From this and

$\lim_{p\to\infty}g_{j}(x^{p})=\lim_{p\to\infty}\dfrac{g_{j}(x^{p})-g_{j}(x^{0})}{t_{p}}=\langle\nabla
g_{j}(x^{0}),u\rangle$

it follows that there exists $N_{4}\in\mathbb{N}$ such that $g_{j}(x^{p})<0$
for all $p\geq N_{4}.$

  3. (c)

$j\in J(x^{0};u)$. An analysis similar to the one made in the proof of (12)
shows that there exist $N_{5}\in\mathbb{N}$ such that $g_{j}(x^{p})<0$ for all
$p\geq N_{5}$.

Put $p_{1}=\max\\{N_{1},N_{2},N_{3},N_{4},N_{5}\\}$. Then we have

$\displaystyle f_{i}(x^{0}+t_{p_{1}}u+\frac{1}{2}t^{2}_{p_{1}}w^{1})$
$\displaystyle<f_{i}(x^{0}),$ $\displaystyle
g_{j}(x^{0}+t_{p_{1}}u+\frac{1}{2}t^{2}_{p_{1}}w^{1})$ $\displaystyle<0,\ \ \
\forall j\in J.$

This means that $x^{0}+t_{p_{1}}u+\frac{1}{2}t^{2}_{p_{1}}w^{1}\in M^{i}.$
Thus, by induction, we could construct a subsequence $\\{t_{p_{h}}\\}$ of
$\\{t_{p}\\}$ satisfying $x^{0}+t_{p_{h}}u+\frac{1}{2}t^{2}_{p_{h}}w^{h}\in
M^{i}$ for all $h\in\mathbb{N}$. From this and
$\displaystyle\lim_{h\to\infty}w^{h}=v$ it follows that $v\in
T^{2}(M^{i};x^{0},u)$. Since $i$ is arbitrary, we have $v\in
T^{2}(M^{i};x^{0},u)$ for all $i\in I$. Thus, the ((ii)) holds at $x^{0}$ for
the direction $u$. ∎

By a similar argument, we have the following result.

######  Proposition 3.2.

Let $x^{0}\in Q_{0}$ and $u\in\mathbb{R}^{n}$. Suppose that, for each $i\in
I$, the following system (in the unknown $w$)

$\displaystyle\langle\nabla f_{k}(x^{0}),w\rangle+\langle\xi^{*k},u\rangle$
$\displaystyle<0,\ \ \ $ $\displaystyle k\in I\setminus\\{i\\},$
$\displaystyle\langle\nabla g_{j}(x^{0}),w\rangle+\langle\zeta^{*j},u\rangle$
$\displaystyle<0,\ \ \ $ $\displaystyle j\in J(x^{0};u),$

has at least one solution, say $w^{i}\in\mathbb{R}^{n}$. Then, the ((i)) holds
at $x^{0}$ for the direction $u$.

##  4\. Strong Second-Order Karush–Kuhn–Tucker Necessary Optimality Conditions

In this section, we provide some strong second-order Karush–Kuhn-Tucker
necessary optimality conditions for Geoffrion properly efficient solutions of
(VP). Properly efficient solution plays a vital role from both theoretical and
practical points of view. This concept has been introduced at first to
eliminate the efficient solutions with unbounded trade-offs. In multiobjective
optimization, properly efficient solutions are efficient solutions in which,
given any objective, the trade-off between that objective and some other
objective is bounded. This notion was introduced initially by Kuhn and Tucker
[3] and was followed thereafter by Geoffrion [28]. Geoffrion’s definition
enjoys economical interpretations, while Kuhn and Tucker’s definition is
useful for numerical and algorithmic purposes. We now recall the definition of
Geoffrion properly efficient solutions from [28].

######  Definition 4.1.

Let $x^{0}\in Q_{0}$. We say that:

  1. (i)

$x^{0}$ is an efficient solution to (VP) iff there is no $x\in Q_{0}$
satisfies $f(x)\leq f(x^{0})$.

  2. (ii)

$x^{0}$ is a Geoffrion properly efficient solution to (VP) iff it is efficient
and there exists $M>0$ such that, for each $i$,

$\frac{f_{i}(x)-f_{i}(x^{0})}{f_{j}(x^{0})-f_{j}(x)}\leq M,$

for some $j$ such that $f_{j}(x^{0})<f_{j}(x)$ whenever $x\in Q_{0}$ and
$f_{i}(x^{0})>f_{i}(x)$.

The following result gives a first-order necessary optimality condition for
(VP) under the $(GAFORC)$.

######  Theorem 4.1.

(see [9, Theorem 4.3]) If $x^{0}\in Q_{0}$ is a Geoffrion properly efficient
solution to problem (VP) and the $(GAFORC)$ holds at $x^{0}$, then the
following system has no solution $u\in\mathbb{R}^{n}$:

$\displaystyle\langle\nabla f_{i}(x^{0}),u\rangle$ $\displaystyle\leq 0,\ \ \
$ $\displaystyle i\in I,$ $\displaystyle\langle\nabla f_{i}(x^{0}),u\rangle$
$\displaystyle<0,\ \ \ $ $\displaystyle\mbox{at least one}\ \ i\in I,$
$\displaystyle\langle\nabla g_{j}(x^{0}),u\rangle$ $\displaystyle\leq 0,\ \ \
$ $\displaystyle j\in J(x^{0}).$

######  Definition 4.2.

We say that the strong second-order Krush–Kuhn-Tucker necessary optimality
conditions $(SSKKT)$ holds at $x^{0}$ for the direction $u$ iff there exist
$\lambda\in\mathbb{R}^{l}$ and $\mu\in\mathbb{R}^{m}$ satisfying

$\displaystyle\sum_{i=1}^{l}\lambda_{i}\nabla
f_{i}(x^{0})+\sum_{j=1}^{m}\mu_{j}\nabla g_{j}(x^{0})=0,$ (13)
$\displaystyle\sum_{i=1}^{l}\lambda_{i}\langle\xi^{*i},u\rangle+\sum_{j=1}^{m}\mu_{j}\langle\zeta^{*j},u\rangle\geq
0,$ (14) $\displaystyle\mu=(\mu_{1},\mu_{2},\ldots,\mu_{m})\geqq 0,\mu_{j}=0,\
\ j\notin J(x^{0};u),$ (15)
$\displaystyle\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{l})>0.$ (16)

The following theorem is crucial for establishing the $(SSKKT)$.

######  Theorem 4.2.

Let $x^{0}$ be a Geoffrion properly efficient solution to problem (VP).
Suppose that the ((ii)) holds at $x^{0}$ for any critical direction at
$x^{0}$. Then, the following system

$\displaystyle F^{2}_{i}(x^{0};u,v)$ $\displaystyle\leq_{\rm lex}(0,0),\ \ \
i\in I,$ (17) $\displaystyle F^{2}_{i}(x^{0};u,v)$ $\displaystyle<_{\rm
lex}(0,0),\ \ \ \mbox{ at least one}\ \ i\in I,$ (18) $\displaystyle
G^{2}_{j}(x^{0};u,v)$ $\displaystyle\leqq_{\rm lex}(0,0),\ \ \ j\in J(x^{0})$
(19)

has no solution $(u,v)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$.

###### Proof.

Arguing by contradiction, assume that the system (17)–(19) admits one solution
$(u,v)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$. Without loss of generality we
may assume that

$F^{2}_{1}(x^{0};u,v)<_{\rm lex}(0,0).$ (20)

From (17) and (19) it follows that $v\in L^{2}(Q;x^{0},u)$ and

$\displaystyle\langle\nabla f_{i}(x^{0}),u\rangle$ $\displaystyle\leq 0,\ \ \
$ $\displaystyle i\in I,$ $\displaystyle\langle\nabla g_{j}(x^{0}),u\rangle$
$\displaystyle\leq 0,\ \ \ $ $\displaystyle j\in J(x^{0}).$

Since the ((ii)) holds at $x^{0}$ for any critical direction, the $(GAFORC)$
holds at $x^{0}$. By Theorem 4.1, we have $\langle\nabla
f_{i}(x^{0}),u\rangle=0$ for all $i\in I$. Thus, $u$ is a critical direction
at $x^{0}$. Since the ((ii)) holds at $x^{0}$ for the critical direction $u$,
we have $v\in T^{2}(M^{i};x^{0},u)$ for all $i\in I$. Consequently, $v\in
T^{2}(M^{1};x^{0},u)$. This implies that there exist a sequence $\\{v^{k}\\}$
converging to $v$ and a positive sequence $\\{t_{k}\\}$ converging to $0$ such
that

$x^{k}:=x^{0}+t_{k}u+\frac{1}{2}t_{k}^{2}v^{k}\in M^{1},\ \ \ \forall
k\in\mathbb{N}.$

Since $M^{1}\subset Q_{0}$, $\\{x^{k}\\}\subset Q_{0}$. By (17) and (20), we
have

$\displaystyle\langle\nabla f_{1}(x^{0}),v\rangle+\langle\xi^{*1},u\rangle$
$\displaystyle<0,$ (21) $\displaystyle\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle$ $\displaystyle\leq 0,\ \ \
\forall i\in\\{2,\ldots,l\\}.$ (22)

An analysis similar to the one made in the proof of Proposition 3.1 shows
that, for each $i\in I$, there exists
$\xi^{i}\in\partial_{S}^{2}f_{i}(x^{0})(u)$ such that

$\lim_{k\to\infty}\dfrac{f_{i}(x^{k})-f_{i}(x^{0})}{\frac{1}{2}t^{2}_{k}}=\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{i},u\rangle.$ (23)

In particular,

$\lim_{k\to\infty}\dfrac{f_{1}(x^{k})-f_{1}(x^{0})}{\frac{1}{2}t^{2}_{k}}=\langle\nabla
f_{1}(x^{0}),v\rangle+\langle\xi^{1},u\rangle.$ (24)

Since (21), we have

$\langle\nabla f_{1}(x^{0}),v\rangle+\langle\xi^{1},u\rangle<0.$ (25)

This and (24) imply that

$f_{1}(x^{k})-f_{1}(x^{0})<0$

for all large enough $k$. Without loss of generality we may assume that

$f_{1}(x^{k})<f_{1}(x^{0}),\ \ \ \forall k\in\mathbb{N}.$

For each $k\in\mathbb{N}$, put

$I_{k}:=\\{i\in I\ :\ i\geq 2\ \ \mbox{and}\ \ f_{i}(x^{k})>f_{i}(x^{0})\\}.$

We claim that $I_{k}$ is nonempty for all $k\in\mathbb{N}$. Indeed, if
$I_{k}=\emptyset$ for some $k\in\mathbb{N}$, then we have

$f_{i}(x^{k})\leq f_{i}(x^{0}),\ \ \ \forall i=2,\ldots,l.$

Using also the fact that $f_{1}(x^{k})<f_{1}(x^{0})$, we arrive at a
contradiction with the efficiency of $x^{0}$.

Since $I_{k}\subset\\{2,\ldots,l\\}$ for all $k\in\mathbb{N}$, without loss of
generality, we may assume that $I_{k}=\bar{I}$ is constant for all
$k\in\mathbb{N}$. By (23), for each $i\in\bar{I}$, we have

$\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{i},u\rangle\geq 0.$

Thus

$\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle\geq\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{i},u\rangle\geq 0.$

This and (22) imply that

$\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{i},u\rangle=\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle=0,$ (26)

for all $i\in\bar{I}$. By (25), we can choose $\delta\in\mathbb{R}$ such that

$\langle\nabla f_{1}(x^{0}),v\rangle+\langle\xi^{1},u\rangle<\delta<0,$

or, equivalently,

$-[\langle\nabla f_{1}(x^{0}),v\rangle+\langle\xi^{1},u\rangle]>-\delta>0.$

From this and (24) it follows that there exists $k_{0}\in\mathbb{N}$ such that

$f_{1}(x^{0})-f_{1}(x^{k})>-\frac{1}{2}\delta t^{2}_{k}>0,$

for all $k\geq k_{0}$. Thus, for any $i\in\bar{I}$ and $k\geq k_{0}$, we have

$0<\dfrac{f_{i}(x^{k})-f_{i}(x^{0})}{f_{1}(x^{0})-f_{1}(x^{k})}\leq\dfrac{f_{i}(x^{k})-f_{i}(x^{0})}{-\frac{1}{2}\delta
t^{2}_{k}}.$

Combining this, (23) and (26), we deduce

$\displaystyle
0\leq\lim_{k\to\infty}\dfrac{f_{i}(x^{k})-f_{i}(x^{0})}{f_{1}(x^{0})-f_{1}(x^{k})}$
$\displaystyle\leq\lim_{k\to\infty}\dfrac{f_{i}(x^{k})-f_{i}(x^{0})}{-\frac{1}{2}\delta
t^{2}_{k}}$ $\displaystyle=-\frac{1}{\delta}[\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{i},u\rangle]=0.$

Thus

$\lim_{k\to\infty}\dfrac{f_{1}(x^{k})-f_{1}(x^{0})}{f_{i}(x^{0})-f_{i}(x^{k})}=+\infty,$

which contradicts that $x^{0}$ is a Geoffrion properly efficient solution to
(VP). ∎

The following theorem gives some strong second-order Karush–Kuhn-Tucker
necessary optimality conditions for Geoffrion properly efficient solutions of
problem (VP). This result generalizes [23, Theorem 3.2] to $C^{1,1}$ vector
optimization problems.

######  Theorem 4.3.

Let $x^{0}\in Q_{0}$ be a Geoffrion properly efficient solution to problem
(VP) and $u\in K(x^{0})$. If the ((ii)) holds at $x^{0}$ for the direction
$u$, then so does the $(SSKKT)$.

###### Proof.

Suppose that $x^{0}$ is a Geoffrion properly efficient solution to problem
(VP) and $u\in K(x^{0})$. Thanks to Theorem 4.2, the system

$\displaystyle\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle$
$\displaystyle\leq 0,\ \ \ \ $ $\displaystyle i\in I,$
$\displaystyle\langle\nabla f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle$
$\displaystyle<0,\ \ \ \ $ $\displaystyle\mbox{at least one }\ \ i\in I,$
$\displaystyle\langle\nabla g_{j}(x^{0}),v\rangle+\langle\zeta^{*j},u\rangle$
$\displaystyle\leq 0,\ \ \ \ $ $\displaystyle j\in J(x^{0};u),$

has no solution $v\in\mathbb{R}^{n}$. This is equivalent to the inconsistency
of the following system

$\displaystyle\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle\epsilon$ $\displaystyle\leq 0,\
\ \ \ i=1,2,...,l,$ (27) $\displaystyle\langle\nabla
f_{i}(x^{0}),v\rangle+\langle\xi^{*i},u\rangle\epsilon$ $\displaystyle<0\ \ \
\ \text{at least one }\ \ i,$ (28) $\displaystyle\langle\nabla
g_{j}(x^{0}),v\rangle+\langle\zeta^{*j},u\rangle\epsilon$ $\displaystyle\leq
0,\ \ \ \ j\in J(x^{0};u),$ (29) $\displaystyle\epsilon$ $\displaystyle>0.$
(30)

Thus, by the Slater theorem [29, p. 27], either the system

$\displaystyle\sum_{i=1}^{l}\lambda_{i}\nabla f_{i}(x^{0})+\sum_{j\in
J(x^{0};u)}\mu_{j}\nabla g_{j}(x^{0})=0,$ (31)
$\displaystyle\sum_{i=1}^{l}\lambda_{i}\xi^{*i}+\sum_{j\in
J}\mu_{j}\zeta^{*j}-\nu=0,$ (32)
$\displaystyle\lambda_{i}>0,i=1,2,...,l,\mu_{j}\geq 0,j\in J(x^{0};u),\nu\geq
0,$ (33)

has solution $\lambda_{i},\mu_{j},\nu\in\mathbb{R}$, or the system (31), (32)
and

$\lambda_{i}\geq 0,i=1,2,...,l,\mu_{j}\geq 0,j\in J(x^{0};u),\nu\geq 0,,$ (34)

has solution $\lambda_{i},\mu_{j},\nu\in\mathbb{R}$. We claim that the system
(31)–(33) has solution $\lambda_{i},\mu_{j},\nu\in\mathbb{R}$. On the
contrary, suppose that the system (31)–(33) has no solution. By the Tucker
theorem [29, p. 29], the system (27)–(29) and $\epsilon\geq 0$ has a solution
$\bar{v}\in\mathbb{R}^{n}$ and $\bar{\epsilon}\geq 0$. It is easily seen that
$\bar{\epsilon}=0$. For each $t>0$, put $u(t)=\bar{v}+tu$. Since $(\bar{v},0)$
is a solution of the system (27)–(29), for $t>0$ sufficiently large, we have

$\displaystyle\langle\nabla f_{i}(x^{0}),u(t)\rangle$
$\displaystyle=\langle\nabla f_{i}(x^{0}),\bar{v}\rangle+t\langle\nabla
f_{i}(x^{0}),u\rangle\leq 0,i=1,2,...,l,$ $\displaystyle\langle\nabla
f_{i}(x^{0}),u(t)\rangle$ $\displaystyle=\langle\nabla
f_{i}(x^{0}),\bar{v}\rangle+t\langle\nabla f_{i}(x^{0}),u\rangle<0,\ \
\text{at least one}\ \ i,$ $\displaystyle\langle\nabla
g_{j}(x^{0}),u(t)\rangle$ $\displaystyle=\langle\nabla
g_{j}(x^{0}),\bar{v}\rangle+t\langle\nabla g_{j}(x^{0}),u\rangle\leq 0,j\in
J(x^{0};u),$ $\displaystyle\langle\nabla g_{j}(x^{0}),u(t)\rangle$
$\displaystyle=\langle\nabla g_{j}(x^{0}),\bar{v}\rangle+t\langle\nabla
g_{j}(x^{0}),u\rangle<0,j\in J(x^{0})\setminus J(x^{0};u),$

by Theorem 4.1, which contradicts the fact that $x^{0}$ is a Geoffrion
properly efficient solution of $(P)$. ∎

Let us illustrate Theorem 4.3.

######  Example 4.1.

Let $f\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ be defined by
$f(x)=(f_{1}(x),f_{2}(x))$, where

$f_{1}(x)=-f_{2}(x)=\begin{cases}2x^{2}_{1}+x^{2}_{2}+x_{2}+x^{2}_{1}\sin(\ln|x_{1}|),&\text{
if }x=(x_{1},x_{2})\in\mathbb{R}^{2},x_{1}\neq 0,\\\ x^{2}_{2}+x_{2},&\text{
if }x_{1}=0,\end{cases}$

and let $g_{1}\colon\mathbb{R}^{2}\to\mathbb{R}$ be given by $g_{1}(x)=-x_{2}$
for $x=(x_{1},x_{2})\in\mathbb{R}^{2}$. Let us consider problem (VP) with the
objective function $f$ and the constraint set

$Q_{0}=\\{x\in\mathbb{R}^{2}\;:\;g_{1}(x)\leq 0\\}.$

It is easy to check that $x^{0}:=(0,0)$ is a Geoffrion properly efficient
solution to problem (VP). An easy computation shows that

$\nabla
f_{1}(x)=\begin{cases}\left(4x_{1}+2x_{1}\sin(\ln|x_{1}|)+x_{1}\cos(\ln|x_{1}|),2x_{2}+1\right),&\mbox{
if }x_{1}\neq 0\\\ (0,2x_{2}+1),&\mbox{ if }x_{1}=0,\end{cases}$

$\nabla f_{2}(x)=-\nabla f_{1}(x)$, $\nabla g_{1}(x)=(0,-1)$, and
$\nabla^{2}g_{1}(x)=(0,0)$ for $x\in\mathbb{R}^{2}$. Thus

$\nabla f_{1}(x^{0})=-\nabla f_{2}(x^{0})=-\nabla g_{1}(x^{0})=(0,1),$

and $K(x^{0})=\\{(u_{1},u_{2})\in\mathbb{R}^{2}\;:\;u_{2}=0\\}$. Let $u=(1,0)$
be a critical direction at $x^{0}$. Then, we have

$\partial_{S}^{2}f_{1}(x^{0})(u)=\partial_{S}\langle u,\nabla
f_{1}(\cdot)\rangle(x^{0})=\partial_{S}(\nabla_{x_{1}}f_{1}(\cdot))(x^{0})=\partial_{S}(\nabla_{x_{1}}f_{1}(\cdot))(x^{0}).$

Since $\nabla_{x_{1}}f_{1}(\cdot)$ does not depend on $x_{2}$, we have
$\partial_{S}(\nabla_{x_{1}}f_{1}(\cdot))(x^{0})=(\partial_{S}\varphi(0),0)$,
where $\varphi(x_{1}):=\nabla_{x_{1}}f_{1}(x)$ for all
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$. Thus
$\partial_{S}^{2}f_{1}(x^{0})(u)=(\partial_{S}\varphi(0),0)$. Thanks to [30,
Corollary 2.3], one has

$\partial_{S}\varphi(0)=\partial_{C}\varphi(0)=\left[\liminf_{x_{1}\to
0,x_{1}\neq 0}\nabla\varphi(x_{1}),\limsup_{x_{1}\to 0,x_{1}\neq
0}\nabla\varphi(x_{1})\right].$

From
$\varphi(x_{1})=\nabla_{x_{1}}f_{1}(x)=4x_{1}+2x_{1}\sin(\ln|x_{1}|)+x_{1}\cos(\ln|x_{1}|)$
it follows that $\nabla\varphi(x_{1})=4+\sin(\ln|x_{1}|)+3\cos(\ln|x_{1}|)$
for all $x_{1}\neq 0$. It is easy to check that

$\liminf_{x_{1}\to 0,x_{1}\neq 0}\nabla\varphi(x_{1})=4-\sqrt{10}\ \
\mbox{and}\ \ \limsup_{x_{1}\to 0,x_{1}\neq
0}\nabla\varphi(x_{1})=4+\sqrt{10}.$

Consequently,

$\partial_{S}^{2}f_{1}(x^{0})(u)=[4-\sqrt{10},4+\sqrt{10}]\times\\{0\\},$

and hence
$\partial_{S}^{2}f_{2}(x^{0})(u)=[-4-\sqrt{10},-4+\sqrt{10}]\times\\{0\\}$. It
follows that

$\langle\xi^{*1},u\rangle=\max\left\\{\langle\xi^{1},u\rangle\,:\,\xi^{1}\in\partial_{S}^{2}f_{1}(x^{0})(u)\right\\}=4+\sqrt{10},$

and

$\langle\xi^{*2},u\rangle=\max\left\\{\langle\xi^{2},u\rangle\,:\,\xi^{2}\in\partial_{S}^{2}f_{2}(x^{0})(u)\right\\}=-4+\sqrt{10}.$

Thus, for each $v=(v_{1},v_{2})\in\mathbb{R}^{2}$, we have

$F^{2}_{1}(x^{0};u,v)=(0,v_{2}+4+\sqrt{10}),F^{2}_{2}(x^{0};u,v)=(0,-v_{2}-4+\sqrt{10}),$

and $G_{1}^{2}(x^{0};u,v)=(0,-v_{2})$. It is easily seen that
$L^{2}(Q;x^{0},u)=\emptyset$. Thus, the ((ii)) holds at $x^{0}$. By Theorem
4.3, the $(SSKKT)$ holds at $x^{0}$ for the direction $u$. Moreover, it is
easy to check that $(\lambda_{1},\lambda_{2})=(1,1)$ and $\mu=0$ satisfy the
conditions (13)–(16). Since the functions $f_{1}$ and $f_{2}$ are not twice
differentiable at $x^{0}$, Theorem 3.2 in [23] cannot be employed.

######  Remark 4.1.

  1. (i)

It is worth noting that Theorem 4.3 embraces also the $(SFKKT)$ optimality
conditions given by Burachik and Rizvi [9, Theorem 4.4], which can be obtained
by just considering the particular case $u=0$.

  2. (ii)

Due to Remark 3.2, the conclusions of Theorem 4.2 and Theorem 4.3 still hold
when the assumption “the ((ii)) holds at $x^{0}$” is replaced by “the ((i))
holds at $x^{0}$”.

  3. (iii)

Theorem 4.2 and Theorem 4.3 cannot be extended to the efficient solution of
(VP). To see this, let us consider the following example.

######  Example 4.2.

Consider problem (VP) with the objective function $f$ from Example 3.1 and the
constraint set $Q_{0}:=\mathbb{R}^{2}$. It is easy to check that
$x^{0}:=(0,0)$ is an efficient solution to problem $\rm(VP)$ and
$Q=\\{x^{0}\\}$. Since

$\nabla f_{1}(x^{0})=(0,1),\nabla f_{2}(x^{0})=(1,0),\nabla
f_{3}(x^{0})=(-1,0),$

we have

$L^{2}(Q;x^{0},0_{\mathbb{R}^{2}})=L(Q;x^{0})=\\{(u_{1},u_{2})\,:\,u_{1}=0,u_{2}\leq
0\\}.$

Thanks to Example 3.1, we have $K(x^{0})=\\{0_{\mathbb{R}^{2}}\\}$ and

$\displaystyle
L^{2}(Q;x^{0},0_{\mathbb{R}^{2}})=L(Q^{0};x^{0})=\bigcap_{i=1}^{3}T(M^{i};x^{0})=\bigcap_{i=1}^{3}T^{2}(M^{i};x^{0},0_{\mathbb{R}^{2}}),$
$\displaystyle
L^{2}(Q;x^{0},0_{\mathbb{R}^{2}})=L(Q^{0};x^{0})\nsubseteq\bigcap_{i=1}^{3}T(Q^{i};x^{0})=\bigcap_{i=1}^{3}T^{2}(Q^{i};x^{0},0_{\mathbb{R}^{2}}),$

where $0_{\mathbb{R}^{2}}:=(0,0)$. Thus the ((ii)) holds at $x^{0}$ but not
the ((i)). Since

$\lambda_{1}\nabla f_{1}(x^{0})+\lambda_{2}\nabla
f_{2}(x^{0})+\lambda_{3}\nabla
f_{3}(x^{0})=0\Leftrightarrow\begin{cases}\lambda_{1}=0\\\
\lambda_{2}=\lambda_{3},\end{cases}$

it follows that the $(SSKKT)$ does not hold at $x^{0}$ for the direction
$0_{\mathbb{R}^{2}}$. Thus Theorem 4.3 does not hold for $x^{0}$. Furthermore,
for each $v=(v_{1},v_{2})\in\mathbb{R}^{2}$, we have

$F_{1}^{2}(x^{0};0_{\mathbb{R}^{2}},v)=(0,v_{2}),F_{2}^{2}(x^{0};0_{\mathbb{R}^{2}},v)=(0,v_{1}),F_{3}^{2}(x^{0};0_{\mathbb{R}^{2}},v)=(0,-v_{1}).$

Clearly, the system

$\begin{cases}F_{1}^{2}(x^{0};0_{\mathbb{R}^{2}},v)=(0,v_{2})&<_{\rm
lex}(0,0)\\\ F_{2}^{2}(x^{0};0_{\mathbb{R}^{2}},v)=(0,v_{1})&\leqq_{\rm
lex}(0,0)\\\ F_{3}^{2}(x^{0};0_{\mathbb{R}^{2}},v)=(0,-v_{1})&\leqq_{\rm
lex}(0,0),\end{cases}$

admits a solution $v=(0,v_{2})$, where $v_{2}<0$. This means that Theorem 4.2
does not hold for efficient solutions, too.

We also note here that $x^{0}$ is not a Geoffrion properly efficient solution.
Indeed, let $x=(0,-a)$, $a>0$, then we have
$f_{1}(x)<f_{1}(x^{0}),f_{2}(x)>f_{2}(x^{0})$ and

$\lim_{a\downarrow
0}\dfrac{f_{1}(x)-f_{1}(x^{0})}{f_{2}(x^{0})-f_{2}(x)}=\lim_{a\downarrow
0}\dfrac{1}{a}=+\infty.$

Therefore $x^{0}$ is not a Geoffrion properly efficient solution.

##  5\. Conclusions

By using the second-order symmetric subdifferential, we introduce some types
of second-order regularity conditions in the sense of Abadie and formulate
strong second-order Karush–Kuhn–Tucker necessary optimality conditions for
Geoffrion properly efficient solutions of $C^{1,1}$ vector optimization
problems. It is meaningful that how to establish second-order
Karush–Kuhn–Tucker-type optimality conditions for efficient solutions of
vector optimization problems purely in the Mordukhovich subdifferential terms.
We aim to investigate this problem in future work.

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