Abstract
The main purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem. Following along with the image space analysis, a Lagrange-type duality for a constrained nonconvex vector optimization problem is proposed by virtue of the class of vector-valued regular weak separation functions in the image space. Simultaneously, some equivalent characterizations to the zero duality gap property are established including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Moreover, an exact penalization is also obtained by means of a local image regularity condition and a class of particular regular weak separation functions in the image space.
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1 Introduction
The present paper is concerned with a Lagrange-type duality and penalty method for a constrained nonconvex vector optimization problem by virtue of the image space analysis (for short, ISA). Just as we know, the traditional Lagrange duality is an important method for constrained scalar and vector convex optimization problems. Among several crucial aspects, the zero duality gap property between the primal problem and the corresponding Lagrange dual problem plays a key role, such as optimality conditions, stability and sensitivity analysis and solution methods [1–3]. However, the zero duality gap property may not hold for nonconvex optimization problems. Simultaneously, the classic Lagrange penalty method may also lose effectiveness. In order to avoid the nonzero duality gap, various kinds of generalized Lagrange-type functions, especially the augmented Lagrangian function, introduced by Hestenes [4, 5] and Powell [6], and the nonlinear Lagrangian function, introduced by Rubinov et al. [7], have been proposed and applied to study scalar and vector nonconvex optimization problems. We refer to [7–10] and references therein for more details.
In the last decades, the ISA has attracted great interest in the academic and professional communities and has been used as a preliminary and auxiliary step for investigating some subjects both in the optimization theory and methodology for constrained optimization problems, such as optimality conditions [11–17], dualities [18–22], variational principles [23], existence of solutions [24], penalty methods [25, 26], regularities and stabilities [27–29]. Since the path-breaking paper [13], it has shown that the ISA is a unified scheme to study any kind of problem, which can be expressed by the form of the impossibility of a parametric system. Simultaneously, the image space (for short, IS) of the constrained optimization problem provides a natural environment for the Lagrange duality and penalty method. By virtue of the ISA, Giannessi [18] showed that the classic Lagrange duality for constrained scalar optimization problems can be derived from regular separations in the IS. Subsequently, this idea was further extended in [19] to investigate relationships between Wolfe and Mond–Weir dualities, especially their equivalence in the IS under some generalized convexity assumptions. Mastroeni [20] studied some duality properties of a constrained scalar optimization problem by means of the ISA and established some equivalent characterizations for the zero duality gap property in the IS. Moreover, Mastroeni [26] analyzed a nonlinear separation scheme in the IS associated with an infinite-dimensional cone constrained scalar optimization problem and established some relationships between the penalty methods and corresponding nonlinear separations. By exploiting the ISA, Zhu and Li [21, 22] proposed a unified duality scheme for a constrained scalar optimization problem in terms of the class of regular weak separation functions and obtained some necessary and sufficient conditions for the zero duality gap property, including the Lagrange saddle point, the Lagrange multiplier, the lower semicontinuity of the perturbation function and a regular separation in the IS, without any convexity assumption.
However, the study on the duality and penalty method for constrained nonconvex vector optimization problems by virtue of the ISA is very limited. In [30, 31], several theoretical and methodological aspects of constrained vector optimization problems and vector variational inequalities, such as optimality conditions, dualities and scalarization, were investigated by virtue of the ISA. For more details, we refer to [32–34] and reference therein.
Motivated by the work reported in [9, 12, 21, 22, 30, 31], the purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem by virtue of the ISA. To the end, we establish a Lagrange-type dual problem via the class of vector-valued regular weak separation functions in the IS. Under some appropriate assumptions, we obtain some equivalent characterizations to the zero duality gap property, including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Simultaneously, we also establish an exact penalization result by means of a local image regularity condition and a class of particular regular weak separation functions in the IS.
The organization of this paper is as follows: In Sect. 2, we recall some preliminaries, particularly the concept of separation functions in the ISA. In Sect. 3, we propose a Lagrange-type dual problem via the class of vector-valued regular weak separation functions in the IS and study the zero duality gap property. In Sect. 4, we consider the applications to exact penalization methods. In Sect. 5, some conclusions are given.
2 Preliminaries and Separation Functions in the ISA
Throughout this paper, all vectors are viewed as column vectors. Let \({\mathbb {R}}^\ell \) be an \(\ell \)-dimensional Euclidean space, \(C={\mathbb {R}}^\ell _+\), and \({\mathrm {int}}\,C\) be the interior of C. When there is no fear of confusion, we always denote by 0 the origin of different spaces. As usual, we denote by \(x^T\) the transpose of x for every \(x\in {\mathbb {R}}^\ell \). In this paper, we use the following orderings: for all \(x, y\in {\mathbb {R}}^\ell \),
In addition, \(y\ge _C x\) (\(y\ngeq _C x\)) means that \(x\le _C y\) (\(x\nleq _C y\)). For other above-mentioned orderings, analogous relations can be defined. Let \(\bar{{\mathbb {R}}}^\ell :={\mathbb {R}}^\ell \cup \{\pm \infty \}\), where \(+\infty \) (\(-\infty \)) is an imaginary point with each coordinate \(+\infty \) (\(-\infty \)). This shows \(-\infty \le _{C\setminus \{0\}} z \le _{C\setminus \{0\}} +\infty \) and \(-\infty \le _{{\mathrm {int}}\,C} z\le _{{\mathrm {int}}\,C} +\infty \) for every \(z\in {\mathbb {R}}^\ell \). Moreover, \(-\infty \le _C z\le _C +\infty \) and \(+\infty \nleq _{C\setminus \{0\}} z\nleq _{C\setminus \{0\}}-\infty \) hold for every \(z\in \bar{{\mathbb {R}}}^\ell \). Without possibility of confusion, we shall not differentiate the \(-\infty \) and \(+\infty \) in \(\bar{{\mathbb {R}}}^\ell \), and the \(-\infty \) and \(+\infty \) in the extended real number space \(\bar{{\mathbb {R}}}\). We extend the addition and the scalar multiplication from \({\mathbb {R}}^\ell \) to \(\bar{{\mathbb {R}}}^\ell \) by using the following conventions:
Definition 2.1
[9, 35] Let \(A\subset {\mathbb {R}}^\ell \) be a nonempty subset. Denote the set of all infimum points of A by \({\mathrm {Inf}}\,A\). For any \(z^*\in {\mathrm {Inf}}\,A\), we mean that
-
(i)
\(z^*\in \bar{{\mathbb {R}}}^\ell \);
-
(ii)
\(z\nleq _{C\setminus \{0\}} z^*, \forall z\in A\);
-
(iii)
there exists a sequence \(\{z^k\}\subset A\) such that \(z^k\rightarrow z^*\) as \(k\rightarrow +\infty \).
The point \(z^*\in {\mathrm {Inf}}\,A\) is said to be an infimum point of A. Similarly, we can define the set of all supremum points of A by \({\mathrm {Sup}}\,A\), and \(z^*\in {\mathrm {Sup}}\,A\) if and only if \(-z^*\in {\mathrm {Inf}}(-A)\).
Remark 2.1
Clearly, \({\mathrm {Sup}}\,A=-{\mathrm {Inf}}\,(-A)\) and \({\mathrm {Sup}}\,(z+A)=z+{\mathrm {Sup}}\,A\) hold for all nonempty subset \(A\subset {\mathbb {R}}^\ell \) and all \(z\in {\mathbb {R}}^\ell \).
Let \(F:X\rightrightarrows {\mathbb {R}}^\ell \) be a set-valued map. We denote the set of all infimum (supremum) points for F on X by \(\inf _{x\in X}F(x)\) (\(\sup _{x\in X}F(x)\)), that is,
Therefore, for every \(z^*\in \inf _{x\in X}F(x)\), it follows from Definition 2.1 that \(z^*\in \bar{{\mathbb {R}}}^\ell \), \((z^*-F(x))\cap (C\setminus \{0\})=\varnothing , \forall x\in X\) and \(\exists \,\{x^k\}\subset X, \exists \,z^k\in F(x^k)\) such that \(z^k\rightarrow z^*\) as \(k\rightarrow +\infty \). Analogously, we can give the similar explanation to \(\sup _{x\in X}F(x)\). Specially, when \(F=f:X\rightarrow {\mathbb {R}}^\ell \) is a vector-valued function, the set of all infimum (supremum) points for f on X is denoted by \(\inf _{x\in X}f(x)\) (\(\sup _{x\in X}f(x)\)).
Next, we recall the following useful concept called nonlinear scalarization function and some of its properties.
Lemma 2.1
[10, 36] Given \(e\in {\mathrm {int}}\,C\), the nonlinear scalarization function \(\xi _e:{\mathbb {R}}^\ell \rightarrow {\mathbb {R}}\), defined by
is convex, strictly \({\mathrm {int}}\,C\)-monotone, C-monotone, nonnegative homogeneous and globally Lipschitz. Simultaneously, for every \(\alpha \in {\mathbb {R}}\), it follows that
Consider the following constrained vector optimization problem
(VOP) \(\min f(x)\), \({\mathrm {s.t.}}\) \(x\in X\), \(g_j(x)\ge 0,\;j=1,2,...,m\),
where \(X\subset {\mathbb {R}}^n\) is a nonempty subset, \(f=(f_1,f_2,...,f_\ell ):X\rightarrow {\mathbb {R}}^\ell \) and \(g=(g_1,g_2,...,g_m):X\rightarrow {\mathbb {R}}^m\) are vector-valued functions with each component function \(f_i:X\rightarrow {\mathbb {R}}, i\in \{1,2,...,\ell \}\), and \(g_j:X\rightarrow {\mathbb {R}}, j\in \{1,2,...,m\}\), respectively. As usual, we denote by \({\mathcal {R}}=\{x\in X: g_j(x)\ge 0, j=1,2,...,m\}\) the feasible set of (VOP). Clearly, one has \({\mathcal {R}}=\{x\in X: g(x)\ge _D 0\}\) where \(D={\mathbb {R}}_+^m\). Throughout this paper, we assume that \({\mathcal {R}}\ne \varnothing \).
Recall that a point \(\hat{x}\in {\mathcal {R}}\) is said to be a strong (weak) vector minimum point (in short, vmp) to (VOP) iff \(f(x)\nleq _{C\setminus \{0\}} f(\hat{x})\) (\(f(x)\nleq _{{\mathrm {int}}\,C} f(\hat{x})\)) for all \(x\in {\mathcal {R}}\). The point \(\hat{x}\in {\mathcal {R}}\) is said to be a local strong (local weak) vmp to (VOP) iff there exists \(\delta >0\) such that \(f(x)\nleq _{C\setminus \{0\}} f(\hat{x})\) (\(f(x)\nleq _{{\mathrm {int}}\,C} f(\hat{x})\)) for all \(x\in {\mathcal {R}}\cap {\mathcal {B}}(\hat{x},\delta )\), where \({\mathcal {B}}(\hat{x},\delta )\) denotes the open ball with center at \(\hat{x}\) and radius \(\delta \). Clearly, every strong vmp (local strong vmp) must be a weak vmp (local weak vmp). Moreover, let \(\hat{x}\in {\mathcal {R}}\). Then, \(\hat{x}\) is a strong vmp to (VOP) if and only if \(f(\hat{x})\in \inf _{x\in {\mathcal {R}}}f(x)\).
Take arbitrary \(\bar{x}\in X\). We consider the vector-valued map \(A:X\rightarrow {\mathbb {R}}^\ell \times {\mathbb {R}}^m\) with
and the sets
Note that \({\mathcal {K}}\) is the image of the map A, that is, \({\mathcal {K}}=A(X)\). Moreover, the sets \({\mathcal {K}}\) and \({\mathcal {H}}\) are subsets of \({\mathbb {R}}^\ell \times {\mathbb {R}}^m\). We recall from [12, 21, 30, 31] that \({\mathbb {R}}^\ell \times {\mathbb {R}}^m\) is said to be the image space associated with (VOP) and \({\mathcal {K}}\) is said to be the image of (VOP). Note that the image \({\mathcal {K}}\) does not generally enjoy some convexity property. To overcome this difficulty, a regularization of the image \({\mathcal {K}}\), called conic extension with respect to the cone \({\mathrm {cl}}\,{\mathcal {H}}=C\times D\), denoted by
was introduced in [12, 30, 31]. As a result, the convexity of \({\mathcal {E}}\) can be verified under some appropriate conditions; for example, A is \(-({\mathrm {cl}}\,{\mathcal {H}})\)-convexlike. We refer to [12, 19, 30, 31] and references therein for more details. Take specially \(\bar{x}\in {\mathcal {R}}\). Then, it is easy to verify that \(\bar{x}\) is a strong vmp to (VOP) if and only if \({\mathcal {K}}\cap {\mathcal {H}}=\varnothing \). Since \({\mathcal {H}}+{\mathrm {cl}}\,{\mathcal {H}}={\mathcal {H}}\), it follows that \({\mathcal {K}}\cap {\mathcal {H}}=\varnothing \) if and only if \({\mathcal {E}}\cap {\mathcal {H}}=\varnothing \), or equivalently, \({\mathcal {E}}\cap {\mathcal {H}}_u=\varnothing \), where
Remark 2.2
It is worth noting that the choice of \(\bar{x}\) is arbitrary throughout this paper, unless otherwise specified. Based on this fact, we use the notations A and \({\mathcal {K}}\), although they seem to be dependent on the point \(\bar{x}\). Just as shown in [12], even if it is a mere formal question, it is not convenient, and it is better to define u as above in \({\mathcal {K}}\); see more details in [12, 19–21, 27, 30, 31].
Consider a vector-valued function \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \), where \(\varPi \) is a set of parameters to be specified case by case. Throughout this paper, we shall always use the same symbol \(\varPi \) to denote the set of parameters unless otherwise specified. For every \(\pi \in \varPi \), the nonnegative level set and the positive level set of the function \(w(\bullet ;\pi ):{\mathbb {R}}^\ell \times {\mathbb {R}}^m\rightarrow {\mathbb {R}}^\ell \) associated with the cone C are, respectively, defined by
and
Next, we recall the following concepts of vector-valued weak separation and regular weak separation functions, respectively.
Definition 2.2
[30, 31] The class of all the vector-valued functions \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \) such that
and
is called the class of weak separation functions and is denoted by \({\mathcal {W}}(\varPi )\).
Definition 2.3
[30, 31] The class of all the vector-valued functions \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \) such that
is called the class of regular weak separation functions and is denoted by \({\mathcal {W}}_R(\varPi )\).
In this paper, we shall pay our attention to establish a Lagrange-type dual problem for (VOP) with applications to exact penalty properties by using the class of regular weak separation functions. To this end, we always assume that the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) enjoys the following two properties:
Assumption
\({\varvec{{\mathcal {A}}}}\) \(w\in {\mathcal {W}}_R(\varPi )\) satisfies \(w(u,0;\pi )=u\), \(\forall \pi \in \varPi \), and
Assumption
\({\varvec{{\mathcal {B}}}}\) \(w\in {\mathcal {W}}_R(\varPi )\) is monotone increasing with respect to the first argument, that is, \(w(u^2,v^2;\pi )\le _C w(u^1,v^1;\pi )\) holds for all \(\pi \in \varPi \) and all \((u^i,v^i)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m, i=1,2\) with \((u^2,v^2)\le _{C\times D}(u^1,v^1)\).
Remark 2.3
Note that Assumption \({\mathcal {A}}\) requires the set of all infimum points for the vector-valued function \(w(u,v;\bullet )\) on \(\varPi \) to be a single point set. Throughout this paper, we denote the single point set \(\{u\}\) by u for simplicity unless otherwise stated.
Next, we consider some important examples for the classes of vector-valued regular weak separation functions which satisfy Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\).
Example 2.1
Recall from [30] that the vector polar of D with respect to C is given by
where \({\mathbb {R}}^{\ell \times m}\) denotes the set of all matrices with real entries and with \(\ell \) rows and m columns. It is easy to verify that
where \(\alpha _j\) is the j-th column of M. Moreover, let \(J:=C\setminus \{0\}\). Then, we have
and
where \(\beta _i\) is the i-th column of M. On the one hand, let \(\varPi =(J^*_C\times D^*_C)\setminus \{0\}\) and the vector-valued function \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \) be defined by
Then, \(w\in {\mathcal {W}}(\varPi )\). On the other hand, let \(\varPi =J^*_{C\setminus \{0\}}\times D^*_C\) and the vector-valued function \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times J^*_{C\setminus \{0\}}\times D^*_C\rightarrow {\mathbb {R}}^\ell \) be defined by
Then, we have \(w\in {\mathcal {W}}_R(\varPi )\). We refer to [30] for more details. Obviously, the function w satisfies Assumption \({\mathcal {B}}\). However, Assumption \({\mathcal {A}}\) does not hold. In fact, by direct calculating, we have \(w(u,0;\varTheta ,\varLambda )=\varTheta u\) and
Specially, let \(\varPi =E_\ell \times D^*_C\), where \(E_\ell \) denotes the identity matrix of order \(\ell \), and let the vector-valued function \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times E_\ell \times D^*_C\rightarrow {\mathbb {R}}^\ell \) be defined by
Then, \(w\in {\mathcal {W}}_R(\varPi )\). Simultaneously, we also have \(w(u,0; E_\ell ,\varLambda )=u\) and
In fact, for every \(v\in D\), we have \(\varLambda v\ge _C 0\) and \(w(u,v; E_\ell ,\varLambda )=u+\varLambda v\ge _C u\) for all \(\varLambda \in D^*_C\). Together with \(0\in D^*_C\) and \(w(u,v; E_\ell ,0)=u\), it follows that \(u\in \inf _{(E_\ell ,\,\varLambda )\in \varPi }\,w(u,v;E_\ell ,\varLambda )\). Moreover, if \(z\in \inf _{(E_\ell ,\,\varLambda )\in \varPi }\,w(u,v;E_\ell ,\varLambda )\), then we have by Definition 2.1 that \(w(u,v;E_\ell ,\varLambda )\nleq _{C\setminus \{0\}}z\) for all \(\varLambda \in D^*_C\) and that there exists a sequence \(w(u,v;E_\ell ,\varLambda ^k)\rightarrow z\) with \(\varLambda ^k\in D^*_C\). Thus, we get \(u=w(u,v; E_\ell ,0)\nleq _{C\setminus \{0\}}z\). Moreover, since C is closed, and \(\varLambda ^k v\ge _C 0\) and \(w(u,v;E_\ell ,\varLambda ^k)\ge _C u\) for all \(\varLambda ^k\in D^*_C\), it follows from \(w(u,v;E_\ell ,\varLambda ^k)\rightarrow z\) that \(z\ge _C u\). Together with \(u\nleq _{C\setminus \{0\}}z\), we have \(z=u\). Thus, we can conclude that
In addition, take arbitrary \(v\notin D\). Then, there exists \(j_0\in \{1,2,...,m\}\) such that \(v_{j_0}<0\). Let \(\varLambda ^k=(0,...,\alpha _{j_0}^k,...,0)\) with the \(j_0\)-th column \(\alpha _{j_0}^k=(k,k,...,k)^T\) and other columns 0. Then, we have \(\varLambda ^k\in D_C^*\) for all \(k\in {\mathbb {N}}\). Moreover, it follows that \(w(u,v;E_\ell ,\varLambda ^k)=u+\varLambda ^k v=u+v_{j_0}\alpha _{j_0}^k\rightarrow -\infty \) as \(k\rightarrow +\infty \), which implies
Therefore, we can conclude that Assumption \({\mathcal {A}}\) holds. Simultaneously, for every \((u^i,v^i)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m, i=1,2\) with \((u^2,v^2)\le _{C\times D}(u^1,v^1)\). Then, we have \(u^2\le _C u^1\) and \(v^2\le _D v^1\), which implies \(u^1-u^2\in C\), \(v^1-v^2\in D\) and \(\varLambda (v^1-v^2)\ge _C 0\) for all \(\varLambda \in D^*_C\). Thus, we get
that is, \(w(u^2,v^2;E_\ell ,\varLambda )\le _C w(u^1,v^1;E_\ell ,\varLambda )\) for all \(\varLambda \in D^*_C\). Therefore, Assumption \({\mathcal {B}}\) also holds.
Example 2.2
Let \(\varPi =D\) and \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \). Given a point \(e\in {\mathrm {int}}\,C\), let
where \(r>0\) is a real constant and the augmented function \(\sigma :{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) satisfies
Then, we have \(w\in {\mathcal {W}}_R(\varPi )\), and Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\) hold. Next, we first show that the vector-valued function w is a regular weak separation function. On the one hand, for every \((u,v)\in {\mathcal {H}}\) and \(\pi \in D\), we get \(u\in C\setminus \{0\}\), \(v\in D\) and
Together with \(e\in {\mathrm {int}}\,C\), we get
which implies
On the other hand, take arbitrary
Then, we have \(w(u,v;\pi )\ge _{C\setminus \{0\}}0\) for all \(\pi \in D\). Specially, let \(\pi =0\in D\). Then, we get
Together with \(r>0\), \(\min _{z\in {\mathbb {R}}^m}\sigma (z)=0\) and \(e\in {\mathrm {int}}\,C\), we have
which implies \(u\ge _{C\setminus \{0\}}0\). Moreover, we have \(v\in D\). Otherwise, there exists \(j_0\in \{1,2,...,m\}\) such that \(v_{j_0}<0\). Since \(e\in {\mathrm {int}}\,C\), there exists \(\rho >0\) such that \(u+\rho v_{j_0}e\le _{C\setminus \{0\}}0\). Specially, let \(\tilde{\pi }\in {\mathbb {R}}^m\) with \(\tilde{\pi }_{j_0}=\rho \) and \(\tilde{\pi }_j=0\) for every \(j\in \{1,2,...,m\}\setminus \{j_0\}\). Then, it follows that \(\tilde{\pi }\in D\), which implies \(w(u,v;\tilde{\pi })\ge _{C\setminus \{0\}}0\) by the choice of (u, v). Moreover, we get from \(r>0\), \(\min _{z\in {\mathbb {R}}^m}\sigma (z)=0\) and \(u+\rho v_{j_0}e\le _{C\setminus \{0\}}0\) that
and
which shows \(w(u,v;\tilde{\pi })\le _{C\setminus \{0\}}0\). This contradicts \(w(u,v;\tilde{\pi })\ge _{C\setminus \{0\}}0\). Then, we have
Thus, we can conclude that
that is, \(w\in {\mathcal {W}}_R(\varPi )\) is a regular weak separation function. Second, we prove that Assumption \({\mathcal {A}}\) holds. Note that
Since \(\sigma (0)=0\), we have \(\sup _{z\le _D\, 0}\Big (\langle \pi ,z\rangle -r\sigma (z)\Big )\ge \langle \pi ,0\rangle -r\sigma (0)=0\) for all \(\pi \in D\). Moreover, for every \(\pi \in D\) and \(z\le _D 0\), we have \(\langle \pi ,z\rangle \le 0\). Together with \(\sigma (z)\ge 0\) and \(r>0\), we get \(\langle \pi ,z\rangle -r\sigma (z)\le 0\) for all \(\pi \in D\) and all \(z\le _D 0\), which implies \(\sup _{z\le _D\, 0}\Big (\langle \pi ,z\rangle -r\sigma (z)\Big )\le 0\) for all \(\pi \in D\). Thus, we immediately have \(\sup _{z\le _D\, 0}\Big (\langle \pi ,z\rangle -r\sigma (z)\Big )=0\) and \(w(u,0;\pi )=u\) for all \(\pi \in D\). Take arbitrary \((u,v)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m\). If \(v\in D\), that is, \(0\le _D v\), then we get \(\sup _{z\le _D\, v}\Big (\langle \pi ,z\rangle -r\sigma (z)\Big )\ge \langle \pi ,0\rangle -r\sigma (0)=0\) for all \(\pi \in D\). Together with \(e\in {\mathrm {int}}\,C\), we have
Specially, it follows that \(w(u,v;0)\ge _C u\). Furthermore, since \(\sigma (z)\ge 0\), \(r>0\) and \(e\in {\mathrm {int}}\,C\), we get
and
Together with \(w(u,v;0)\ge _C u\), we have \(w(u,v;0)=u\). Simultaneously, it follows from \(w(u,v;\pi )\ge _C u\) for all \(\pi \in D\) that \(u\in \inf _{\pi \in D}w(u,v;\pi )\). Moreover, we can conclude by the similar method to Example 2.1 that
Simultaneously, if \(v\notin D\), then there exists \(j_0\in \{1,2,...,m\}\) such that \(v_{j_0}<0\). Let the sequence \(\{\pi ^k\}\subset D\) with \(\pi ^k_{j_0}=k\) and \(\pi ^k_j=0\) for every \(j\in \{1,2,...,m\}\) with \(j\ne j_0\). Then, it follows from \(\sigma (z)\ge 0\) and \(r>0\) that
Together with \(e\in {\mathrm {int}}\,C\), we have
which implies
Thus, \(w\in {\mathcal {W}}_R(\varPi )\) satisfies Assumption \({\mathcal {A}}\). Lastly, we show that Assumption \({\mathcal {B}}\) holds. For all \((u^i,v^i)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m, i=1,2\) with \((u^2,v^2)\le _{C\times D}(u^1,v^1)\). Then, we have \(u^2\le _C u^1\) and \(v^2\le _D v^1\). Thus, it follows that \(\{z\in {\mathbb {R}}^m: z\le _D v^2\}\) is a subset of \(\{z\in {\mathbb {R}}^m: z\le _D v^1\}\) and
Together with \(u^2\le _C u^1\) and \(e\in {\mathrm {int}}\,C\), we have
Therefore, \(w\in {\mathcal {W}}_R(\varPi )\) satisfies Assumption \({\mathcal {B}}\).
Example 2.3
Let \(\varPi =D\) and \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \). Given a point \(e\in {\mathrm {int}}\,C\), then the following functions \(w\in {\mathcal {W}}_R(\varPi )\) satisfy Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\) (see more details in Sect. 4):
3 Lagrange-Type Duality and Zero Duality Gap Property
In this section, we pay our attention to establish a Lagrange-type dual problem for (VOP) and study the zero duality gap property by virtue of \(W_R(\varPi )\). Given the class \({\mathcal {W}}_R(\varPi )\) and a regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\), we consider the vector-valued function \({\mathcal {L}}_w:X\times \varPi \rightarrow {\mathbb {R}}^\ell \), defined by
The function \({\mathcal {L}}_w\) will be called a Lagrange-type function for (VOP) with respect to w.
The Lagrange-type dual function, which is a set-valued function from the parameter set \(\varPi \) to \(\bar{{\mathbb {R}}}^\ell \), for (VOP) with respect to w is defined by
The Lagrange-type dual optimization problem (DOP) to the primal problem (VOP) with respect to w is
(DOP) \(\sup _{\pi \in \varPi }\,q_w(\pi )\).
By symmetry, we are also interested in the set-valued map from X to \(\bar{{\mathbb {R}}}^\ell \), for (VOP) with respect to w, which is defined by
The primal problem (VOP) related to this map with respect to w is
(\(\overline{\mathrm {VOP}}\)) \(\inf _{x\in X}\,r_w(x)\).
Now, we first explain the relationships between the primal problem (VOP) and the related problem \(({\mathrm {\overline{VOP}}})\) under Assumption \({\mathcal {A}}\) and then further establish a weak dual relationship for (VOP) and the dual problem (DOP) under Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\).
Lemma 3.1
Let the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) satisfy Assumption \({\mathcal {A}}\). Then, the following results hold:
-
(i)
\(r_w(x)=f(x)\) for all feasible point \(x\in {\mathcal {R}}\) and \(r_w(x)=+\infty \) otherwise.
-
(ii)
\(\hat{x}\in X\) is a strong vmp to \({\mathrm {(VOP)}}\) if and only if \(\hat{x}\) is a strong vmp to \({\mathrm {(\overline{VOP})}}\).
Proof
(i) It follows from (4), (6) and Assumption \({\mathcal {A}}\) that
Note that \(g(x)\in D\) if \(x\in {\mathcal {R}}\) and \(g(x)\notin D\) otherwise. Together with Assumption \({\mathcal {A}}\), we get \(r_w(x)=f(\bar{x})-\inf _{\pi \in \varPi }w(f(\bar{x})-f(x), g(x); \pi )=f(x)\) for all feasible point \(x\in {\mathcal {R}}\) and \(r_w(x)=+\infty \) otherwise.
(ii) It follows from (i) that \(\hat{x}\in X\) is a strong vmp to \({\mathrm {(VOP)}}\), that is, \(\hat{x}\in {\mathcal {R}}\) and \(f(x)\nleq _{C\setminus \{0\}}f(\hat{x})\) for all \(x\in {\mathcal {R}}\), if and only if \(r_w(x)\nleq _{C\setminus \{0\}}r_w(\hat{x})\) for all \(x\in X\). This shows that \(\hat{x}\) is a strong vmp to \(\mathrm {(\overline{VOP})}\). This completes the proof. \(\square \)
Remark 3.1
Clearly, Lemma 3.1 shows that the primal problem (VOP) coincides with the related problem \(({\mathrm {\overline{VOP}}})\) under Assumption \({\mathcal {A}}\), in the sense that they have the same feasible sets and the same objective functions. This always holds for the scalar case when \(\ell =1\) and \(C={\mathbb {R}}_+\).
Theorem 3.1
(Weak duality theorem) Let the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) satisfy Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\). Then, the following assertions hold:
-
(i)
For every feasible point \(x\in {\mathcal {R}}\) and every \(z\in q_w(\pi )\) with \(\pi \in \varPi \), one has \(z\ngeq _{C\setminus \{0\}}f(x)\).
-
(ii)
For every \(\bar{z}\in \inf _{x\in {\mathcal {R}}}f(x)\) and every \(\tilde{z}\in \sup _{\pi \in \varPi }q_w(\pi )\), one has \(\tilde{z}\ngeq _{\mathrm {int\,C}}\bar{z}\).
Proof
(i) Assume that there exist \(x_0\in {\mathcal {R}}\) and \(z_0\in q_w(\pi _0)\) with \(\pi _0\in \varPi \), such that
Since \(x_0\in {\mathcal {R}}\), we get \((f(\bar{x})-f(x_0),g(x_0))-(f(\bar{x})-f(x_0),0)\in \ C\times D\). Together with Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\), we have \(w(f(\bar{x})-f(x_0),0;\pi _0)=f(\bar{x})-f(x_0)\) and
This shows \(f(x_0)\ge _C f(\bar{x})-w(f(\bar{x})-f(x_0),g(x_0);\pi _0)\). Thus, it follows from Assumption \({\mathcal {A}}\) and (4) that \(f(\bar{x})=w(f(\bar{x}),0;\pi _0)\) and \(f(x_0)\ge _C{\mathcal {L}}_w(x_0,\pi _0)\). Combining (7), we have \(z_0\ge _{C\setminus \{0\}}{\mathcal {L}}_w(x_0,\pi _0)\), which implies \(z_0\notin \inf _{x\in X}\,{\mathcal {L}}_w(x,\pi _0)\). This is a contradiction to \(z_0\in q_w(\pi _0)\) by (5).
(ii) Suppose that there exist \(z_1\in \inf _{x\in {\mathcal {R}}}f(x)\) and \(z_2\in \sup _{\pi \in \varPi }q_w(\pi )\), such that
Then, there exist a sequence \(f(x^k)\rightarrow z_1\) with \(x^k\in {\mathcal {R}}\), and a sequence \(z^k\rightarrow z_2\) with \(z^k\in q_w(\pi ^k)\) and \(\pi ^k\in \varPi \). Clearly, (i) implies that
Note that \(z^k-f(x^k)\rightarrow z_2-z_1\). Together with (8), one has \(z^k-f(x^k)\in {\mathrm {int}}\,C\) for sufficiently large \(k\in {\mathbb {N}}\). This is a contradiction to (9). \(\square \)
It is worth noting that the conclusion (i) coincides with conclusion (ii) in Theorem 3.1 for the scalar case when \(\ell =1\) and \(C={\mathbb {R}}_+\). However, they does not coincide for the vector case. Most pertinently, the conclusion (ii) shall not hold if \({\mathrm {int}}\,C\) is replaced by \(C\setminus \{0\}\). We give the following example to explain the case.
Example 3.1
Consider the following vector optimization problem
where \(X=[-1,1]\), \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}^2\) with \(f(x)=(x+1,x-1)^T\) is a vector-valued function and \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) with \(g(x)=x\) is a real-valued function. Obviously, we get \({\mathcal {R}}=[0,1]\). Let \(\ell =2\), \(m=1\), \(C={\mathbb {R}}_+^2\) and \(D={\mathbb {R}}_+\). It follows from Example 2.1 that \(D_C^*={\mathbb {R}}_+^2\). Let \(\varPi =E_2\times D_C^*\). Then, the vector-valued function \(w:{\mathbb {R}}^2\times {\mathbb {R}}\times \varPi \rightarrow {\mathbb {R}}^2\) with \(w(u,v;E_2,\varLambda )=u+\varLambda v\), \(\forall (u,v)\in {\mathbb {R}}^2\times {\mathbb {R}},~\forall \varLambda \in D_C^*\) is a regular weak separation function satisfying Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\). It follows from (4) and (5) that for all \(x\in {\mathbb {R}}\) and all \(\varLambda =(\lambda _1, \lambda _2)^T\in {\mathbb {R}}_+^2\), we have
and for all \(\varLambda =(\lambda _1,\lambda _2)^T\in {\mathbb {R}}_+^2\), we have
It is easy to verify that \(f(x)\ge _C (1,-1)^T\) for all \(x\in {\mathcal {R}}\) , and \(z\ngeq _{C\setminus \{0\}}(1,-1)^T\) for all \(z\in q_w(E_2,\varLambda )\) and all \(\varLambda \in D_C^*\). Thus, for every feasible point \(x\in {\mathcal {R}}\) and every \(z\in q_w(E_2,\varLambda )\) with \(\varLambda \in D_C^*\), one has \(z\ngeq _{C\setminus \{0\}}f(x)\), that is, the conclusion (i) holds. Simultaneously, by direct calculating, we get
and
Clearly, for every \(\bar{z}\in \inf _{x\in {\mathcal {R}}}f(x)\) and every \(\tilde{z}\in \sup _{\varLambda \in D_C^*}q_w(E_2,\varLambda )\), one has \(\bar{z}=(1,-1)^T\) and \(\tilde{z}\ngeq _{\mathrm {int\,C}}\bar{z}\). Thus, the conclusion (ii) also holds. Note that if we take \(\lambda _1=0\), \(x=1\) and \(\lambda _2=1\), then \((2,-1)^T\in \sup _{\varLambda \in D_C^*}q_w(E_2,\varLambda )\). Obviously, we still have \((2,-1)^T\ngeq _{{\mathrm {int}}\,C} (1,-1)^T\). However, \((2,-1)^T\ge _{C\setminus \{0\}} (1,-1)^T\). This shows that the conclusion (ii) does not hold when we replace \({\mathrm {int}}\,C\) by \(C\setminus \{0\}\).
The following corollary establishes a further relationship between the primal problem (VOP) and the dual problem (DOP).
Corollary 3.1
Let the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) satisfy Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\). If there exists a feasible point \(\hat{x}\in {\mathcal {R}}\) of \(\mathrm {(VOP)}\) such that \(f(\hat{x})\in \sup _{\pi \in \varPi }q_w(\pi )\), then \(\hat{x}\) is a weak vmp to \(\mathrm {(VOP)}\).
Proof
Assume that \(\hat{x}\) is not a weak vmp to \({\mathrm {(VOP)}}\). Then, there exists \(\bar{x}\in {\mathcal {R}}\) satisfying \(f(\bar{x})\le _{{\mathrm {int}}\,C}f(\hat{x})\). Note that \(f(\hat{x})\in \sup _{\pi \in \varPi }q_w(\pi )\), that is, there exists a sequence \(z^k\rightarrow f(\hat{x})\) with \(z^k\in q_w(\pi ^k)\) and \(\pi ^k\in \varPi \). Thus, we can conclude that \(z^k-f(\bar{x})\rightarrow f(\hat{x})-f(\bar{x})\). Together with \(f(\bar{x})\le _{{\mathrm {int}}\,C}f(\hat{x})\), we have \(z^k-f(\bar{x})\in {\mathrm {int}}\,C\) for sufficiently large \(k\in {\mathbb {N}}\). This is a contradiction from Theorem 3.1(i). \(\square \)
Next, we focus our attention to investigate some equivalent characterizations of the zero duality gap property between the primal problem (VOP) and the dual problem (DOP). We first follow the classic approach, namely the Lagrange multiplier and the Lagrange saddle point, and then apply the regular separation associated with \({\mathcal {K}}\) and \({\mathcal {H}}\) in the IS to discuss the zero duality gap property.
Let \(\varDelta _1\) be the set of all the infimum values of (VOP) and \(\varDelta _2\) the set of all the supremum values of (DOP), that is,
We recall from [30] that \(\varDelta :=\varDelta _1-\varDelta _2\) is said to be the duality gap. In the sequel, we call that the zero duality gap property with respect to w holds iff \(0\in \varDelta \), that is, \(\varDelta _1\cap \varDelta _2\ne \varnothing \).
First of all, we introduce some standard notions associated with the classes of separation functions and the Lagrange-type function for (VOP).
Definition 3.1
[30, 31] Given the classes \({\mathcal {W}}(\varPi )\) and \({\mathcal {W}}_R(\varPi )\), the sets \({\mathcal {K}}\) and \({\mathcal {H}}\) admit a separation with respect to \(w\in {\mathcal {W}}(\varPi )\) and \(\hat{\pi }\in \varPi \) iff
Moreover, the separation is said to be regular iff \(w\in {\mathcal {W}}_R(\varPi )\).
We observe from Definitions 2.2 and 2.3 that if either \({\mathcal {K}}\) or \({\mathcal {H}}\) admits a separation, and (10) is fulfilled when \(C\setminus \{0\}\) is replaced by C, or \({\mathcal {K}}\) and \({\mathcal {H}}\) admit a regular separation, then \({\mathcal {K}}\cap {\mathcal {H}}=\varnothing \). In addition, let \(\bar{x}\in {\mathcal {R}}\). Then, \(\bar{x}\) is a strong vmp to (VOP).
Definition 3.2
Given the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) and the Lagrange-type function \({\mathcal {L}}_w\) for \(\mathrm {(VOP)}\) with respect to w, then \(\hat{\pi }\in \varPi \) is said to be a generalized Lagrange multiplier for \(\mathrm {(VOP)}\) with respect to w iff
Together with (5), the above inequality is equivalent to
Definition 3.3
[30] Given the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) and the Lagrange-type function \({\mathcal {L}}_w\) for \({\mathrm {(VOP)}}\) with respect to w, the pair \((\hat{x},\hat{\pi })\in X\times \varPi \) is said to be a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w iff
Now, we establish some equivalent characterizations to the zero duality gap property for (VOP) by virtue of the classic approach, namely the Lagrange multiplier and the Lagrange saddle point.
Theorem 3.2
Given the pair \((\hat{x},\hat{\pi })\in X\times \varPi \), and the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) satisfying Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\), then the following assertions are equivalent:
-
(i)
\((\hat{x},\hat{\pi })\in X\times \varPi \) is a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w.
-
(ii)
\(\hat{x}\in {\mathcal {R}}\) is a feasible point and \(\hat{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w, and moreover, \(f(\hat{x})\in f({\mathcal {R}})\cap q_w(\hat{\pi })\).
In addition, if any condition \(\mathrm {(i)}\) or \({\mathrm {(ii)}}\) above fulfills, then \(\hat{x}\) is a strong vmp of \({\mathrm {(VOP)}}\) and \((\hat{\pi },f(\hat{x}))\in \mathrm {graph}\,q_w\) is a strong vector maximum point of \(\mathrm {(DOP)}\) in the set-valued case, that is, \(f(\hat{x})\nleq _{C\setminus \{0\}}z\) for all \(z\in q_w(\pi )\) with \(\pi \in \varPi \). Simultaneously, the zero duality gap property with respect to w holds and \({\mathcal {L}}_w(\hat{x},\hat{\pi })=f(\hat{x})\in \varDelta _1\cap \varDelta _2\). Moreover, the generalized complementary slackness condition \(w(f(\bar{x})-f(\hat{x}),g(\hat{x});\hat{\pi })=f(\bar{x})-f(\hat{x})\) also fulfills.
Proof
(i) \(\Rightarrow \) (ii) Since \((\hat{x},\hat{\pi })\in X\times \varPi \) is a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w, it follows that
Then, \({\mathcal {L}}_w(\hat{x},\hat{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\pi )\) holds for all \(\pi \in \varPi \). First, we show \(\hat{x}\in {\mathcal {R}}\), that is, \(\hat{x}\) is a feasible point of (VOP). In fact, it follows from Assumption \({\mathcal {A}}\) and (4) that the above inequality implies
Suppose that \(\hat{x}\notin {\mathcal {R}}\), that is, \(g(\hat{x})\notin D\). By Assumption \({\mathcal {A}}\), we get
which implies that there exists a sequence \(w(f(\bar{x})-f(\hat{x}),g(\hat{x});\pi ^k)\rightarrow -\infty \) with \(\pi ^k\in \varPi \), which implies \(w(f(\bar{x})-f(\hat{x}),g(\hat{x});\pi ^k)\le _{C\setminus \{0\}}w(f(\bar{x})-f(\hat{x}),g(\hat{x});\hat{\pi })\) for sufficiently large \(k\in {\mathbb {N}}\). This is a contradiction to (12). Thus, we get \(\hat{x}\in {\mathcal {R}}\). Second, we show that
Note that \({\mathcal {L}}_w(\hat{x},\hat{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\pi ),~\forall \pi \in \varPi \) implies \({\mathcal {L}}_w(\hat{x},\hat{\pi })\in \sup _{\pi \in \varPi }{\mathcal {L}}_w(\hat{x},\pi )\). Together with (6), \(\hat{x}\in {\mathcal {R}}\) and Lemma 3.1 (i), we have
which implies that (13) holds. At last, we prove that \(\hat{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w and \(f(\hat{x})\in f({\mathcal {R}})\cap q_w(\hat{\pi })\). It follows from (11) that \({\mathcal {L}}_w(x,\hat{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\hat{\pi })\) holds for all \(x\in X\). Together with (5), we get \({\mathcal {L}}_w(\hat{x},\hat{\pi })\in \inf _{x\in X}{\mathcal {L}}_w(x,\hat{\pi })=q_w(\hat{\pi })\). Note that \(\hat{x}\in {\mathcal {R}}\) and \(f(\hat{x})={\mathcal {L}}_w(\hat{x},\hat{\pi })\). Then, it follows that \(f(\hat{x})\in f({\mathcal {R}})\cap q_w(\hat{\pi })\) and \(\hat{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w.
(ii) \(\Rightarrow \) (i) Since \(\hat{x}\in {\mathcal {R}}\) is a feasible point, we have \(g(\hat{x})\in {\mathbb {R}}_+^m\). Together with Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\), we get
and
Applying (4) and Assumption \({\mathcal {A}}\), we have
In addition, since \(\hat{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w, and moreover, \(f(\hat{x})\in f({\mathcal {R}})\cap q_w(\hat{\pi })\), it follows from (5) that
Thus, \(f(\hat{x})\ngeq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\hat{\pi })\). Together with (14), we have
On the one hand, combining (15) and (16), we have \({\mathcal {L}}_w(x,\hat{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\hat{\pi })\) for all \(x\in X\). On the other hand, it follows from \(\hat{x}\in {\mathcal {R}}\), Lemma 3.1 (i), (6) and (16) that \({\mathcal {L}}_w(\hat{x},\hat{\pi })=f(\hat{x})=r_w(\hat{x})=\sup _{\pi \in \varPi }{\mathcal {L}}_w(\hat{x},\pi )\), which implies \({\mathcal {L}}_w(\hat{x},\hat{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\pi )\) for all \(\pi \in \varPi \). Therefore, we can conclude that
that is, \((\hat{x},\hat{\pi })\in X\times \varPi \) is a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w.
In addition, let any condition (i) or (ii) be fulfilled. We firstly show that \(\hat{x}\in {\mathcal {R}}\) is a strong vmp to (VOP). Otherwise, there exists \(\tilde{x}\in {\mathcal {R}}\) such that
Note that (11) implies \({\mathcal {L}}_w(\tilde{x},\hat{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\hat{x},\hat{\pi })\). Applying \(f(\hat{x})={\mathcal {L}}_w(\hat{x},\hat{\pi })\), Assumption \({\mathcal {A}}\) and (4), we get
Moreover, it follows from \(\tilde{x}\in {\mathcal {R}}\), and Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\), we have
and
Together with (17), we have
which contradicts (18). Secondly, we show that \((\hat{\pi },f(\hat{x}))\in {\mathrm {graph}}\,q_w\) is a strong vector maximum point of \({\mathrm {(DOP)}}\). Obviously, we have \(f(\hat{x})\in q_w(\hat{\pi })\) from (ii). Moreover, it follows from \(\hat{x}\in {\mathcal {R}}\) and Theorem 3.1(i) (weak duality theorem) that \(f(\hat{x})\nleq _{C\setminus \{0\}} z\) for all \(z\in q_w(\pi )\) with \(\pi \in \varPi \). Thus, \((\hat{\pi }, f(\hat{x}))\in {\mathrm {graph}}\,q_w\) is a strong vector maximum point to (DOP). Lastly, since \(\hat{x}\in {\mathcal {R}}\) is a strong vmp to (VOP), we have \(f(\hat{x})\in \inf _{x\in {\mathcal {R}}}f(x)=\varDelta _1\). Furthermore, since \(f(\hat{x})\in q_w(\hat{\pi })\) and \((\hat{\pi },f(\hat{x}))\in {\mathrm {graph}}\,q_w\) is a strong vector maximum point of \({\mathrm {(DOP)}}\), we get \(f(\hat{x})\in \sup _{\pi \in \varPi }q_w(\pi )=\varDelta _2\). Combining \({\mathcal {L}}_w(\hat{x},\hat{\pi })=f(\hat{x})\), we can conclude that the zero duality gap property with respect to w holds, and moreover, \({\mathcal {L}}_w(\hat{x},\hat{\pi })=f(\hat{x})\in \varDelta _1\cap \varDelta _2\). Simultaneously, it follows from \({\mathcal {L}}_w(\hat{x},\hat{\pi })=f(\hat{x})\), (4) and Assumption \({\mathcal {A}}\) that \(f(\bar{x})-w(f(\bar{x})-f(\hat{x}),g(\hat{x});\hat{\pi })=f(\hat{x})\), which means that the generalized complementary slackness condition \(w(f(\bar{x})-f(\hat{x}),g(\hat{x});\hat{\pi })=f(\bar{x})-f(\hat{x})\) fulfills. This completes the proof. \(\square \)
Specially, let \(\bar{x}\in {\mathcal {R}}\). Then, we immediately have the following characterization to the zero duality gap property for (VOP) by means of the regular separation associated with \({\mathcal {K}}\) and \({\mathcal {H}}\) in the IS.
Theorem 3.3
Given the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) satisfying Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\), then the following assertions are equivalent:
-
(i)
\(\bar{x}\in {\mathcal {R}}\), and the sets \({\mathcal {K}}\) and \({\mathcal {H}}\) admit a regular separation with respect to w and \(\bar{\pi }\in \varPi \).
-
(ii)
\((\bar{x},\bar{\pi })\in X\times \varPi \) is a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w.
-
(iii)
\(\bar{x}\in {\mathcal {R}}\) is a feasible point and \(\bar{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w, and moreover, \(f(\bar{x})\in f({\mathcal {R}})\cap q_w(\bar{\pi })\).
In addition, if any condition \(\mathrm {(i)}\), \({\mathrm {(ii)}}\) or \({\mathrm {(iii)}}\) above fulfills, then \(\bar{x}\) is a strong vmp of \({\mathrm {(VOP)}}\) and \((\bar{\pi },f(\bar{x}))\in {\mathrm {graph}}\,q_w\) is a strong vector maximum point of \({\mathrm {(DOP)}}\) in the set-valued case. Simultaneously, the zero duality gap property with respect to w holds and \({\mathcal {L}}_w(\bar{x},\bar{\pi })=f(\bar{x})\in \varDelta _1\cap \varDelta _2\). Moreover, the generalized complementary slackness condition \(w(0,g(\bar{x});\bar{\pi })=0\) also fulfills.
Proof
Obviously, we only need to prove that (i) \(\Rightarrow \) (ii) and (iii) \(\Rightarrow \) (i) from Theorem 3.2.
(i) \(\Rightarrow \) (ii) Since \(\bar{x}\in {\mathcal {R}}\), and the sets \({\mathcal {K}}\) and \({\mathcal {H}}\) admit a regular separation with respect to w and \(\bar{\pi }\in \varPi \), we have \(w(f(\bar{x})-f(x),g(x);\bar{\pi })\ngeq _{C\setminus \{0\}}0\) for all \(x\in X\). This shows \(f(\bar{x})-w(f(\bar{x})-f(x),g(x);\bar{\pi })\nleq _{C\setminus \{0\}}f(\bar{x})\) for all \(x\in X\). Together with (4) and Assumption \({\mathcal {A}}\), we get
Take specially \(x=\bar{x}\). Then, we have
Moreover, since \(\bar{x}\in {\mathcal {R}}\), that is, \(g(\bar{x})\in D\), it follows from Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\) that \((0,g(\bar{x}))\in C\times D\) and \(w(0,g(\bar{x});\bar{\pi })\ge _C w(0,0;\bar{\pi })=0\). Together with (4) and Assumption \({\mathcal {A}}\), we get \(f(\bar{x})\ge _C f(\bar{x})-w(0,g(\bar{x});\bar{\pi })={\mathcal {L}}_w(\bar{x},\bar{\pi })\). Applying (20), it follows that
On the one hand, combining (19) and (21), we get \({\mathcal {L}}_w(x,\bar{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\bar{x},\bar{\pi })\) for all \(x\in X\). On the other hand, since \(\bar{x}\in {\mathcal {R}}\), it follows from Lemma 3.1 (i), (6) and (21) that \({\mathcal {L}}_w(\bar{x},\bar{\pi })=f(\bar{x})=r_w(\bar{x})=\sup _{\pi \in \varPi }{\mathcal {L}}_w(\bar{x},\pi )\), which implies \({\mathcal {L}}_w(\bar{x},\bar{\pi })\nleq _{C\setminus \{0\}}{\mathcal {L}}_w(\bar{x},\pi )\) for all \(\pi \in \varPi \). Therefore, we can conclude that \((\bar{x},\bar{\pi })\in X\times \varPi \) is a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w.
(iii) \(\Rightarrow \) (i) Since \(\bar{x}\in {\mathcal {R}}\) is a feasible point, and \(\bar{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w and \(f(\bar{x})\in f({\mathcal {R}})\cap q_w(\bar{\pi })\), it follows that \(f(\bar{x})\in \inf _{x\in X}{\mathcal {L}}_w(x,\bar{\pi })\), which implies \({\mathcal {L}}_w(x,\bar{\pi })\nleq _{C\setminus \{0\}} f(\bar{x})\) for all \(x\in X\). Together with (4) and Assumption \({\mathcal {A}}\), we can conclude that for all \(x\in X\), one has \(f(\bar{x})-w(f(\bar{x})-f(x),g(x);\bar{\pi })\nleq _{C\setminus \{0\}} f(\bar{x})\), namely \(w(f(\bar{x})-f(x),g(x);\bar{\pi })\ngeq _{C\setminus \{0\}}0\). Thus, the sets \({\mathcal {K}}\) and \({\mathcal {H}}\) admit a regular separation with respect to w and \(\bar{\pi }\in \varPi \). \(\square \)
To end this section, we give the following example to explain Theorem 3.3.
Example 3.2
Consider Example 3.1. Let \(\bar{x}=0\) and \(\bar{\pi }=(E_2,\bar{\varLambda })\in \varPi \) with \(\bar{\varLambda }=(1,1)^T\). Then, it follows that \(\bar{x}\in {\mathcal {R}}\), \(f(\bar{x})=(1,-1)^T\) and
Thus, we have
which implies that the sets \({\mathcal {K}}\) and \({\mathcal {H}}\) admit a regular separation with respect to w and \(\bar{\pi }\in \varPi \). Note that the Lagrange-type function
where \(\varLambda =(\lambda _1,\lambda _2)^T\in {\mathbb {R}}_+^2\). Then, we get
Obviously, we have
that is, \((\bar{x},\bar{\pi })\in X\times \varPi \) is a generalized Lagrange saddle point for \({\mathrm {(VOP)}}\) with respect to w. Moreover, for all \(\pi =(E_2,\varLambda )\in \varPi \), we have
Clearly, we get \(f(\bar{x})=(1,-1)^T\in f({\mathcal {R}})\cap q_w(\bar{\pi })\). Thus, \(\bar{\pi }\in \varPi \) is a generalized Lagrange multiplier for \({\mathrm {(VOP)}}\) with respect to w. In addition, it is easy to verify that \(\bar{x}\) is a strong vmp of \({\mathrm {(VOP)}}\) and \((\bar{\pi },f(\bar{x}))\in {\mathrm {graph}}\,q_w\) is a strong vector maximum point of \({\mathrm {(DOP)}}\). Simultaneously, the zero duality gap property with respect to w holds and \({\mathcal {L}}_w(\bar{x},\bar{\pi })=f(\bar{x})\in \varDelta _1\cap \varDelta _2\). Moreover, the generalized complementary slackness condition \(w(0,g(\bar{x});\bar{\pi })=\bar{\varLambda }g(\bar{x})=(0,0)^T\) also fulfills.
4 Exact Penalization
In this section, we first consider a special class of regular weak separation functions, which are separable with respect to the objective and the constraints. Subsequently, we focus on establishing a local exact penalization result for (VOP) by virtue of the special regular separation functions and a local image regularity condition in the IS.
Let the parameter set \(\varPi =D\) and \(\pi \in \varPi \). The vector-valued function \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \), which is separable with respect to the objective and the constraints, is defined by
where \(e\in {\mathrm {int}}\,C\) is a fixed element and the function \(\underline{w}:{\mathbb {R}}^m\times D\rightarrow {\mathbb {R}}\) must be such that
Clearly, it follows from (b) and (d) that
Theorem 4.1
The vector-valued functions given by conditions \(\mathrm {(a}\)–\(\mathrm {e)}\) are regular weak separation functions satisfying Assumptions \({\mathcal {A}}\) and \({\mathcal {B}}\).
Proof
We firstly prove that the vector-valued functions given by conditions \(\mathrm {(a}\)–\(\mathrm {e)}\) are regular weak separation functions. It is easy to verify that
In fact, for every \((u,v)\in {\mathcal {H}}\) and every \(\pi \in D\), we get \(u\in C\setminus \{0\}\), and \(v\in D\). Moreover, since \(e\in {\mathrm {int}}\,C\), it follows from (b) that \(\underline{w}(v,\pi )\ge 0\) and \(\underline{w}(v,\pi )e\in C\). Then, we have \(w(u,v;\pi )=u+\underline{w}(v,\pi )e\in C\setminus \{0\}+C\subset C\setminus \{0\}\), which implies \((u,v)\in {\mathrm {lev}}_{C\setminus \{0\}}\,w(\bullet ;\pi )\). Next, we prove the converse inclusion
Take arbitrary \((u,v)\in \cap _{\pi \in D}\,{\mathrm {lev}}_{C\setminus \{0\}}\,w(\bullet ;\pi )\), that is,
Then, it follows from (g) that there exists \(\bar{\pi }\in \varPi \) such that
that is, \(u\in C\setminus \{0\}\). Moreover, we get \(v\in D\). Otherwise, it follows from (f) that there exists \(\hat{\pi }\in D\) satisfying \(\underline{w}(v,\hat{\pi })<0\). Let \(\theta >0\) and \(\theta \rightarrow +\infty \). Then, we get from (d) that there exists \(\hat{\pi }_{\theta }\in D\) such that \(\underline{w}(v,\hat{\pi }_\theta )=\theta \underline{w}(v,\hat{\pi })\rightarrow -\infty \). Together with \(e\in {\mathrm {int}}\,C\), it follows that \(w(u,v;\hat{\pi }_\theta )=u+\underline{w}(v,\hat{\pi }_\theta )e\notin C\setminus \{0\}\) for sufficiently large \(\theta >0\). This is a contradiction to (22). Thus, we can conclude that \(u\in C\setminus \{0\}\) and \(v\in D\), that is, \((u,v)\in {\mathcal {H}}\). This shows that the vector-valued functions given by conditions \(\mathrm {(a}\)–\(\mathrm {e)}\) are regular weak separation functions. Secondly, we show that Assumption \({\mathcal {A}}\) holds. On the one hand, it follows from (c) and (e) that
and \(\underline{w}(v,\pi )\ge \underline{w}(0,\pi )=0,~\forall v\in D,~\forall \pi \in D\). Together with \(e\in {\mathrm {int}}\,C\), we have
Furthermore, (g) implies that there exists \(\bar{\pi }\in \pi \) such that
Thus, we have \(u\in \inf _{\pi \in {\mathbb {R}}_+^m}\,w(u,v;\pi )\) for all \(v\in D\). Moreover, we can conclude that
In fact, take arbitrary \(v\in D\) and \(z\in \inf _{\pi \in D}\,w(u,v;\pi )\). Then, it follows that
and there exists a sequence \(w(u,v;\pi ^k)\rightarrow z\) with \(\pi ^k\in D\). Together with (24), we get \(w(u,v;\pi ^k)\ge _C u\) and \(w(u,v;\pi ^k)-u\rightarrow z-u\), which implies \(z-u\in C\), i.e., \(z\ge _C u\), since C is closed. Simultaneously, it follows from (25) and (27) that \(z\ngeq _{C\setminus \{0\}}w(u,v;\bar{\pi })=u\). Thus, we have \(z=u\). This shows that (26) holds since \(v\in D\) and \(z\in \inf _{\pi \in D}\,w(u,v;\pi )\) are arbitrary. On the other hand, take arbitrary \(v\notin D\). Then, it follows from (f) that there exists \(\hat{\pi }\in D\) satisfying \(\underline{w}(v,\hat{\pi })<0\). Let \(\theta >0\) and \(\theta \rightarrow +\infty \). Then, we get from (d) that there exists \(\hat{\pi }_{\theta }\in D\) such that \(\underline{w}(v,\hat{\pi }_\theta )=\theta \underline{w}(v,\hat{\pi })\rightarrow -\infty \). Note that \(e\in {\mathrm {int}}\,C\). Thus, we have \(w(u,v;\hat{\pi }_\theta )=u+\underline{w}(v,\hat{\pi }_\theta )e\rightarrow -\infty \), which implies
Together with (23) and (26), it follows that Assumption \({\mathcal {A}}\) holds. Lastly, we prove that Assumption \({\mathcal {B}}\) also holds. For all \((u^i,v^i)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m, i=1,2\) with \((u^1,v^1)\ge _{C\times D}(u^2,v^2)\), that is, \(u^1\ge _C u^2\) and \(v^1\ge _D v^2\), it follows from \(e\in {\mathrm {int}}\,C\) and (e) that \(\underline{w}(v^1,\pi )\ge \underline{w}(v^2,\pi )\) and
which implies \(w(u^1,v^1;\pi )\ge _C w(u^2,v^2;\pi )\) for all \(\pi \in D\). Therefore, Assumption \({\mathcal {B}}\) holds. This completes the proof. \(\square \)
Next, we consider the following penalization optimization problem
(POP) \(\min f(x)\), \({\mathrm {s.t.}}\) \(x\in X\), \([g_j(x)]_-\ge 0, j=1,2,...,m\),
where \([g_j(x)]_-:=\min \{0,g_j(x)\}\). Note that (POP) is equivalent to (VOP) in the sense that (POP) and (VOP) have the same feasible sets and the same objective functions. For simplicity, let \([g(x)]_-:=\Big ([g_1(x)]_-,[g_2(x)]_-,...,[g_m(x)]_-\Big )\) for all \(x\in X\). In the sequel, we always assume that \(X\subset {\mathbb {R}}^n\) is nonempty and closed, and moreover, every component function \(f_i:X\rightarrow {\mathbb {R}}, i=1,2,...,\ell \) of the vector-valued function \(f=(f_1,f_2,...,f_\ell ):X\rightarrow {\mathbb {R}}^\ell \) is continuous. It is easy to verify that the feasible set \({\mathcal {R}}=\{x\in X: g_j(x)\ge 0, j=1,2,...,m\}\) is closed under the assumption that \(g_j\) is continuous or upper semi-continuous for each \(j\in \{1,2,...,m\}\).
Given the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) with conditions \(\mathrm {(a}\)–\(\mathrm {e)}\), then the Lagrange-type function \({\mathcal {L}}_w^\diamond :X\times \varPi \rightarrow {\mathbb {R}}^\ell \) for (POP) with respect to w has the form
As usual, \({\mathcal {L}}_w^\diamond \) is said to be the Lagrange-type penalty function and the following optimization problem
(VPP) \(\min _{x\in X} {\mathcal {L}}^\diamond _w(x,\pi )\)
is called the vector penalty problem for (VOP) with respect to w. Clearly, it follows from Theorem 4.1 and (a) that
Now, we are in the position to extend the local image regularity condition from scalar case to (VOP) in the IS. Let \(\bar{x}\in {\mathcal {R}}\) and \(\delta >0\). The localization of the image \({\mathcal {K}}\) for (VOP) with respect to \(\bar{x}\) is defined by
Definition 4.1
Suppose that \(\bar{x}\in {\mathcal {R}}\) is a local strong vmp to \({\mathrm {(VOP)}}\). The local image regularity condition holds at \(\bar{x}\) for \({\mathrm {(VOP)}}\) iff there exists \(\delta >0\) such that
Remark 4.1
Specially, the local image regularity condition at \(\bar{x}\in {\mathcal {R}}\) for \({\mathrm {(VOP)}}\) coincides with the local image regularity condition in [27, 28] for the scalar case when \(\ell =1\) and \(C={\mathbb {R}}_+\).
In the sequel, we discuss the existence of a class of exact penalty functions for \({\mathrm {(VOP)}}\) by virtue of the local image regularity condition in the IS. To this end, we shall consider the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) given by conditions (a–e) satisfying the following assumption:
Assumption
\({\varvec{{\mathcal {C}}}}\) There exists some \(\alpha >0\) such that
holds for all \(v=(v_1,v_2,...,v_m)^T\in -D\) and all \(\pi =(\pi _1,\pi _2,...,\pi _m)^T\in D\).
Example 4.1
Let \(\varPi =D\) and \(e\in {\mathrm {int}}\,C\). It is easy to verify that the following vector-valued functions \(w:{\mathbb {R}}^\ell \times {\mathbb {R}}^m\times \varPi \rightarrow {\mathbb {R}}^\ell \) satisfy conditions (a–e) and Assumption \({\mathcal {C}}\):
-
(i)
\(w(u,v;\pi )=u+\langle \pi ,v\rangle e,~~\forall (u,v)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m,~\forall \pi \in D\) where \(\underline{w}(v,\pi )=\langle \pi ,v\rangle \);
-
(ii)
\(w(u,v;\pi )=u+\min \{\pi _1v_1,\pi _2v_2,...,\pi _mv_m\}e,~\forall (u,v)\in {\mathbb {R}}^\ell \times {\mathbb {R}}^m,~\forall \pi \in D\) where \(\underline{w}(v,\pi )=\min \{\pi _1v_1,\pi _2v_2,...,\pi _mv_m\}\).
Thus, all the functions defined by (i) or (ii) are regular weak separation functions and satisfy Assumptions \({\mathcal {A}}\), \({\mathcal {B}}\) and \({\mathcal {C}}\).
Theorem 4.2
(Exact penalization) Let the regular weak separation function \(w\in {\mathcal {W}}_R(\varPi )\) given by conditions (a–e) satisfy Assumption \({\mathcal {C}}\) and let \(\bar{x}\in {\mathcal {R}}\) be a local strong vmp to \({\mathrm {(VOP)}}\). If the local image regularity condition holds at \(\bar{x}\) for \({\mathrm {(VOP)}}\), then the Lagrange-type penalty function \({\mathcal {L}}^\diamond _w(x,\pi )\) is an exact penalty function for \({\mathrm {(VOP)}}\), that is, there exists \(\tilde{\pi }\in D\) such that \(\bar{x}\) is a local strong vmp to \(\mathrm {(VPP)}\) with respect to w for all \(\pi \in D\) satisfying \(\pi \ge _D \tilde{\pi }\).
Proof
Assume that the Lagrange-type penalty function \({\mathcal {L}}^\diamond _w(x,\pi )\) is not an exact penalty function for \({\mathrm {(VOP)}}\). Then, for all \(\pi ^k=(k,k,...,k)\in D\) with \(k\in {\mathbb {N}}\), there exist \(\bar{\pi }\in D\) satisfying \(\bar{\pi }\ge _D \pi ^k\) and \(x^k\in {\mathcal {B}}(\bar{x},\frac{1}{k})\cap X\) such that
which implies \(f(x^k)-\underline{w}([g(x^k)]_-,\bar{\pi })e\in f(\bar{x})-\underline{w}([g(\bar{x})]_-,\bar{\pi })e-C\setminus \{0\}\). Since \(\bar{x}\in {\mathcal {R}}\), that is, \(g_j(\bar{x})\ge 0\) for all \(j\in \{1,2,...,m\}\), we get \([g(\bar{x})]_-=0\) and \(\underline{w}([g(\bar{x})]_-,\bar{\pi })=0\) by (c). Thus, we have
Let \(w^k_j=[g_j(x^k)]_-,j=1,2,...,m\). Then, \(w^k=(w^k_1,w^k_2,...,w^k_m)^T=[g(x^k)]_-\) and \(w^k\in -D\). It follows from Assumption \({\mathcal {C}}\) that there exists \(\alpha >0\) such that
Together with \(\bar{\pi }\ge _D \pi ^k\), we get \(\bar{\pi }_j\ge k\) for all \(j\in \{1,2,...,m\}\) and
Since \(\Vert w^k\Vert =\sqrt{|w^k_1|^2+|w^k_2|^2+...+|w^k_m|^2}\le \sqrt{m(\max \{|w^k_1|,|w^k_2|,...,|w^k_m|\})^2}\), it follows that
This shows \(-\underline{w}([g(x^k)]_-,\bar{\pi })\ge \alpha k\frac{\Vert w^k\Vert }{\sqrt{m}}\). Together with \(e\in {\mathrm {int}}\,C\), we have
Therefore, we can conclude from (28) and (29) that
that is,
Together with \(e\in {\mathrm {int}}\,C\), it follows that \(\frac{\alpha k}{\sqrt{m}}\Vert w^k\Vert e\in C\). Simultaneously, we have \(f(\bar{x})-f(x^k)\in C+C\setminus \{0\}\subset C\setminus \{0\}\), which implies \(f(\bar{x})\ne f(x^k)\) for all \(k\in {\mathbb {N}}\). Let \(t^k:=\Vert f(\bar{x})-f(x^k)\Vert \). Then, we have \(t^k>0\) and \(t^k\rightarrow 0\) since \(x^k\rightarrow \bar{x}\) as \(k\rightarrow +\infty \) and \(f_i, i=1,2,...,\ell \) are continuous. Note that \(\{\frac{f(\bar{x})-f(x^k)}{t^k}\}\subset C\setminus \{0\}\) since C is a cone and \(\Vert \frac{f(\bar{x})-f(x^k)}{t^k}\Vert =1\). Therefore, there exists \(\tilde{u}\in C\setminus \{0\}\) with \(\Vert \tilde{u}\Vert =1\) and a subsequence of \(\frac{f(\bar{x})-f(x^k)}{t^k}\) such that it converges to \(\tilde{u}\). Without loss of generality, we may suppose that
Moreover, it follows from (30) that \(f(x^k)-f(\bar{x})\in -\frac{\alpha k}{\sqrt{m}}\Vert w^k\Vert e-C\setminus \{0\}\). Since C is a cone, we have
Applying Lemma 2.1 (nonlinear scalar function), we get
which implies
since \(\xi _e\) is nonnegative homogeneous. Note that \(\Big \Vert \frac{f(x^k)-f(\bar{x})}{kt^k}\Big \Vert =\frac{1}{k}\rightarrow 0\) and \(\xi _e\) is continuous. Then, we have \(\xi _e\Big (\frac{f(x^k)-f(\bar{x})}{kt^k}\Big )\rightarrow \xi _e(0)=0\) and \(\Big \Vert \frac{w^k}{t^k}\Big \Vert \rightarrow 0\). Together with (31), it follows that
Note that \(w^k_j=[g_j(x^k)]_-,j=1,2,...,m\). Then, it follows that \(g_j(x^k)\ge w_j^k\) for all \(j\in \{1,2,...,m\}\). Thus, we have \(w^k\in g(x^k)-D\), and from \(x^k\in X\cap {\mathcal {B}}(\bar{x},\frac{1}{k})\) and \({\mathrm {cl}}\,{\mathcal {H}}=C\times D\) that
Take arbitrary \(\delta >0\). Then, we get
for all sufficiently large \(k\in {\mathbb {N}}\). Together with (32) and \(\tilde{u}\in C\setminus \{0\}\), we can conclude that \((\tilde{u},0)\in {\mathcal {H}}_u\cap {\mathrm {cl}}\,{\mathrm {cone}}\,({\mathcal {K}}^\delta -{\mathrm {cl}}\,{\mathcal {H}})\), which is a contradiction to the local image regularity condition at \(\bar{x}\) for (VOP). This completes the proof. \(\square \)
Remark 4.2
Note that Theorem 4.4 in [26] also established an existence result for a exact penalty function, where the separable function \(\underline{w}\) was particularly defined by a distance function, for the scalar case when \(\ell =1\) and \(C={\mathbb {R}}_+\). In Theorem 4.2, we extend the results to vector cases by virtue of a more general class of regular weak separation functions.
5 Conclusions
In this paper, we established a Lagrange-type duality for a constrained nonconvex vector optimization problem by virtue of the class of vector-valued regular weak separation functions in the IS. Specially, we obtained some equivalent characterizations to the zero duality gap property including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Moreover, we also established a local exact penalization result by means of a local image regularity condition and a class of particular regular weak separation functions in the IS.
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Acknowledgments
The author is grateful to the anonymous referees and the Editor-in-Chief for their valuable comments and suggestions, which help to improve the paper. This research was supported by the National Natural Science Foundation of China (Grants: 11526165 and 11601437), the Scientific Research Fund of Sichuan Provincial Science and Technology Department (Grant: 2015JY0237) and the Fundamental Research Funds for the Central Universities (Grant: JBK160129).
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Zhu, S. Image Space Analysis to Lagrange-Type Duality for Constrained Vector Optimization Problems with Applications. J Optim Theory Appl 177, 743–769 (2018). https://doi.org/10.1007/s10957-016-1027-6
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DOI: https://doi.org/10.1007/s10957-016-1027-6
Keywords
- Image space analysis
- Vector optimization
- Lagrange-type duality
- Exact penalization
- Image regularity condition