Some new properties of the Lagrange function and its applications

Abstract

Using a dual problem in Wolfe type, the Lagrange function of an inequality constrained nonconvex programming problem is proved to be constant not only on its optimal solution set but also on a wider set. In addition, it is also constant on the set of Lagrange multipliers corresponding to solutions of the dual problem.

MSC:90C46, 49N15, 49K30.

1 Introduction

In mathematical programming, Lagrange functions play a key role in finding maxima or minima of the problems subject to constraint functions. In several papers, to establish characterizations of solution sets of inequality constrained programming problems, Lagrange functions which were associated to the problems were proved to be constant on their optimal solution sets [1â€“5]. The aim of this paper is to show some more properties of Lagrange functions. Concretely, we will show that such Lagrange functions can be constant not only on optimal solution sets but also on wider sets.

Let us consider the following nonconvex problem:

where $f,{f}_{t}:Xâ†’\mathbb{R}$, $tâˆˆT$, are locally Lipschitz functions on a Banach space X, T is an arbitrary (possibly infinite) index set, and C is a closed convex subset of X. Our new results on the Lagrange function of (P) will be obtained via its dual problem (D) in Wolfe type.

where the Lagrange function L is formulated by

$L\left(y,\mathrm{Î»}\right)=\left\{\begin{array}{cc}f\left(y\right)+{âˆ‘}_{tâˆˆT}{\mathrm{Î»}}_{t}{f}_{t}\left(y\right),\hfill & \left(y,\mathrm{Î»}\right)âˆˆCÃ—{\mathbb{R}}_{+}^{\left(T\right)},\hfill \\ +\mathrm{âˆž},\hfill & \text{otherwise}.\hfill \end{array}$

We denote by G the feasible set of (D). Let $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ be a solution of (D). We will prove that the function $L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)$ is constant on a subset of X which is wider than the solution set of (P) and the function $L\left({y}^{âˆ—},â‹\dots \right)$ is constant on the set of Lagrange multipliers corresponding to solutions of (P).

Our main results are divided into two parts. In the first one, we present some new properties of a Lagrange function. The second one is devoted to finding saddle points. Some remarks and further developments will be given.

2 Preliminaries

Let ${\mathbb{R}}^{\left(T\right)}$ be a linear space of generalized finite sequences $\mathrm{Î»}={\left(\mathrm{Î»}\right)}_{tâˆˆT}$ such that ${\mathrm{Î»}}_{t}âˆˆ\mathbb{R}$ for all $tâˆˆT$, but only finitely many . For each $\mathrm{Î»}âˆˆ{\mathbb{R}}^{\left(T\right)}$, the corresponding supporting set is a finite subset of T. We denote by ${\mathbb{R}}_{+}^{\left(T\right)}:=\left\{\mathrm{Î»}={\left({\mathrm{Î»}}_{t}\right)}_{tâˆˆT}âˆˆ{\mathbb{R}}^{\left(T\right)}âˆ£{\mathrm{Î»}}_{t}â‰¥0,tâˆˆT\right\}$ the nonnegative cone of ${\mathbb{R}}^{\left(T\right)}$. For $\mathrm{Î»}âˆˆ{\mathbb{R}}^{\left(T\right)}$ and ${\left\{{z}_{t}\right\}}_{tâˆˆT}âŠ‚Z$, Z being a real linear space, we understand that

The following concepts can be found in Clarkeâ€™s book [6, 7]. Let $CâŠ‚X$ be a convex set, and let $zâˆˆC$. The normal cone to C at z, denoted by $N\left(C,z\right)$, is defined by

$N\left(C,z\right):=\left\{vâˆˆ{X}^{âˆ—}âˆ£v\left(xâˆ’z\right)â‰¤0,\mathrm{âˆ€}xâˆˆC\right\},$

where ${X}^{âˆ—}$ is the dual space of X. Let $g:Xâ†’\mathbb{R}$ be a locally Lipschitz function. The directional derivative and the Clarke generalized directional derivative of g at $zâˆˆX$ in direction $dâˆˆX$ are defined respectively by

The Clarke subdifferential of g at $zâˆˆX$, denoted by ${\mathrm{âˆ‚}}^{c}g\left(z\right)$, is defined by

${\mathrm{âˆ‚}}^{c}g\left(z\right):=\left\{vâˆˆ{X}^{âˆ—}âˆ£v\left(d\right)â‰¤{g}^{c}\left(z;d\right),\mathrm{âˆ€}dâˆˆX\right\}.$

A locally Lipschitz function g is said to be quasidifferentiable (or regular in the sense of Clarke) at $zâˆˆX$ if the directional derivative ${g}^{\mathrm{â€²}}\left(z;d\right)$ exists and

${g}^{c}\left(z;d\right)={g}^{\mathrm{â€²}}\left(z;d\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}dâˆˆX.$

Definition 2.1 [8]

Let C be a subset of X. A function $g:Xâ†’\mathbb{R}$ is said to be semiconvex at $zâˆˆC$ if g is locally Lipschitz, regular at z, and

$dâˆˆX,z+dâˆˆC,\phantom{\rule{1em}{0ex}}{g}^{\mathrm{â€²}}\left(z;d\right)â‰¥0\phantom{\rule{1em}{0ex}}â‡’\phantom{\rule{1em}{0ex}}g\left(z+d\right)â‰¥f\left(z\right).$

The function g is said to be semiconvex on C if g is semiconvex at every $zâˆˆC$.

3 Main results

Let us denote by $Sol\left(\mathrm{P}\right)$ the solution set of (P) and by A the feasible set of (P). Suppose that . For $zâˆˆSol\left(\mathrm{P}\right)$, we assume that, under some constraint qualification condition (see [9]), there exists $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$ such that

$0âˆˆ{\mathrm{âˆ‚}}^{c}f\left(z\right)+\underset{tâˆˆT}{âˆ‘}{\mathrm{Î»}}_{t}{\mathrm{âˆ‚}}^{c}{f}_{t}\left(z\right)+N\left(C,z\right),\phantom{\rule{1em}{0ex}}{\mathrm{Î»}}_{t}{f}_{t}\left(z\right)=0,\mathrm{âˆ€}tâˆˆT.$
(3.1)

Note that in [9], T is a compact topological space. We denote by $V\left(\mathrm{P}\right)$ and $V\left(\mathrm{D}\right)$ the optimal values of (P) and (D), respectively. The following lemma is needed for our further research.

Lemma 3.1 For the problem (P), suppose that $f,{f}_{t},tâˆˆT$, are regular on C and the function $L\left(â‹\dots ,\mathrm{Î»}\right)$ is semiconvex on C for every $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$. Let z be a solution of (P) and $\stackrel{Â¯}{\mathrm{Î»}}$ be such that (3.1) holds. Then $\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)$ is a solution of (D) and $V\left(\mathrm{P}\right)=V\left(\mathrm{D}\right)$.

Proof Suppose that z is a solution of (P) and $\stackrel{Â¯}{\mathrm{Î»}}$ is such that (3.1) holds. We get

$0âˆˆ{\mathrm{âˆ‚}}^{c}f\left(z\right)+\underset{tâˆˆT}{âˆ‘}{\stackrel{Â¯}{\mathrm{Î»}}}_{t}{\mathrm{âˆ‚}}^{c}{f}_{t}\left(z\right)+N\left(C,z\right),\phantom{\rule{1em}{0ex}}{\stackrel{Â¯}{\mathrm{Î»}}}_{t}{f}_{t}\left(z\right)=0,\mathrm{âˆ€}tâˆˆT.$
(3.2)

Hence, $\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)$ is a feasible solution of (D). Since ${\stackrel{Â¯}{\mathrm{Î»}}}_{t}{f}_{t}\left(z\right)=0$, for all $tâˆˆT$,

$L\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)=f\left(z\right).$

On the other hand, since ${\stackrel{Â¯}{\mathrm{Î»}}}_{t}{f}_{t}\left(z\right)=0$, for all $tâˆˆT$,

$L\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)âˆ’L\left(x,\mathrm{Î»}\right)=f\left(z\right)âˆ’L\left(x,\mathrm{Î»}\right).$

By the weak duality between (P) and (D), $f\left(z\right)âˆ’L\left(x,\mathrm{Î»}\right)â‰¥0$ for all feasible point $\left(x,\mathrm{Î»}\right)$ of (D). Consequently, $L\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)â‰¥L\left(x,\mathrm{Î»}\right)$ for all feasible point $\left(x,\mathrm{Î»}\right)$ of (D). The desired results follow.â€ƒâ–¡

3.1 Some new results of the Lagrange function

Theorem 3.2 Suppose that $f,{f}_{t},tâˆˆT$, are regular on C and $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ is a solution of (D). Suppose further that the function $L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)$ is semiconvex on C. The following holds:

$L\left(y,{\mathrm{Î»}}^{âˆ—}\right)=L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆ{G}_{1}:=\left\{yâˆˆCâˆ£\left(y,{\mathrm{Î»}}^{âˆ—}\right)âˆˆG\right\}.$
(3.3)

Proof Let $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ be a solution of (D). We obtain $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)âˆˆCÃ—{\mathbb{R}}_{+}^{\left(T\right)}$ and

$0âˆˆ{\mathrm{âˆ‚}}^{c}f\left({y}^{âˆ—}\right)+\underset{tâˆˆT}{âˆ‘}{\mathrm{Î»}}_{t}^{âˆ—}{\mathrm{âˆ‚}}^{c}{f}_{t}\left({y}^{âˆ—}\right)+N\left(C,{y}^{âˆ—}\right).$

Thus, there exist $uâˆˆ{\mathrm{âˆ‚}}^{c}f\left({y}^{âˆ—}\right)$, ${u}_{t}âˆˆ{\mathrm{âˆ‚}}^{c}{f}_{t}\left({y}^{âˆ—}\right)$, $tâˆˆT$, and $wâˆˆN\left(C,{y}^{âˆ—}\right)$ such that

$u\left(yâˆ’{y}^{âˆ—}\right)+\underset{tâˆˆT}{âˆ‘}{\mathrm{Î»}}_{t}^{âˆ—}{u}_{t}\left(yâˆ’{y}^{âˆ—}\right)=âˆ’w\left(yâˆ’{y}^{âˆ—}\right)â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC.$

Since $f,{f}_{t},tâˆˆT$, are regular on C and $L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)$ is semiconvex on C, it follows that $L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¤L\left(y,{\mathrm{Î»}}^{âˆ—}\right)$ for all $yâˆˆC$. Hence,

$L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¤L\left(y,{\mathrm{Î»}}^{âˆ—}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆ{G}_{1}.$
(3.4)

On the other hand, we have that ${inf}_{{G}_{1}}L\left(y,{\mathrm{Î»}}^{âˆ—}\right)â‰¤{sup}_{{G}_{1}}L\left(y,{\mathrm{Î»}}^{âˆ—}\right)â‰¤{sup}_{\left(x,\mathrm{Î»}\right)âˆˆG}L\left(x,\mathrm{Î»}\right)$. Combining this and (3.4), we get

$L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¤\underset{yâˆˆ{G}_{1}}{inf}L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)â‰¤\underset{yâˆˆ{G}_{1}}{sup}L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)â‰¤\underset{\left(x,\mathrm{Î»}\right)âˆˆG}{sup}L\left(x,\mathrm{Î»}\right)=L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right).$

We obtain the desired result.â€ƒâ–¡

Corollary 3.3 Suppose that $f,{f}_{t},tâˆˆT$, are regular on C, z is a solution of (P), and there exists ${\mathrm{Î»}}^{âˆ—}$ such that (3.1) holds. If the function $L\left(â‹\dots ,\mathrm{Î»}\right)$ is semiconvex on C for every $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$, then

$L\left(y,{\mathrm{Î»}}^{âˆ—}\right)=f\left({y}^{âˆ—}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆ{G}_{1}.$

In addition, ${\mathrm{Î»}}_{t}^{âˆ—}{f}_{t}\left(y\right)=0$ for all $yâˆˆSol\left(\mathrm{P}\right)$.

Proof Suppose that ${y}^{âˆ—}$ is a solution of (P) and the condition (3.1) holds for $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$. Then by Lemma 3.1, $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ is a solution of (D). Note that ${\mathrm{Î»}}_{t}^{âˆ—}{f}_{t}\left({y}^{âˆ—}\right)=0$ for all $tâˆˆT$. By Theorem 3.2, we obtain $L\left(y,{\mathrm{Î»}}^{âˆ—}\right)=f\left({y}^{âˆ—}\right)$ for all $yâˆˆ{G}_{1}$. If $yâˆˆSol\left(\mathrm{P}\right)$, then $f\left(y\right)=f\left({y}^{âˆ—}\right)$. From the equality above, we can deduce that ${\mathrm{Î»}}_{t}^{âˆ—}{f}_{t}\left(y\right)=0$ for all $tâˆˆT$.â€ƒâ–¡

Remark 3.4

1. (1)

Corollary 3.3 covers Lemma 3.1 in [3]. It also shows that the Lagrange function can be constant on a subset of X which is wider than a solution set.

2. (2)

If the involved functions of (P) are convex, Corollary 3.3 covers Lemma 3.1 in [5].

3. (3)

Using the same method as above, we can establish the results which cover Theorem 2.1 in [2] and Theorem 3.2. in [4].

There exists a question: Which behavior does the function $L\left({y}^{âˆ—},â‹\dots \right)$ achieve for ${y}^{âˆ—}âˆˆ{G}_{1}$? The question will be adapted below.

Theorem 3.5 Let $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ be a solution of (D). Suppose that $f,{f}_{t},tâˆˆT$, are regular on C and the function $L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)$ is semiconvex on C. If $f\left({y}^{âˆ—}\right)â‰¥V\left(\mathrm{P}\right)$, then the function $L\left({y}^{âˆ—},â‹\dots \right)$ is constant on ${G}_{2}$, where

${G}_{2}=\left\{\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}|\left({y}^{âˆ—},\mathrm{Î»}\right)âˆˆG,{\mathrm{Î»}}_{t}{f}_{t}\left({y}^{âˆ—}\right)â‰¥0\right\}.$

Proof Since $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ is a solution of (D) and $f\left({y}^{âˆ—}\right)â‰¥V\left(\mathrm{P}\right)$,

$L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¥L\left({y}^{âˆ—},\mathrm{Î»}\right)â‰¥f\left({y}^{âˆ—}\right)â‰¥V\left(\mathrm{P}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}\mathrm{Î»}âˆˆ{G}_{2}.$

On the other hand, since $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)âˆˆG$, using an argument as in the proof of Theorem 3.2, we get $L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¤L\left(y,{\mathrm{Î»}}^{âˆ—}\right)$ for all $yâˆˆC$. This implies that $L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¤V\left(\mathrm{P}\right)$. The desired result follows.â€ƒâ–¡

Corollary 3.6 Assume that $f,{f}_{t},tâˆˆT$, are regular on C, ${y}^{âˆ—}$ is a solution of (P) and there exists ${\mathrm{Î»}}^{âˆ—}$ such that (3.1) holds. If $L\left(â‹\dots ,\mathrm{Î»}\right)$ is semiconvex on C, every $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$, then the function $L\left({y}^{âˆ—},â‹\dots \right)$ is constant on ${G}_{2}$.

Proof If ${y}^{âˆ—}$ is a solution of (P) and there exists ${\mathrm{Î»}}^{âˆ—}$ such that (3.1) holds, then by Lemma 3.1, $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ is a solution of (D). By Theorem 3.5, we have that $L\left({y}^{âˆ—},â‹\dots \right)$ is constant on ${G}_{2}$.â€ƒâ–¡

In this part, by applying the results above, we can determine saddle points of the function L.

Definition 3.7 For the problem (P), a point $\left(\stackrel{Â¯}{z},\stackrel{Â¯}{\mathrm{Î»}}\right)âˆˆCÃ—{\mathbb{R}}_{+}^{\left(T\right)}$ is said to be a saddle point of the function L if

$L\left(\stackrel{Â¯}{z},\mathrm{Î»}\right)â‰¤L\left(\stackrel{Â¯}{z},\stackrel{Â¯}{\mathrm{Î»}}\right)â‰¤L\left(x,\stackrel{Â¯}{\mathrm{Î»}}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}\left(x,\mathrm{Î»}\right)âˆˆCÃ—{\mathbb{R}}_{+}^{\left(T\right)}.$

We need the following lemma.

Lemma 3.8 Let $\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)âˆˆG$ be a saddle point of the function L. Suppose that the function $L\left(â‹\dots ,\stackrel{Â¯}{\mathrm{Î»}}\right)$ is semiconvex on C. Then z is a solution of (P), ${\stackrel{Â¯}{\mathrm{Î»}}}_{t}{f}_{t}\left(z\right)=0$, and $\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)$ is a solution of (D). Moreover, $V\left(\mathrm{P}\right)=V\left(\mathrm{D}\right)$.

Theorem 3.9 Assume that $f,{f}_{t},tâˆˆT$, are regular on C and $L\left(â‹\dots ,\mathrm{Î»}\right)$ is semiconvex on C for every $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$. Let $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)âˆˆG$ be a saddle point of the function L. Then,

1. (i)

For every $\stackrel{Â¯}{\mathrm{Î»}}âˆˆ{G}_{2}$, if $\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)âˆˆG$, then it is a saddle point for L, and

2. (ii)

For every $zâˆˆSol\left(\mathrm{P}\right)$, if $\left(z,{\mathrm{Î»}}^{âˆ—}\right)âˆˆG$, then it is a saddle point for L.

Proof Suppose that $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)âˆˆG$ is a saddle point of the function L. We get

$L\left({y}^{âˆ—},\mathrm{Î»}\right)â‰¤L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)â‰¤L\left(x,{\mathrm{Î»}}^{âˆ—}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}\left(x,\mathrm{Î»}\right)âˆˆCÃ—{\mathbb{R}}_{+}^{\left(T\right)}.$
(3.5)

Since $L\left(â‹\dots ,{\mathrm{Î»}}^{âˆ—}\right)$ is semiconvex on C, by Lemma 3.8, ${y}^{âˆ—}$ is a solution of (P), ${\mathrm{Î»}}_{t}^{âˆ—}{f}_{t}\left({y}^{âˆ—}\right)=0$, and $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ is a solution of $\left(\mathrm{D}\right)$, and $V\left(\mathrm{P}\right)=V\left(\mathrm{D}\right)$.

1. (i)

$\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)$ is a saddle point. For ${y}^{âˆ—}$ above, by Corollary 3.6, we obtain $L\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)=L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ for all $\stackrel{Â¯}{\mathrm{Î»}}âˆˆ{G}_{2}$. Note that by (3.5), $L\left({y}^{âˆ—},\mathrm{Î»}\right)â‰¤L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ for all $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$. Hence,

$L\left({y}^{âˆ—},\mathrm{Î»}\right)â‰¤L\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}.$
(3.6)

Since $L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)=L\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)$ and $\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)âˆˆG$, it is a solution of (D). So,

$0âˆˆ{\mathrm{âˆ‚}}^{c}f\left({y}^{âˆ—}\right)+\underset{tâˆˆT}{âˆ‘}{\stackrel{Â¯}{\mathrm{Î»}}}_{t}{\mathrm{âˆ‚}}^{c}{f}_{t}\left({y}^{âˆ—}\right)+N\left(C,{y}^{âˆ—}\right).$

From this, it is easy to deduce that $L\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)â‰¤L\left(x,\stackrel{Â¯}{\mathrm{Î»}}\right)$ for all $xâˆˆC$. We obtain

$L\left({y}^{âˆ—},\mathrm{Î»}\right)â‰¤L\left({y}^{âˆ—},\stackrel{Â¯}{\mathrm{Î»}}\right)â‰¤L\left(x,\stackrel{Â¯}{\mathrm{Î»}}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}\left(x,\mathrm{Î»}\right)âˆˆCÃ—{\mathbb{R}}_{+}^{\left(T\right)}.$
1. (ii)

$\left(z,{\mathrm{Î»}}^{âˆ—}\right)$ is a saddle point. For ${\mathrm{Î»}}^{âˆ—}$ above, by Corollary 3.3, we get $L\left(z,{\mathrm{Î»}}^{âˆ—}\right)=L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ for all $zâˆˆSol\left(\mathrm{P}\right)$. Then by (3.5), we obtain $L\left(z,{\mathrm{Î»}}^{âˆ—}\right)â‰¤L\left(x,{\mathrm{Î»}}^{âˆ—}\right)$ for all $xâˆˆC$ and for all $zâˆˆSol\left(\mathrm{P}\right)$. It remains to prove that $L\left(z,\mathrm{Î»}\right)â‰¤L\left(z,{\mathrm{Î»}}^{âˆ—}\right)$ for all $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$. Indeed, since $zâˆˆSol\left(\mathrm{P}\right)$, $L\left(z,\mathrm{Î»}\right)â‰¤f\left(z\right)$ for all $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$. By Lemma 3.8, $V\left(\mathrm{P}\right)=V\left(\mathrm{D}\right)$. Hence, $f\left(z\right)=L\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)=L\left(z,{\mathrm{Î»}}^{âˆ—}\right)$. Thus, $L\left(z,\mathrm{Î»}\right)â‰¤L\left(z,{\mathrm{Î»}}^{âˆ—}\right)$ for all $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$.â€ƒâ–¡

The following corollary can be deduced directly from the theorem above.

Corollary 3.10 Assume that $f,{f}_{t},tâˆˆT$, are regular on C. If there exists a feasible point $\left({y}^{âˆ—},{\mathrm{Î»}}^{âˆ—}\right)$ of $\left(\mathrm{D}\right)$ being a saddle point of the function L and $L\left(â‹\dots ,\mathrm{Î»}\right)$ is semiconvex on C for every $\mathrm{Î»}âˆˆ{\mathbb{R}}_{+}^{\left(T\right)}$, then every point $\left(z,\stackrel{Â¯}{\mathrm{Î»}}\right)âˆˆSol\left(\mathrm{P}\right)Ã—{G}_{2}$ is also the saddle point of the function L.

4 Further developments

1. (1)

Using Theorem 3.2, we can re-establish the characterizations of a solution set of the problem given in [3]via its dual problem in Wolfe type.

2. (2)

The characterizations of solution sets of the problem considered in [5] can be rebuilt via its dual problem.

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Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no.2010-0012780) and the National Foundation for Science and Technology Development (NAFOSTED), Vietnam. The authors are thankful to the anonymous referees whose suggestions have enhanced the presentation of the paper.

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Kim, D.S., Son, T.Q. Some new properties of the Lagrange function and its applications. Fixed Point Theory Appl 2012, 192 (2012). https://doi.org/10.1186/1687-1812-2012-192