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Some new properties of the Lagrange function and its applications
Fixed Point Theory and Applications volume 2012, Article number: 192 (2012)
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.
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 [1–5]. 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 , , 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
We denote by G the feasible set of (D). Let be a solution of (D). We will prove that the function is constant on a subset of X which is wider than the solution set of (P) and the function 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.
Let be a linear space of generalized finite sequences such that for all , but only finitely many . For each , the corresponding supporting set is a finite subset of T. We denote by the nonnegative cone of . For and , Z being a real linear space, we understand that
where is the dual space of X. Let be a locally Lipschitz function. The directional derivative and the Clarke generalized directional derivative of g at in direction are defined respectively by
The Clarke subdifferential of g at , denoted by , is defined by
A locally Lipschitz function g is said to be quasidifferentiable (or regular in the sense of Clarke) at if the directional derivative exists and
Definition 2.1 
Let C be a subset of X. A function is said to be semiconvex at if g is locally Lipschitz, regular at z, and
The function g is said to be semiconvex on C if g is semiconvex at every .
3 Main results
Let us denote by the solution set of (P) and by A the feasible set of (P). Suppose that . For , we assume that, under some constraint qualification condition (see ), there exists such that
Note that in , T is a compact topological space. We denote by and 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 , are regular on C and the function is semiconvex on C for every . Let z be a solution of (P) and be such that (3.1) holds. Then is a solution of (D) and .
Proof Suppose that z is a solution of (P) and is such that (3.1) holds. We get
Hence, is a feasible solution of (D). Since , for all ,
On the other hand, since , for all ,
By the weak duality between (P) and (D), for all feasible point of (D). Consequently, for all feasible point of (D). The desired results follow. □
3.1 Some new results of the Lagrange function
Theorem 3.2 Suppose that , are regular on C and is a solution of (D). Suppose further that the function is semiconvex on C. The following holds:
Proof Let be a solution of (D). We obtain and
Thus, there exist , , , and such that
Since , are regular on C and is semiconvex on C, it follows that for all . Hence,
On the other hand, we have that . Combining this and (3.4), we get
We obtain the desired result. □
Corollary 3.3 Suppose that , are regular on C, z is a solution of (P), and there exists such that (3.1) holds. If the function is semiconvex on C for every , then
In addition, for all .
Proof Suppose that is a solution of (P) and the condition (3.1) holds for . Then by Lemma 3.1, is a solution of (D). Note that for all . By Theorem 3.2, we obtain for all . If , then . From the equality above, we can deduce that for all . □
Corollary 3.3 covers Lemma 3.1 in . It also shows that the Lagrange function can be constant on a subset of X which is wider than a solution set.
If the involved functions of (P) are convex, Corollary 3.3 covers Lemma 3.1 in .
There exists a question: Which behavior does the function achieve for ? The question will be adapted below.
Theorem 3.5 Let be a solution of (D). Suppose that , are regular on C and the function is semiconvex on C. If , then the function is constant on , where
Proof Since is a solution of (D) and ,
On the other hand, since , using an argument as in the proof of Theorem 3.2, we get for all . This implies that . The desired result follows. □
Corollary 3.6 Assume that , are regular on C, is a solution of (P) and there exists such that (3.1) holds. If is semiconvex on C, every , then the function is constant on .
Proof If is a solution of (P) and there exists such that (3.1) holds, then by Lemma 3.1, is a solution of (D). By Theorem 3.5, we have that is constant on . □
3.2 Finding saddle points
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 is said to be a saddle point of the function L if
We need the following lemma.
Lemma 3.8 Let be a saddle point of the function L. Suppose that the function is semiconvex on C. Then z is a solution of (P), , and is a solution of (D). Moreover, .
Theorem 3.9 Assume that , are regular on C and is semiconvex on C for every . Let be a saddle point of the function L. Then,
For every , if , then it is a saddle point for L, and
For every , if , then it is a saddle point for L.
Proof Suppose that is a saddle point of the function L. We get
Since is semiconvex on C, by Lemma 3.8, is a solution of (P), , and is a solution of , and .
is a saddle point. For above, by Corollary 3.6, we obtain for all . Note that by (3.5), for all . Hence,(3.6)
Since and , it is a solution of (D). So,
From this, it is easy to deduce that for all . We obtain
is a saddle point. For above, by Corollary 3.3, we get for all . Then by (3.5), we obtain for all and for all . It remains to prove that for all . Indeed, since , for all . By Lemma 3.8, . Hence, . Thus, for all . □
The following corollary can be deduced directly from the theorem above.
Corollary 3.10 Assume that , are regular on C. If there exists a feasible point of being a saddle point of the function L and is semiconvex on C for every , then every point is also the saddle point of the function L.
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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.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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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
- Wolfe-type duality
- Lagrange function
- saddle points