Some new properties of the Lagrange function and its applications
© Kim and Son; licensee Springer 2012
Received: 21 May 2012
Accepted: 15 October 2012
Published: 29 October 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.
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.
Definition 2.1 
The function g is said to be semiconvex on C if g is semiconvex at every .
3 Main results
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 .
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
We obtain the desired result. □
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.
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.
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, .
For every , if , then it is a saddle point for L, and
For every , if , then it is a saddle point for L.
- (i)is a saddle point. For above, by Corollary 3.6, we obtain for all . Note that by (3.5), for all . Hence,(3.6)
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.
4 Further developments
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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