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We give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints (MPVC for short), which is proposed in Achtziger et al. (2013). We show that the Mangasarian-Fromovitz constraints qualification for the smoothing-regularization problem still holds under the VC-MFCQ (see Definition 5) which is weaker than the VC-LICQ (see Definition 7) and the condition of asymptotic nondegeneracy. We also analyze the convergence behavior of the smoothing-regularization method and prove that any accumulation point of a sequence of stationary points for the smoothing-regularization problem is still strongly-stationary under the VC-MFCQ and the condition of asymptotic nondegeneracy.

We consider the following mathematical program with vanishing constraints:

The MPVC was firstly introduced to the mathematical community in [

In this note, we give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints in [

The rest of the note is organized as follows. In Section

For convenience of discussion, some notations to be used in this paper are given. The

Firstly, we will introduce some definitions about the following optimization problem:

A point

A feasible point

A finite set of vectors

If the above system only has a solution

By using Motzkin’s theorem of the alternatives in [

A point

Now, we borrow notations from mathematical programs with complementarity constraints to define the following sets of active constraints in an arbitrary

A feasible point

Similar to Lemma

A point

A feasible point

It is easy to see that the VC-LICQ implies the VC-MFCQ. Moreover, the VC-LICQ (VC-MFCQ) is weaker than the MPVC-LICQ (MPVC-MFCQ) (See [

Let

Finally, we give the smoothing-regularization method of Problem (

Let

In this section, we will consider the improved convergence properties of a sequence of stationary points for the smoothing-regularization problem (

For convenience of discussion, we give the following notations:

Let

Since

Based on the above lemma, we can show the following theorem.

Let

Taking Lemma

We now prove that the standard MFCQ holds at

In Theorem

To establish the relations between the solutions of the original problem and those of the smoothing-regularization problem under the VC-MFCQ and the condition of asymptotic nondegeneracy, we give the following key lemma.

Let

It follows from Theorem

Assume that the sequence

By Lemma

Again, noting the condition of asymptotic nondegeneracy and the definitions of

Based on Lemma

Let

In Theorem

In this note, we have shown that the VC-LICQ assumption can be replaced by the weaker VC-MFCQ condition in order to get the strong stationarity for the smoothing-regularization approach to mathematical programs with vanishing constraints, which is proposed in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by NNSF (nos. 11371073, 11461015, 11361018) of China, Guangxi Natural Science Foundation (no. 2014GXNSFFA118001), and Guangxi Fund for Distinguished Young Scholars (no. 2012GXSFFA060003).