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In this paper, a tri-dimensional filter method for nonlinear programming was proposed. We add a parameter into the traditional filter for relaxing the criterion of iterates. The global convergent properties of the proposed algorithm are proved under some appropriate conditions.

This paper is concerned with finding a solution of a Nonlinear Programming (NLP) problem, as following

where

where

we also say that

Traditionally, this question has been answered by using penalty function. But it is difficult to find a suitable penalty parameter. In order to avoid the pitfalls of penalty function, Nonlinear programming problems (NLP) filter methods were first proposed by Fletcher in a plenary talk at the SIAM Optimization Conference in Victoria in May 1996; the methods are described in [

Motivated by the ideas of filter methods above, a tri-dimensional filter method for nonliner programming was proposed as acceptance criterion to judge whether to accept a trial step in our algorithm. We have following advantages:

1) By enhancing the flexibility of filter, motivated by [

2) The Maratos effect that makes good progress toward the solution may be rejected and has been avoided by using tri-dimensional filter method as acceptance criterion.

3) Tri-dimensional filter method can make full use of the information we get along the algorithm process.

This paper is divided into 4 sections. The next section introduces the concept of a Modified tri-dimensional filter and the NCP function. In Section 3, an algorithm of line search filter is given. The global convergence properties are proved in the last section.

The method that based on the Fischer-Burmeister NCP function are efficient, both theoretical results and computational experience. The Fischer-Burmeister function has a very simple structure

We know that:

if

Let

We denote

Clearly, the KKT optimality conditions (2) can be equivalently reformulated as the nonsmooth equations

If

where

If

and

We may reformulated the KKT (at point

where

Replace the violation constrained function

violation constrained function

If

otherwise we denote

Let

where H^{k} is a positive matrix which may be modified by BFGS update.

Definition 1.1 [

Definition 1.2 [

Definition 1.3 NCP pair and NCP functions [

Denote

A two dimensional filter is often used in traditional filter method, some information about convergent like the positions of iterates are neglected. Therefore, we aim to enhance its flexibility of filter. Motivated by [

We use pairs

If

We say that the algorithm does not make good improvement since we do not want to accept points with larger constraint violation. Thus, we try to impose stricter acceptance criterion. Meanwhile, we do not permit

If s_{k} moved into region P which is defined as

We say that the algorithm makes good improvement since it reduces not only the constraint violation, but also the penalty function value. So, we may loosen the acceptance criterion to wish more improvement. Here, we achieve this goal by reducing

In our algorithm, the trial step s_{k} is accepted by filter if

For all

then we say

In this section we hope that the Lagrange multiplier

Now, we consider how to update the penalty parameter. Let

satisfied, and the second order sufficient conditions are satisfied. Then when

minimizer of penalty function. So we force the condition at each iteration:

And also, since the penalty term aims to reduce the constraint violation we double the penalty parameter if the constraint violation could not reduce by half, that is

To summarize, we update the penalty parameter in the following formula:

The improved algorithm is presented as following.

Algorithm

Step 0. Initialization: Give a starting point

Step 1. Terimination test. If _{k} as a solution and stop.

Step 2. Computation of the search direction. compute

where

If

where

Step3. Liner search with filter

If

and let

Step 4. Acceptance criterion of the trial step

Let

Step 5. Paramenters update

Update

To present a proof of global convergence of algorithm, in this section, we always assume that the following conditions hold.

A1 The level set

A2

where

A3

for all

Lemma 1. If

Proof. If

and

From the definition of

Putting (14) into (12), we have

The fact that

(14).

The lemma 2 hold (see [

Lemma 2. If

Lemma 3. Consider an infinite sequence iterations on which

Proof. Suppose the theorem is not true, then exists an

The following lemma 4 - 5 hold (see [

Lemma 4.

Lemma 5. If

and

Theorem 1. If

It is obviously to prove the conclusion holds according to the above lemmas.

We thank the Editor and the referee for their comments. This work is supported by the National Natural Science Foundation of China (No. 11101115), the Natural Science Foundation of Hebei Province (No. 2014201033) and the Science and Technology project of Hebei province (No. 13214715).