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The problems studied are the separable variational inequalities with linearly coupling constraints. Some existing decomposition methods are very problem specific, and the computation load is quite costly. Combining the ideas of proximal point algorithm (PPA) and augmented Lagrangian method (ALM), we propose an asymmetric proximal decomposition method (AsPDM) to solve a wide variety separable problems. By adding an auxiliary quadratic term to the general Lagrangian function, our method can take advantage of the separable feature. We also present an inexact version of AsPDM to reduce the computation load of each iteration. In the computation process, the inexact version only uses the function values. Moreover, the inexact criterion and the step size can be implemented in parallel. The convergence of the proposed method is proved, and numerical experiments are employed to show the advantage of AsPDM.

The original model considered here is the convex minimization problem with linearly coupling constraints:

One of the best-known algorithms for solving convex programming or equivalent VI is the

Note that the process in SALM for

To our best knowledge, there are few dedicated methods for solving inequality constraints problems (

Here,

The separable VI(

We summarize some basic properties and related definitions which will be used in the following discussions.

(i) The mapping

(ii) A function

The projection onto a closed convex set is a basic concept in this paper. Let

Let

See [

Let

See [

Hence, solving VI

In each iteration, by our proper construction, our method solves

Phase I: For arbitrary

Phase II: After the

We make the standard assumptions to guarantee that the problem under consideration is solvable and the proposed methods are well defined.

By this assumption, it is easy to get that

In this section, the inexact version of the AsPDM method is present, and some remarks are briefly made.

For later analysis convenience, we denote

The first phase of our method works as follows. At the beginning of

Once one of the above criteria fails to be satisfied, we will increase

In what follows, let us describe the second phase. We require

In the inexact AsPDM, the main task of Phase I is to find a solution for (

Combining (

Recalling that

Since

The update form (

Note that

Let

According to Definition (

Set

Note that, in our proposed method, problems VI (

In the proposed methods, the first phase (accomplished by the local administrators) offers a descent direction of the unknown distance function, and the second phase (accomplished by the central authority) determines the “optimal” step length along this direction. This section gives more theory analysis.

For any

Assume that

Since

Now, we state the main properties of

Let

Recalling that

Since

Let

It follows from Definition (

In fact,

Assume that

Using the fact that square matrix

Now, we are in the stage to prove the main convergence theorem of this paper.

For any

First, it follows from (

Since we use

This section describes experiments testifying to the good performance of proposed method. The algorithms were written in Matlab (version 7.0) and complied on a notebook with CPU of Intel Core 2 Duo (2.01 GHz and RAM of 0.98 GB).

To evaluate the behavior of the proposed method, we construct examples about convex separable quadratic programming (CSQP) with linearly coupling constraints. The convex separable quadratic programming was generated as follows:

In the first experiment, we employ AsPDM with update

AsPDM for CSQP with 3 separable operators.

Its. | |||||
---|---|---|---|---|---|

100 | 50 | 50 | 50 | 674 | 4066 |

100 | 100 | 100 | 100 | 1581 | 9508 |

150 | 150 | 150 | 150 | 1775 | 10672 |

200 | 200 | 200 | 200 | 2108 | 12670 |

Next, we compared the computational efficiency of AsPDM against the method in [

Its. and function eval. for different problem sizes.

AsPDM | PCM | |||||

Its. | Its. | |||||

100 | 100 | 100 | 1089 | 4371 | 1343 | 5393 |

200 | 200 | 200 | 1265 | 5075 | 1649 | 6621 |

300 | 300 | 300 | 1655 | 6635 | 1765 | 7087 |

400 | 400 | 400 | 1573 | 6307 | 1834 | 7365 |

500 | 500 | 500 | 2299 | 9211 | 2218 | 8901 |

600 | 600 | 600 | 2267 | 9083 | 2289 | 9187 |

100 | 80 | 80 | 568 | 2287 | 616 | 2485 |

100 | 50 | 150 | 1192 | 4787 | 1231 | 4945 |

200 | 150 | 100 | 567 | 2283 | 572 | 2311 |

200 | 100 | 200 | 744 | 2991 | 846 | 3407 |

Error versus iteration number for the method and for with

In addition to being fast, AsPDM can solve the problem separately; that is the most significant advantage over other methods. Hence, AsPDM is more suitable to solve the real-life separable problems.

We have proposed AsPDM for solving separable problems. It decomposes the original problem to independent low-dimension subproblems and solves those subproblems in parallel. Only the function values is required in the process, and the total computational cost is very small. AsPDM is easy to implement and does not appear to require application-specific tuning. The numerical results also evidenced the efficiency of our method. Thus, the new method is applicable and recommended in practice.

The author was supported by the NSFC Grant 70901018.