# American Institute of Mathematical Sciences

January  2019, 15(1): 59-79. doi: 10.3934/jimo.2018032

## The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints

 a, b. School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, China c. School of Finance, Zhejiang University of Finance and Economics, Zhejiang 310018, China

* Corresponding author: Li-Ping Pang

Received  February 2017 Revised  November 2017 Published  April 2018

Fund Project: The first author is supported by Huzhou science and technology plan on No.2016GY03.

We study the convergence of the log-exponential regularization method for mathematical programs with vertical complementarity constraints (MPVCC). The previous paper assume that the sequence of Lagrange multipliers are bounded and it can be found by software. However, the KKT points can not be computed via Matlab subroutines exactly in many cases. We note that it is realistic to compute inexact KKT points from a numerical point of view. We prove that, under the MPVCC-MFCQ assumption, the accumulation point of the inexact KKT points is Clarke (C-) stationary point. The idea of inexact KKT conditions can be used to define stopping criteria for many practical algorithms. Furthermore, we introduce a feasible strategy that guarantees inexact KKT conditions and provide some numerical examples to certify the reliability of the approach. It means that we can apply the inexact regularization method to solve MPVCC and show the advantages of the improvement.

Citation: Liping Pang, Na Xu, Jian Lv. The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints. Journal of Industrial & Management Optimization, 2019, 15 (1) : 59-79. doi: 10.3934/jimo.2018032
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##### References:
The numerical results for Example 2
 t $(y^t_1, y^t_2, y^t_3, y^t_4, z^t_1, z^t_2)$ $f^t$ 0.2 (0.0592, -0.4868, 0.3863, 0.2741, 1.0058, 0.4937) 1.7496 0.01 (-0.0000, -0.4999, 0.4011, 0.1994, 1.0001, 0.4999) 1.6929 0.005 ( 0.0000, -0.5000, 0.3997, 0.1998, 1.0000, 0.5000) 1.6901
 t $(y^t_1, y^t_2, y^t_3, y^t_4, z^t_1, z^t_2)$ $f^t$ 0.2 (0.0592, -0.4868, 0.3863, 0.2741, 1.0058, 0.4937) 1.7496 0.01 (-0.0000, -0.4999, 0.4011, 0.1994, 1.0001, 0.4999) 1.6929 0.005 ( 0.0000, -0.5000, 0.3997, 0.1998, 1.0000, 0.5000) 1.6901
The numerical results for Example 4, 5
 Example Algorithm $z$ $f$ $Gap$ 4 Algorithm 2 (0.0000, 2.0000) 0.0000 100 % fmincon (0.0004, 2.0000) 0.0000 99.98 % ADH (0.0000, 1.9988) 0.0000 99.94 % AH (-0.0000, 1.9999) 0.0000 100 % $Polak^1$ (0.0000, 1.8708) 0.0167 93.54 % 5 Algorithm 2 (0.7500, 0.0000) 0.0625 100 % fmincon (0.7500, 0.0003) 0.0625 99.96 % ADH (0.7500, 0.0000) 0.0625 100 % AH (0.7500, 0.0000) 0.0625 100 % $Polak^1$ (0.7500, 0.0000) 0.0625 100 %
 Example Algorithm $z$ $f$ $Gap$ 4 Algorithm 2 (0.0000, 2.0000) 0.0000 100 % fmincon (0.0004, 2.0000) 0.0000 99.98 % ADH (0.0000, 1.9988) 0.0000 99.94 % AH (-0.0000, 1.9999) 0.0000 100 % $Polak^1$ (0.0000, 1.8708) 0.0167 93.54 % 5 Algorithm 2 (0.7500, 0.0000) 0.0625 100 % fmincon (0.7500, 0.0003) 0.0625 99.96 % ADH (0.7500, 0.0000) 0.0625 100 % AH (0.7500, 0.0000) 0.0625 100 % $Polak^1$ (0.7500, 0.0000) 0.0625 100 %
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