| 摘要: |
| 针对损失函数为最小一乘问题，惩罚项由基数函数定义的绝对值优化问题，提出用MCP (Minimax Concave Penalty)非凸正则来连续逼近基数罚，得到一个精确连续的绝对值优化松弛问题。首先，证明了带基数罚的绝对值优化问题的全局最优解；其次，研究了带基数罚的绝对值优化问题与带MCP罚的绝对值优化松弛问题之间全局最优解的等价性；最后，证明了在一定的条件下这两个绝对值优化问题具有相同的全局最优解。 |
| 关键词: 绝对值优化问题 MCP 最优解 等价性 |
| DOI： |
| 分类号: |
| 基金项目: |
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| The Equivalence of Solutions of Two Absolute Value Optimization Problems |
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LUO Xiao-min, PENG Ding-tao
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School of Mathametics and Statistics, Guizhou University, Guiyang 550025,China
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| Abstract: |
| This paper focuses on the absolute value optimization problem, where the loss function is least absolute deviation and the penalty term is defined by the cardinality function. In order to obtain an exact continuous relaxation problem, we use MCP (minimax concave penalty) to approximate the cardinality penalty. Firstly, we prove the global optimal solution of the absolute value optimization problem with cardinal penalty. Then we discuss the equivalence of the global optimal solution between the absolute value optimization problem with cardinal penalty and the absolute value optimization relaxation problem with MCP penalty. Finally, under some mild conditions, we proved that the two problems have the same optimal solutions. |
| Key words: absolute value optimization problem MCP optimal solution equivalence |
