羅孝敏 彭定濤

摘 要：針對(duì)損失函數(shù)為最小一乘，懲罰項(xiàng)由基數(shù)函數(shù)定義的稀疏回歸問(wèn)題，用SCAD（smoothly clipped absolute deviation）罰來(lái)連續(xù)逼近基數(shù)罰，得到一個(gè)連續(xù)的松弛問(wèn)題，研究SCAD罰問(wèn)題與原基數(shù)罰問(wèn)題之間解的等價(jià)性。首先，證明了SCAD罰松弛模型的下界性質(zhì)，并借助此下界性質(zhì)分析了原問(wèn)題與松弛問(wèn)題之間解的等價(jià)性，證明了在一定條件下兩個(gè)問(wèn)題具有相同的全局最優(yōu)解以及最優(yōu)值。此外，證明了松弛模型的局部最優(yōu)解是原問(wèn)題的局部最優(yōu)解并且在局部極小值點(diǎn)處松弛模型與原問(wèn)題的目標(biāo)值相等。

關(guān)鍵詞：基數(shù)罰問(wèn)題;SCAD;解的等價(jià)性

中圖分類號(hào)：O224?? 文獻(xiàn)標(biāo)識(shí)碼： A

參考文獻(xiàn)：

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（責(zé)任編輯：曾 晶）

Smoothly Clipped Absolute Deviation （SCAD） for Least Absolute

Deviation Regession Regularied Problem

LUO Xiaomin， PENG Dingtao*

（School of Mathemetics and Statistics， Guizhou University， Guiyang 550025，China）

Abstract：

For the sparse regression problem where the loss function is the least absolute deviation and the penalty term is the cardinal penalty， we use SCAD（smoothly clipped absolute deviation） penalty to relax the cardinal penalty. We focus on the equivalence of solutions between the relaxed problem and the original problem. Firstly， the lower bound theory property of the relaxed model is proved， and the equivalence between the original problem and the relaxed problem is analyzed under the lower bound property. It is proved that the two problems have the same global optimal solution and optimal value under certain conditions. In addition， it is proved that the local optimal solution of the relaxed model is the local optimal solution of the original problem， and the relaxed module is at the local minimum point. The optimal value of type A is equal to that of the original problem.

Key words：

cardinal penalty problem; SCAD; equivalence of solutions

收稿日期：2020-01-14

基金項(xiàng)目：國(guó)家自然科學(xué)基金資助項(xiàng)目（11861020）;貴州省高層次留學(xué)人才創(chuàng)新創(chuàng)業(yè)擇優(yōu)資助重點(diǎn)項(xiàng)目（[2018]03）;貴州省科技計(jì)劃資助項(xiàng)目（[2018]5781）;貴州省青年科技人才成長(zhǎng)資助項(xiàng)目（[2018]121）

作者簡(jiǎn)介：羅孝敏（1993-），女，在讀碩士，研究方向：稀疏優(yōu)化，Email：lxm2440775499@163.com.

通訊作者：彭定濤，Email：dingtaopeng@126.com.