張量廣義高次特征值互補(bǔ)問(wèn)題解的一個(gè)刻劃

常肖蕊，凌　晨

(杭州電子科技大學(xué)理學(xué)院，浙江 杭州 310018)

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張量廣義高次特征值互補(bǔ)問(wèn)題解的一個(gè)刻劃

常肖蕊，凌晨

(杭州電子科技大學(xué)理學(xué)院，浙江 杭州 310018)

提出了一類張量廣義高次特征值互補(bǔ)問(wèn)題與非線性規(guī)劃之間的等價(jià)關(guān)系.進(jìn)一步給出了相應(yīng)非線性規(guī)劃問(wèn)題的穩(wěn)定點(diǎn)是張量廣義高次特征值互補(bǔ)問(wèn)題解的充要條件，最后，在特征值次數(shù)滿足一定條件下，證明了張量廣義高次特征值互補(bǔ)問(wèn)題可被轉(zhuǎn)化為張量高次特征值互補(bǔ)問(wèn)題.

高階張量；高次特征值互補(bǔ)問(wèn)題；非線性規(guī)劃；穩(wěn)定點(diǎn)

0　引　言

矩陣特征值互補(bǔ)問(wèn)題是一類特殊的非線性互補(bǔ)問(wèn)題，它具有廣泛的應(yīng)用背景[1-2].張量特征值互補(bǔ)問(wèn)題是矩陣特征值互補(bǔ)問(wèn)題的推廣，其求解是一個(gè)困難問(wèn)題.通常將張量特征值互補(bǔ)問(wèn)題轉(zhuǎn)化為等價(jià)的非線性規(guī)劃問(wèn)題來(lái)求解[3].本文提出了一類更為一般的張量廣義高次特征值互補(bǔ)問(wèn)題，并刻劃該問(wèn)題解的特征，從而說(shuō)明在一定條件下求解此問(wèn)題可轉(zhuǎn)化成求解一類非線性優(yōu)化問(wèn)題的穩(wěn)定點(diǎn).進(jìn)一步，在若干特殊情形下，張量廣義高次特征值互補(bǔ)問(wèn)題可轉(zhuǎn)化為張量高次特征值互補(bǔ)問(wèn)題.

1　概念與問(wèn)題

本文考慮以下形式的張量廣義高次特征值互補(bǔ)問(wèn)題(簡(jiǎn)記TGHDEiCP)，存在實(shí)數(shù)λ∈R和向量x∈Rn{0}，使得

(1)

若(λ,x)滿足式(1)，則稱λ為(A,B,C)的(k,l)次特征值，x為(A,B,C)的屬于λ的特征向量.此時(shí)，也稱(λ,x)為(A,B,C)的(k,l)次特征對(duì).這里，m是偶數(shù)，而k和l為滿足m≥k>l≥1的自然數(shù).顯然，若m=2，則k=2和l=1，此時(shí)上述問(wèn)題即為矩陣的二次特征值互補(bǔ)問(wèn)題.若A=0，記上述問(wèn)題為張量高次特征值互補(bǔ)問(wèn)題(THDEiCP)，它與張量特征值互補(bǔ)問(wèn)題(TEiCP)密切相關(guān)，此時(shí)稱(λ,x)為(B,C)的l次特征對(duì).

眾所周知，張量特征值問(wèn)題與多項(xiàng)式優(yōu)化關(guān)系密切.下面，研究TGHDEiCP的非線性規(guī)劃轉(zhuǎn)化形式并以此刻劃TGHDEiCP解的特征.

2　主要結(jié)果

針對(duì)A，B,C∈Tm,n考慮非線性規(guī)劃

minf(x,y,z,w,λ)

s.t.w-A zm-1-Bym-1-C xm-1=0,

xTe=1,

x≥0,w≥0.

(2)

(3)

稱滿足式(3)的(x,y,z,w,λ)為式(2)的穩(wěn)定點(diǎn)，并稱(α,η,δ,β,γ)為相應(yīng)的Lagrange乘子.

(4)

(5)

(6)

下面討論TGHDEiCP與式(2)的解之間的關(guān)系.

定理1設(shè)A, B, C∈Tm,n.則(A, B, C)有(k,l)次特征對(duì)，當(dāng)且僅當(dāng)式(2)有目標(biāo)函數(shù)值為0的全局最優(yōu)解.

上述定理表明，求解TGHDEiCP可等價(jià)轉(zhuǎn)化成求解非線性規(guī)劃的全局最優(yōu)解.然而，求解非線性規(guī)劃的全局最優(yōu)解仍是一件困難任務(wù)，下面的定理進(jìn)一步建立了全局最優(yōu)解與穩(wěn)定點(diǎn)之間的關(guān)系.

(7)

張量特征值互補(bǔ)問(wèn)題和通常的張量高次特征值互補(bǔ)問(wèn)題都是特殊的互補(bǔ)問(wèn)題，利用投影算法和交替方向算法可分別有效求得它們的解[3,7].下面討論TGHDEiCP與THDEiCP的關(guān)系.

即

(8)

由式(8)知

(9)

進(jìn)一步由式(9)知

(10)

從而由式(10)中第2式知

(11)

3　結(jié)束語(yǔ)

本文首先研究了一類張量廣義高次特征值互補(bǔ)問(wèn)題的非線性規(guī)劃轉(zhuǎn)化形式，將所考慮的張量廣義高次特征值互補(bǔ)問(wèn)題的求解轉(zhuǎn)化為相應(yīng)多項(xiàng)式優(yōu)化問(wèn)題的全局最優(yōu)解的求解.進(jìn)一步，刻劃了優(yōu)化問(wèn)題的穩(wěn)定點(diǎn)是張量廣義高次特征值互補(bǔ)問(wèn)題解的充要條件.同時(shí)，在特征值次數(shù)滿足k=2l的情形下，本文證明了張量廣義高次特征值互補(bǔ)問(wèn)題可以被轉(zhuǎn)化為相對(duì)較易求解的張量高次特征值互補(bǔ)問(wèn)題.這些結(jié)果均為以后設(shè)計(jì)求解張量廣義高次特征值互補(bǔ)問(wèn)題算法提供了新途徑.

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A Characterization of Solutions of Tensor Generalized Higher-degree Eigenvalue Complementarity Problem

CHANG Xiaorui, LING Chen

(SchoolofScience,HangzhouDianziUniversity,HangzhouZhejiang310018,China)

This paper proposes an equivalence relation between the considered tensor generalized higher-degree eigenvalue complementarity problem and nonlinear programming problems. Furthermore, a necessary and sufficient condition for the stationary point of the corresponding nonlinear programming problem being the solution of the tensor generalized higher-degree eigenvalue complementarity problem is given. Finally, under the degree of eigenvalues satisfies certain conditions, it is proved that the tensor generalized higher-degree eigenvalue complementarity problem can be transformed into the tensor higher-degree eigenvalue complementary problem.

higher-order tensor; higher-degree eigenvalue complementarity problem; nonlinear programming; stationary point

10.13954/j.cnki.hdu.2016.05.017

2016-03-11

國(guó)家自然科學(xué)基金資助項(xiàng)目(11571087)

常肖蕊(1991-)，女，河南濮陽(yáng)人，碩士研究生，非線性優(yōu)化.通信作者：凌晨教授，E-mail:macling@hdu.edu.cn.

O221.2

A

1001-9146(2016)05-0087-05