# 非线性半定规划一个新的全局收敛算法A New Globally Convergent Algorithm for Nonlinear Semidefinite Programming

DOI: 10.12677/AAM.2018.74056, PDF, HTML, XML, 下载: 821  浏览: 1,316  国家自然科学基金支持

Abstract: In this paper, we present a sequence quadratic semidefinite programming (SSDP) algorithm for nonlinear semidefinite programming. At each iteration, the search direction is determined by solving two semidefinite programming sub problems; by introducing a distance function, a merit function is constructed for line search. Under some appropriate conditions, any accumulation point of iterative point sequence is either an infeasible stationary point, or a KKT point of the problem.

1. 引言

$\begin{array}{l}\mathrm{min}\begin{array}{cc}& f\left(x\right)\end{array}\\ \text{s}\text{.t}\text{.}\text{ }\text{ }\mathcal{A}\left(x\right)\underset{_}{\prec }0,\\ \begin{array}{ccc}& & \text{ }\text{\hspace{0.17em}}h\left(x\right)\end{array}=0,\end{array}$ (1)

· 产生搜索方向的子问题是相容的；

· 初始点任意，并且算法的线搜索保证效益函数的充分下降；

· 在一定条件下算法或收敛到问题的不可行稳定点，或收敛到问题的KKT点。

2. 算法

${ℝ}^{m×n}$ 为所有 $m×n$ 矩阵构成的集合， ${\mathbb{S}}_{+}^{m}\left({\mathbb{S}}_{++}^{m}\right)$ 为所有对称半正定(正定)矩阵构成的集合， ${\mathbb{S}}_{-}^{m}\left({\mathbb{S}}_{--}^{m}\right)$ 为所有对称半负定(负定)矩阵构成的集合。记NLSDP (0.1)的可行集为S。

$〈A,B〉:=\text{Tr}\left({B}^{\text{T}}A\right)=\underset{i=1}{\overset{m}{\sum }}\underset{j=1}{\overset{n}{\sum }}{a}_{ij}{b}_{ij},$

$Dh\left(x\right)=\left(\begin{array}{ccc}\frac{\partial {h}_{1}\left(x\right)}{\partial {x}_{1}}& \cdots & \frac{\partial {h}_{1}\left(x\right)}{\partial {x}_{n}}\\ ⋮& \ddots & ⋮\\ \frac{\partial {h}_{{m}_{2}}\left(x\right)}{\partial {x}_{1}}& \cdots & \frac{\partial {h}_{{m}_{2}}\left(x\right)}{\partial {x}_{n}}\end{array}\right).$ (1.1)

$D\mathcal{A}\left(x\right)={\left(\frac{\partial \mathcal{A}\left(x\right)}{\partial {x}_{1}},\cdots ,\frac{\partial \mathcal{A}\left(x\right)}{\partial {x}_{n}}\right)}^{\text{T}},$

$D\mathcal{A}\left(x\right)d:=\underset{i=1}{\overset{n}{\sum }}{d}_{i}\frac{\partial \mathcal{A}\left(x\right)}{\partial {x}_{i}}.$ (1.2)

${V}^{*}Z={\left(〈{V}_{1},Z〉,\cdots ,〈{V}_{n},Z〉\right)}^{\text{T}},\forall Z\in {\mathbb{S}}^{{m}_{1}}.$

NLSDP (0.1)的Lagrange函数定义为

(1.3)

(1.4a)

(1.4b)

，定义

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

，则算法停止；否则，转步骤3。

(1.10)

(1.11)

(1.2a)

(1.12b)

(1.12c)

(1.12d)

A1：函数是连续可微的。

A2：存在正数a和使得

A3：由算法A产生的迭代点列是有界的。

A4：对任意的，Mangasarian-Fromovitz约束规格(MFCQ)成立，即

1)行满秩；

2) 存在非零向量，使得

，此时若为NLSDP (0.1)的不可行点，由不可行稳定点的定义知为NLSDP (0.1)的不可行稳定点。因此综合以上所述引理结论得证。,

(1.13a)

(1.13b)

(1.13c)

(1.13d)

,

3. 全局收敛性

(2.1)

(2.2)

(2.3)

a) 证明存在，使得

(2.4)

b) 基于，将导出矛盾。

(2.5a)

(2.5b)

(2.5c)

(2.5d)

(2.5e)

4. 结束语

NOTES

*通讯作者。

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