多目标优化问题解的概念与数值优化问题解的概念有着本质的不同.这是因为序线性空间中的序关系是非完全的偏序关系.因此, 如何界定多目标优化问题解的概念是多目标优化理论的一个首要问题.对此, 人们引进了各种意义下解的概念.主要有:有效解, 弱有效解和各种真有效解.由于这些解在非紧的情况下往往不一定存在, 而近似解在很弱的情况下都可能存在(Ekelend变分原理), 并且通过数值算法得到的大多都是近似解.因此, 研究近似解不仅有理论价值而且有实际意义. 1979年, Kutateladze ^[3]首次提出了数值优化问题$\varepsilon$-近似解的概念.随后, 1984年, Loridan ^[4]在文[3]的基础上, 引进了多目标优化问题的$\varepsilon$-有效解的概念, 并研究了$\varepsilon$-有效解的一些性质. 1986年, White ^[5]研究了多目标优化问题的六种$\varepsilon$-近似有效解.随后一些学者相继提出其它新的近似解的概念, 并对此进行了理论研究, 见文献[6-10].

在求解多目标优化问题的方法中, 将多目标优化问题转化为数值优化问题是一个重要途径.即, 对多目标优化问题进行标量化刻画.近几年来, 关于多目标优化问题近似解的标量化研究取得了很多有意义的结果(见文[8-14, 17]). 2007年, Engau和Wiecek ^[11]给出了多目标优化问题近似解在凸性条件下的线性标量化和在没有任何凸性条件下的几种非线性标量化. 2008年, Miruna Beldiman ^[12]等人给出了多目标优化问题$(\varepsilon, \overline{\varepsilon})$ -拟近似(弱, 真)有效解的概念, 并在择一性定理的基础上, 利用范数对多目标优化问题$(\varepsilon, \overline{\varepsilon})$ -拟近似(弱, 真)有效解进行了非线性标量化刻画. 2011年, Ghaznavi和Khorram ^[13]提供了$\varepsilon$ -(弱, 真)有效解的一些充分和必要条件. 2013年, Ghaznavi, Khorram和Soleimani-Damaneh ^[14]进一步提出了$\varepsilon$ -(弱, 真)有效解的一些充分和必要条件. 2014年, Rastegar and Khorram ^[1]在文[11, 13, 14]的基础上, 给出了统一的非线性标量化模型, 考虑了多目标优化问题$\varepsilon$ -拟近似(弱, 真)有效解的一些充分和必要条件.本文在文[1]的基础上, 在没有任何凸性条件下, 研究了多目标优化问题$(\varepsilon, \overline{\varepsilon})$ -拟近似(弱, 真)有效解的一些充分和必要条件.最后, 利用文献[2]中给出的范数, 对多目标优化问题的$(\varepsilon, \overline{\varepsilon})$ -拟近似解进行了非线性标量化刻画.

本文的结构如下:在第2节, 给出了一些基本定义和结论, 在第3节, 利用文献[1]中统一的非线性标量化刻画了多目标优化问题$(\varepsilon, \overline{\varepsilon})$ -拟近似(弱, 真)有效解; 在第4节, 利用已有的范数(见文献[2])给出了一种非线性标量化问题, 并给出了多目标优化问题$(\varepsilon, \overline{\varepsilon})$ -拟近似解的充分条件, 举例说明了其逆命题不成立.

在本文中, 给出以下符号:对任意的$x, y\in R^{n}, $

$x<y\Leftrightarrow y-x\in \text{int}R_{+}^{n},\\x\le y\Leftrightarrow y-x\in R_{+}^{n}\backslash \{0\},\\x\leqq y\Leftrightarrow y-x\in R_{+}^{n}.$

考虑如下的多目标优化问题

$\begin{array}{*{35}{l}} (\text{MOP}) & \min & f(x)=({{f}_{1}}(x),{{f}_{2}}(x),\cdots ,{{f}_{p}}(x)), \\ {} & \text{s}.\text{t}. & {{g}_{j}}(x)\leqq 0,j=1,2,\cdots ,m, \\ {} & {} & {{h}_{k}}(x)=0,k=1,2,\cdots ,s, \\ \end{array}$

其中, $\emptyset\neq\Omega\subset R^{n}, f_{i}, g_{j}, h_{k}:\Omega\rightarrow R, i=1,2,\cdots,p, j=1,2,\cdots,m, k=1,2,\cdots,s,$ $X$表示$(\rm MOP)$的可行域.

定义2.1^[12] 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}, \widehat{x}\in X $.

(1) $ \widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解, 若不存在$x\in X,$使得

$f(x)\le f(\hat{x})-\varepsilon ||x-\hat{x}||-\bar{\varepsilon },\forall x\in X.$

(2) $ \widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似弱有效解, 若不存在$x\in X,$使得

$f(x)<f(\hat{x})-\varepsilon ||x-\hat{x}||-\bar{\varepsilon },\forall x\in X.$

(3) $ \widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解, 若存在$M>0$, 使得对于满足$ f_{i}(x)<f_{i}(\widehat{x})-\varepsilon_{i}||x-\widehat{x}||-\overline{\varepsilon}_{i}$的任何$i\in \{1,2,\cdots,p\}$和$x\in X ,$存在$j \in \{1,\cdots,p\}\setminus\{i\} , $使得$f_{j}(x)>f_{j}(\widehat{x})-\varepsilon_{j}||x-\widehat{x}||-\overline{\varepsilon}_{j}$且

$\frac{{{f}_{i}}(\hat{x})-{{f}_{i}}(x)-{{\varepsilon }_{i}}||x-\hat{x}||-{{{\bar{\varepsilon }}}_{i}}}{{{f}_{j}}(x)-{{f}_{j}}(\hat{x})+{{\varepsilon }_{j}}||x-\hat{x}||+{{{\bar{\varepsilon }}}_{j}}}\leqq M.$

考虑如下单目标优化问题

$\begin{array}{*{35}{l}} (\text{P}) & \min & \varphi (x), \\ {} & \text{s}.\text{t}. & x\in X. \\ \end{array}$

定义2.2^[12] 设$(\epsilon,\overline{\epsilon})\in R_{+}\times R_{+}, \widehat{x}\in X,$称$\widehat{x}$是$(\rm P)$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 若$\forall x\in X$, 有

$\varphi (\hat{x})\leqq \varphi (x)+\epsilon ||x-\hat{x}||+\bar{\epsilon }.$

Rastegar和Khorram在文献[1]中考虑了如下统一的非线性标量化问题,

$\begin{array}{*{35}{l}} (SO{{P}_{\lambda ,\mu ,\gamma }}) & \min \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}s_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}s_{i}^{-}, \\ {} & {{f}_{i}}(x)+s_{i}^{+}-s_{i}^{-}\le {{\alpha }_{i}},i=1,\cdots ,p, \\ {} & x\in X,{{s}^{+}},{{s}^{-}}\ge 0, \\ \end{array}$

其中, $\lambda_{i}, \mu_{i}, \gamma_{i}\geq0,$ $\alpha_{i}$是$f_{i}(x)+s_{i}^{+}-s_{i}^{-}$的一个上界, $\forall i=1,\cdots,p$. $X^{'}$表示$(\rm SOP_{\lambda,\mu,\nu})$的可行域.

特别地当$\gamma=0$或$s^{+}=0$时, $(\rm SOP_{\lambda,\mu,\nu})$可化为$(\rm SOP_{\lambda,\mu}).$

$\begin{array}{*{35}{l}} (\text{SO}{{\text{P}}_{\lambda ,\mu }}) & \min \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{s}_{i}}, \\ {} & {{f}_{i}}(x)-{{s}_{i}}\le {{\alpha }_{i}},i=1,\cdots ,p, \\ {} & x\in X,s\ge 0, \\ \end{array}$

其中, $\lambda_{i}, \mu_{i}\geq0,$ $\alpha_{i}$是$f_{i}(x)-s_{i}$的一个上界, $\forall i=1,\cdots,p$. $X^{'}$表示$(\rm SOP_{\lambda,\mu})$的可行域.

下面给出$(\rm SOP_{\lambda,\mu,\gamma})$和$ (\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似解的定义.

定义2.3 设$(\epsilon,\overline{\epsilon})\in R_{+}\times R_{+}, \widehat{x}\in X $.

(1) $ (\widehat{x},\widehat{s}^+,\widehat{s}^-) $是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 若

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\\le \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}s_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}s_{i}^{-}+\epsilon ||x-\hat{x}||+\bar{\epsilon },\forall (x,{{s}^{+}},{{s}^{-}})\in {{X}^{'}}.$

(2) $ (\widehat{x},\widehat{s}) $是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 若

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}}\le \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{s}_{i}}+\epsilon ||x-\hat{x}||+\bar{\epsilon },\forall (x,s)\in {{X}^{'}}.$

引理2.4^[6] 设$\lambda\in R^p,\lambda>0, \sum\limits_{i=1}^{p}\lambda_{i}=1, \epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}.$若$\overline{x}$是$(\rm P_{\lambda})$的$\epsilon $ -近似最优解, 则$\overline{x}$是(MOP)的$\varepsilon$ -近似真有效解.

$\begin{array}{*{35}{l}} ({{\text{P}}_{\lambda }}) & \min & {{\lambda }^{T}}f(x), \\ {} & \text{s}.\text{t}. & x\in X. \\ \end{array}$

由引理2.4容易得出以下结果.

引理2.5  2.设$\lambda\in R^p,\lambda>0, \sum\limits_{i=1}^{p}\lambda_{i}=1, \epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}\lambda_{i} \overline{\varepsilon}_{i}$.若$\overline{x}$是$(\rm P_{\lambda})$的$(\epsilon,\overline{\epsilon} )$ -拟近似最优解, 则$\overline{x}$是(MOP)的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.

在文献[1]中, Rastegar和Khorram考虑了(MOP)问题的$\varepsilon$ -近似解的一些充分和必要条件.本节, 将在没有任何凸性条件下, 运用非线性标量化问题$(\rm SOP_{\lambda,\mu,\gamma})$和$ (\rm SOP_{\lambda,\mu})$来刻画(MOP)的$(\varepsilon,\overline{\varepsilon})$ -拟近似(弱, 真)有效解.

定理3.1 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}, \epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}(\lambda_{i}+\gamma_{i})\overline{\varepsilon_{i}}, (\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解,

(i)若$\lambda+\gamma>0$, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(ii)若$0\leq\varepsilon<\widehat{s}^{-},\lambda+\mu>0$, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

证 因为$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 故

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\\le \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}s_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}s_{i}^{-}+\epsilon ||x-\hat{x}||+\bar{\epsilon },\forall (x,{{s}^{+}},{{s}^{-}})\in \text{ }{{X}^{'}},$

且

${{f}_{i}}(x)+s_{i}^{+}-s_{i}^{-}\le {{\alpha }_{i}},i=1,\cdots ,p,\hat{x}\in X,{{\hat{s}}^{+}},{{\hat{s}}^{-}}\ge 0.$

(i)若$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解, 则存在$ x\in X$使得

$f(x)\le f(\hat{x})-\varepsilon ||x-\hat{x}||-\bar{\varepsilon },$

即对任意的$i\in\{1,2,\cdots,p\}$有

${{f}_{i}}(x)\le {{f}_{i}}(\hat{x})-{{\varepsilon }_{i}}||x-\hat{x}||-{{\bar{\varepsilon }}_{i}},\text{ }$

(3.1)

存在$j\in\{1,\cdots,p\}$满足

${{f}_{j}}(x)<{{f}_{j}}(\hat{x})-{{\varepsilon }_{j}}||x-\hat{x}||-{{\bar{\varepsilon }}_{j}}.\text{ }$

(3.2)

从而对满足(3.1) 式的任意的$i$有

${{f}_{i}}(x)+{{\varepsilon }_{i}}||x-\hat{x}||+{{\bar{\varepsilon }}_{i}}+\hat{s}_{i}^{+}-\hat{s}_{i}^{-}\le {{f}_{i}}(\hat{x})+\hat{s}_{i}^{+}-\hat{s}_{i}^{-}\le {{\alpha }_{i}},$

且对满足(3.2) 式的任意的$j$有

${{f}_{j}}(x)+{{\varepsilon }_{j}}||x-\hat{x}||+{{\bar{\varepsilon }}_{j}}+\hat{s}_{j}^{+}-\hat{s}_{j}^{-}+{{v}_{j}}\le {{f}_{j}}(\hat{x})+\hat{s}_{j}^{+}-\hat{s}_{j}^{-}\le {{\alpha }_{j}},~{{v}_{j}}>0.$

令$s_{i}^{+}=\widehat{s}_{i}^{+}+\overline{\varepsilon}_{i},s_{j}^{+}=\widehat{s}_{j}^{+}+\overline{\varepsilon}_{j}+v,$则$(x,s^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的可行解.又由假设$\lambda_{j}+\gamma_{j}>0, j=1,2,\cdots,p$可知

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}s_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\=\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}(\hat{s}_{i}^{+}+{{{\bar{\varepsilon }}}_{i}})-{{\gamma }_{j}}v+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\=\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{{\bar{\varepsilon }}}_{i}}-{{\gamma }_{j}}v\\<\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}({{f}_{i}}(\hat{x})-{{\varepsilon }_{i}}||x-\hat{x}||-{{{\bar{\varepsilon }}}_{i}})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{{\bar{\varepsilon }}}_{i}}\\=\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\varepsilon }_{i}}||x-\hat{x}||-\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\gamma }_{i}})}{{{\bar{\varepsilon }}}_{i}}\\\le \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\epsilon ||x-\hat{x}||-\bar{\epsilon },$

这与$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解矛盾.因此, $\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(ii)若$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$拟近似有效解, 则(3.1) 和(3.2) 式成立.类似与(i), 令$s_{i}^{-}=\widehat{s}_{i}^{-}-\varepsilon_{i}, s_{j}^{-}=\widehat{s}_{j}^{-}-\varepsilon_{j}-v. $则$(x,\widehat{s}^{+},s^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的可行解, 且有

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}s_{i}^{-}\\ <\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\varepsilon }_{i}}||x-\hat{x}||-\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\mu }_{i}})}{{{\bar{\varepsilon }}}_{i}}\\ \le \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\epsilon ||x-\hat{x}||-\bar{\epsilon },$

这与$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解矛盾.因此, $\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

定理3.2 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}$,

(i)设$\epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}(\lambda_{i}+\gamma_{i})\overline{\varepsilon}_{i}, $ $\lambda+\gamma\geq0$.若$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似弱有效解.

(ii)设$\epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}(\lambda_{i}+\mu_{i})\overline{\varepsilon}_{i}, $ $0\leqq\varepsilon<\widehat{s}^{-}, \lambda+\mu\geq0$.若$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -(严格)拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -(拟近似有效解)拟近似弱有效解.

(iii)设$\epsilon<\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}<\sum\limits_{i=1}^{p}(\lambda_{i}+\gamma_{i})\overline{\varepsilon}_{i}, $ $\lambda+\gamma\geq0$.若$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(iv)设$\epsilon<\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}<\sum\limits_{i=1}^{p}(\lambda_{i}+\mu_{i})\overline{\varepsilon}_{i}, $ $0\leq\varepsilon<\widehat{s}^{-}, \lambda+\mu\geq0$.若$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(v)设$\epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}(\lambda_{i}+\gamma_{i})\overline{\varepsilon}_{i},$ $\lambda+\gamma\geq0$.若$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -严格拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

证 与定理3.1的证明类似.

定理3.3 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}, 0\leq\epsilon\leq\sum\limits_{i=1}^{p}(\lambda_{i}+\gamma_{i})\varepsilon_{i}, 0\leq\overline{\epsilon}\leq\sum\limits_{i=1}^{p}(\lambda_{i}+\gamma_{i})\overline{\varepsilon_{i}}, \gamma>0, \sum\limits_{i=1}^{p}\lambda_{i}=1.$若$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, $f_{i}(\widehat{x})+\widehat{s}_{i}^{+}-\widehat{s}_{i}^{-}<\alpha_{i}, 1\leq i\leq p,$则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.

证 由定理3.1, $\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

下面证明$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.若$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解, 则存在满足$\lim\limits_{\beta\to\infty}M_{\beta}=\infty$的正数列$\{M_{\beta}\}$, 使得对任意的$M_{\beta}, $存在满足$f_{i}(x_{\beta})<f_{i}(\widehat{x})-\varepsilon_{i}||x_{\beta}-\widehat{x}||-\overline{\varepsilon}_{i}$的$ x_{\beta}\in X$和$ i\in\{1,2,\cdots,p\}$, 使得对任意满足$f_{j}(\widehat{x})-\varepsilon_{j}||x_{\beta}-\widehat{x}||-\overline{\varepsilon}_{j}<f_{j}(x_{\beta})$的$j\in\{1,\cdots,p\}\setminus\{i\}$, 总有

$\begin{eqnarray} \frac{f_{i}(\widehat{x})-f_{i}(x_{\beta})-\varepsilon_{i}||x_{\beta}-\widehat{x}||-\overline{\varepsilon}_{i}}{f_{j}(x_{\beta})-f_{j}(\widehat{x})+\varepsilon_{j}||x_{\beta}-\widehat{x}||+\overline{\varepsilon}_{j}}>M_{\beta}.\nonumber \end{eqnarray}$

设$Q=\{j:f_{j}(x_{\beta})>f_{j}(\widehat{x})-\varepsilon_{j}||x_{\beta}-\widehat{x}||-\overline{\varepsilon}_{j}\}\neq\varnothing$.任取$j\in\{1,2,\cdots,p\}$, 下面分两种情况讨论:

情形1 若$j\notin Q $, 则

${{f}_{j}}({{x}_{\beta }})\le {{f}_{j}}(\hat{x})-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}}<{{\alpha }_{j}}-\hat{s}_{j}^{+}+\hat{s}_{j}^{-}-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}},$

从而

${{f}_{j}}({{x}_{\beta }})+\hat{s}_{j}^{+}-\hat{s}_{j}^{-}+{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||+{{\bar{\varepsilon }}_{j}}<{{\alpha }_{j}}.$

因此存在$ v_{\beta j}>0$使得

${{f}_{j}}({{x}_{\beta }})+\hat{s}_{j}^{+}-\hat{s}_{j}^{-}+{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||+{{\bar{\varepsilon }}_{j}}+{{v}_{\beta j}}\le {{\alpha }_{j}}.$

情形2 若$ j\in Q,$则$f_{j}(x_{\beta})>f_{j}(\widehat{x})-\varepsilon_{j}||x_{\beta}-\widehat{x}||-\overline{\varepsilon}_{j}.$因为$f(X)$有界, $\lim\limits_{\beta\to\infty}M_{\beta}=\infty,$故$f_{j}(x_{\beta})-f_{j}(\widehat{x})+\varepsilon_{j}||x_{\beta}-\widehat{x}||+\overline{\varepsilon}_{j}\to 0,(\beta\to \infty),$因此有

$\underset{\beta \to \infty }{\mathop{\lim }}\,{{f}_{j}}({{x}_{\beta }})={{f}_{j}}(\hat{x})-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}}\text{ }{{\alpha }_{j}}-\hat{s}_{j}^{+}+\hat{s}_{j}^{-}-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}},$

从而存在$\beta_{0}>0$使得$f_{j}(x_{\beta})+\widehat{s}_{j}^{+}-\widehat{s}_{j}^{-}+\varepsilon_{j}||x_{\beta}-\widehat{x}||+\overline{\varepsilon}_{j}<\alpha_{j}, \forall\beta\geq\beta_{0}.\nonumber $所以对任意的$\beta\geq\beta_{0},$存在$ v_{\beta j}$使得

${{f}_{j}}({{x}_{\beta }})+\hat{s}_{j}^{+}-\hat{s}_{j}^{-}+{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||+{{\bar{\varepsilon }}_{j}}+{{v}_{\beta j}}\le {{\alpha }_{j}}.$

令$s_{\beta j}^{+}=\widehat{s}_{j}^{+}+\varepsilon_{j}||x_{\beta}-\widehat{x}||+\overline{\varepsilon}_{j}+v_{\beta j}, j=1,2,\cdots,p, \beta\geq\beta_{0}, $则对任意的$\beta\geq\beta_{0},$ $(x_{\beta},s_{\beta}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的可行解.

另一方面, 当$j\in Q,\beta\geq\beta_{0}$时, 有$\lim\limits_{\beta\to\infty}f_{j}(x_{\beta})=f_{j}(\widehat{x})-\varepsilon_{j}||x_{\beta}-\widehat{x}||-\overline{\varepsilon}_{j},$从而

$\underset{\beta \to \infty }{\mathop{\lim }}\,(-{{f}_{j}}({{x}_{\beta }})+{{f}_{j}}(\hat{x})-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}}+{{\gamma }_{1}}{{v}_{\beta 1}})={{\gamma }_{1}}{{v}_{\beta 1}}>0.$

因此存在$\beta_{0}^{'}>\beta_{0},$对任意的$\beta>\beta_{0}^{'},$有

${{f}_{j}}({{x}_{\beta }})<{{f}_{j}}(\hat{x})-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}}+{{\gamma }_{1}}{{v}_{\beta 1}}.\text{ }$

(3.3)

当$j\notin Q$时,

${{f}_{j}}({{x}_{\beta }})\le {{f}_{j}}(\hat{x})-{{\varepsilon }_{j}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{j}}.\text{ }$

(3.4)

取$\beta^{*}>\beta_{0}^{'},$令$(\overline{x},\overline{s}^{+},\widehat{s}^{-})=(x_{\beta^{*}},s_{\beta^{*}j}^{+},\widehat{s}^{-}).$由(3.3), (3.4) 式和条件$\sum\limits_{i\in Q}\lambda_{i}\leq1$可知

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\bar{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\bar{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\=\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\bar{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}(\hat{s}_{i}^{+}+{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||+{{\bar{\varepsilon }}_{i}})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\nu }_{{{\beta }^{*}}i}}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\=\sum\limits_{i\notin Q}{{{\lambda }_{i}}}{{f}_{i}}(\bar{x})+\sum\limits_{i\in Q}{{{\lambda }_{i}}}{{f}_{i}}(\bar{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||\\-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{{\bar{\varepsilon }}}_{i}}-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\nu }_{{{\beta }^{*}}i}}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\\le \sum\limits_{i\notin Q}{{{\lambda }_{i}}}({{f}_{i}}(\hat{x})-{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{i}})+\sum\limits_{i\in Q}{{{\lambda }_{i}}}({{f}_{i}}(\hat{x})-{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||-{{\bar{\varepsilon }}_{i}}+{{\gamma }_{1}}{{v}_{{{\beta }^{*}}1}})\\-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\bar{\varepsilon }}_{i}}-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\nu }_{{{\beta }^{*}}i}}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}\\=\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\gamma }_{i}})}{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||\\-\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\gamma }_{i}})}{{\bar{\varepsilon }}_{i}}+{{\gamma }_{1}}{{v}_{{{\beta }^{*}}1}}(\sum\limits_{i\in Q}{{{\lambda }_{i}}})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}{{\nu }_{{{\beta }^{*}}i}}\\<\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\gamma }_{i}})}{{\varepsilon }_{i}}||{{x}_{\beta }}-\hat{x}||-\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\gamma }_{i}})}{{\bar{\varepsilon }}_{i}}\\\le \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\gamma }_{i}}}\hat{s}_{i}^{+}+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}\hat{s}_{i}^{-}-\epsilon ||{{x}_{\beta }}-\hat{x}||-\bar{\epsilon }.$

这与$(\widehat{x},\widehat{s}^{+},\widehat{s}^{-})$是$(\rm SOP_{\lambda,\mu,\gamma})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解矛盾.所以$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.

定理3.4 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}, \epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}\lambda_{i}\overline{\varepsilon}_{i}.$若$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似弱有效解.

证 若$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似弱有效解, 则存在$ x\in X,$使得

$f(x)<f(\hat{x})-\varepsilon ||x-\hat{x}||-\bar{\varepsilon }.\text{ }$

(3.5)

从而$f_{i}(x)-\widehat{s}_{i}<f_{i}(\widehat{x})-\widehat{s}_{i}\leq\alpha_{i}.$所以$(x,\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的可行解.又由(3.5) 式可知

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}}+\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\varepsilon }_{i}}||x-\hat{x}||+\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\bar{\varepsilon }}_{i}}<\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}},$

从而

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}}+\epsilon ||x-\hat{x}||+\bar{\epsilon }<\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}}.$

这与$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解矛盾.所以$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似弱有效解.

定理3.5 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+},$

(i)若$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -严格拟近似最优解, $\epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}\lambda_{i}\overline{\varepsilon}_{i}.$则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(ii)若$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, $\epsilon<\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}<\sum\limits_{i=1}^{p}\lambda_{i}\overline{\varepsilon}_{i}.$则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

证 与定理3.4的证明类似.

定理3.6 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}, \epsilon\leq\sum\limits_{i=1}^{p}\lambda_{i}\varepsilon_{i}, \overline{\epsilon}\leq\sum\limits_{i=1}^{p}\lambda_{i}\overline{\varepsilon}_{i}.$若$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$-拟近似最优解, 其中$\lambda+\mu>0, \widehat{s} >0, $则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.

证 先证$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.不失一般性, 不妨设$\widehat{s}_{i}=\max\{0,f_{i}(\widehat{x})-\alpha_{i}\}, i=1,\cdots,p.$因为$\widehat{s}>0,$所以$\widehat{s}_{i}=f_{i}(\widehat{x})-\alpha_{i}>0,i=1,2,\cdots,p.$若$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解, 与定理3.1 (i)的证明类似, 对满足(1) 式的任意的$i$和满足(2) 式的任意的$j$, 令$s_{i} =\widehat{s}_{i}, s_{j}=\widehat{s}_{j}-\nu,$则$s\leq\widehat{s}$, $s_{j}<\widehat{s}_{j} $且$(x,s)$是$(\rm SOP_{\lambda,\mu})$的可行解.因为$\lambda_{j}+\mu_{j}>0,$从而有

${{\lambda }_{j}}{{f}_{j}}(x)+{{\lambda }_{j}}{{\varepsilon }_{j}}||x-\hat{x}||+{{\lambda }_{j}}\bar{\varepsilon }+{{\mu }_{j}}{{s}_{j}}<{{\lambda }_{j}}{{f}_{j}}(\hat{x})+{{\mu }_{j}}{{\hat{s}}_{j}},$

且

${{\lambda }_{i}}{{f}_{i}}(x)+{{\lambda }_{i}}{{\varepsilon }_{i}}||x-\hat{x}||+{{\lambda }_{i}}\bar{\varepsilon }+{{\mu }_{i}}{{s}_{i}}\le {{\lambda }_{i}}{{f}_{i}}(\hat{x})+{{\mu }_{i}}{{\hat{s}}_{i}},\forall i\ne j,$

这表明

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{s}_{i}}+\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\varepsilon }_{i}}||x-\hat{x}||+\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}\bar{\varepsilon }<\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}},$

因此

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{s}_{i}}+\epsilon ||x-\hat{x}||+\bar{\epsilon }<\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\hat{s}}_{i}}.$

这与$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解矛盾.所以$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

下面证明$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.事实上, 由于$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 从而$\widehat{x}$是下面问题的$(\epsilon,\overline{\epsilon})$ -拟近似最优解.

$\min \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}({{f}_{i}}(x)-{{\alpha }_{i}})=\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\mu }_{i}}){{f}_{i}}(x)}-\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\alpha }_{i}},\\{{f}_{i}}(x)>{{\alpha }_{i}},\forall i\in \{1,\cdots ,p\},\\x\in X.$

因为$\lambda+\mu>0,$由引理2.6可知$\widehat{x}$是(MOP)在$ f_{i}(x)>\alpha_{i}, i=1,\cdots,p $条件下的$(\varepsilon,\overline{\varepsilon})$ -真有效解.又由文献[2]中的引理3.3可知, $\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解.

定理3.7 若$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似真有效解, $\lambda\geq0$, $\sum\limits_{i=1}^{p}\lambda_{i}=1, $则存在$\alpha, \widehat{s}, \widehat{\mu},$使得$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 对任意的$\mu\geqq\widehat{\mu},$其中$\epsilon=\sum\limits_{i=1}^{p}(\lambda_{i}+\mu_{i})\varepsilon_{i}, \overline{\epsilon}=\sum\limits_{i=1}^{p}\lambda_{i}\overline{\varepsilon}_{i}.$

证 设$\alpha_{i}=f_{i}(\widehat{x})-\overline{\varepsilon}_{i}, \widehat{s}_{i}=\varepsilon_{i}, i=1,2,\cdots,p.$任取$x\in X,$令$s_{i}=\max\{0,f_{i}(x)+\varepsilon_{i}||x-\widehat{x}||-\alpha_{i}\} =\max\{0,f_{i}(x)+\varepsilon_{i}||x-\widehat{x}||-f_{i}(\widehat{x})+\overline{\varepsilon}_{i}\}, \widehat{\mu}_{i}=M, i=1,\cdots,p.$对$k\in\{1,\cdots,p\},$若$f_{k}(x)\geq f_{k}(\widehat{x})-\varepsilon_{k}||x-\widehat{x}||-\overline{\varepsilon}_{k},$则

${{f}_{k}}(x)+\sum\limits_{i=1}^{p}{{{{\hat{\mu }}}_{i}}}{{s}_{i}}\ge {{f}_{k}}(\hat{x})-{{\varepsilon }_{k}}||x-\hat{x}||-{{\bar{\varepsilon }}_{k}}.\text{ }$

(3.6)

若$f_{k}(x)< f_{k}(\widehat{x})-\varepsilon_{k}||x-\widehat{x}||-\overline{\varepsilon}_{k},$设$I^{*}=\{i:f_{i}(x)>f_{i}(\widehat{x})-\varepsilon_{i}||x-\widehat{x}||-\overline{\varepsilon}_{i}\},$则$I^{*}\neq\varnothing,$且

${{f}_{k}}(x)+\sum\limits_{i=1}^{p}{{{{\hat{\mu }}}_{i}}}{{s}_{i}}={{f}_{k}}(x)+\sum\limits_{i=1}^{p}{{{{\hat{\mu }}}_{i}}}\max \{0,{{f}_{i}}(x)+{{\varepsilon }_{i}}||x-\hat{x}||-{{f}_{i}}(\hat{x})+{{\bar{\varepsilon }}_{i}}\}\\={{f}_{k}}(x)+\sum\limits_{i\in {{I}^{*}}}{{{{\hat{\mu }}}_{i}}}({{f}_{i}}(x)+{{\varepsilon }_{i}}||x-\hat{x}||-{{f}_{i}}(\hat{x})+{{\bar{\varepsilon }}_{i}}),$

因为$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -的真有效解且$f_{k}(x)< f_{k}(\widehat{x})-\varepsilon_{k}||x-\widehat{x}||-\overline{\varepsilon}_{k},$故存在$ k^{*}\in I^{*},$使得

$\frac{{{f_k}(\hat x) - {f_k}(x) - {\varepsilon _k}||x - \hat x|| - {{\bar \varepsilon }_k}}}{{{f_{{k^*}}}(x) - {f_{{k^*}}}(\hat x) + {\varepsilon _{{k^*}}}||x - \hat x|| + {{\bar \varepsilon }_{{k^*}}}}} < {\rm{ }}M.$

从而有

${{f}_{k}}(x)+\sum\limits_{i\in {{I}^{*}}}{{{{\hat{\mu }}}_{i}}}({{f}_{i}}(x)+{{\varepsilon }_{i}}||x-\hat{x}||-{{f}_{i}}(\hat{x})+{{\bar{\varepsilon }}_{i}})\\ \ge {{f}_{k}}(x)+{{\hat{\mu }}_{{{k}^{*}}}}({{f}_{{{k}^{*}}}}(x)-{{f}_{{{k}^{*}}}}(\hat{x})+{{\varepsilon }_{{{k}^{*}}}}||x-\hat{x}||+{{\bar{\varepsilon }}_{{{k}^{*}}}})\\ >{{f}_{k}}(x)+\frac{{{f}_{k}}(\hat{x})-{{f}_{k}}(x)-{{\varepsilon }_{k}}||x-\hat{x}||-{{{\bar{\varepsilon }}}_{k}}}{{{f}_{{{k}^{*}}}}(x)-{{f}_{{{k}^{*}}}}(\hat{x})+{{\varepsilon }_{{{k}^{*}}}}||x-\hat{x}||+{{{\bar{\varepsilon }}}_{{{k}^{*}}}}}({{f}_{{{k}^{*}}}}(x)-{{f}_{{{k}^{*}}}}(\hat{x})+{{\varepsilon }_{{{k}^{*}}}}||x-\hat{x}||+{{\bar{\varepsilon }}_{{{k}^{*}}}})\\ ={{f}_{k}}(\hat{x})-{{\varepsilon }_{k}}||x-\hat{x}||-{{\bar{\varepsilon }}_{k}}.$

再由(3.6) 式可知

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}(\sum\limits_{i=1}^{p}{{{{\hat{\mu }}}_{i}}}{{s}_{i}})\ge \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})-\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\varepsilon }_{i}}||x-\hat{x}||-\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{\bar{\varepsilon }}_{i}}.$

又因为$\sum\limits_{i=1}^{p}\lambda_{i}=1,$ $\epsilon=\sum\limits_{i=1}^{p}(\lambda_{i}+\mu_{i})\varepsilon_{i}, \overline{\epsilon}=\sum\limits_{i=1}^{p}\lambda_{i}\overline{\varepsilon}_{i},$从而有

$\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(x)+\sum\limits_{i=1}^{p}{{{{\hat{\mu }}}_{i}}}{{s}_{i}}\\\ge \sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\varepsilon }_{i}}-[\sum\limits_{i=1}^{p}{({{\lambda }_{i}}+{{\mu }_{i}})}{{\varepsilon }_{i}}||x-\hat{x}||]-\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{{\bar{\varepsilon }}}_{i}}\\=\sum\limits_{i=1}^{p}{{{\lambda }_{i}}}{{f}_{i}}(\hat{x})+\sum\limits_{i=1}^{p}{{{\mu }_{i}}}{{\varepsilon }_{i}}-\epsilon ||x-\hat{x}||-\bar{\epsilon }.$

所以$(\widehat{x},\widehat{s})$是$(\rm SOP_{\lambda,\mu})$的$(\epsilon,\overline{\epsilon})$ -拟近似最优解.

注3.8 当$\overline{\varepsilon}=0,$时, 定理3.1--3.7分别退化为文献[1]中的定理6.1, 定理6.2, 定理6.3, 定理6.7, 定理6.9, 定理6.11和定理6.13.

设$y\in R^{n},$文献[2]中考虑了如下范数, $ ||y||_{\alpha}=||y||_{\infty}+(\frac{1}{\alpha})(\sum\limits_{i=1}^{p}|y_{i}|),~\alpha\in [1,\infty),$其中$||y||_{\infty}=\max\limits_{i=1,\cdots,n}|y_{i}|.$

利用范数$||\cdot||_{\alpha}$给出如下非线性标量化问题:

$(\text{SOP})\underset{x\in X}{\mathop{\min }}\,||f(x)-r|{{|}_{\alpha }}=\underset{i=1,\cdots ,p}{\mathop{\max }}\,\{{{f}_{i}}(x)-{{r}_{i}}\}+\frac{1}{\alpha }(\sum\limits_{i=1}^{p}{({{f}_{i}}(}x)-{{r}_{i}})),$

其中$r=(r_{1},r_{2},\cdots,r_{p}), r_{i}\leq\inf\limits_{x\in X}f_{i}(x), i=1,2,\cdots,p.$即$r$为问题$(\rm MOP)$的理想点.

定理4.1 设$(\varepsilon,\overline{\varepsilon})\in R^{p}_{+}\times R^{p}_{+}, \epsilon\leq\min\limits_{i=1,\cdots,p}\{\varepsilon_{i}\}+\frac{1}{\alpha}\sum\limits_{i=1}^{p}\varepsilon_{i}, \overline{\epsilon}\leq\min\limits_{i=1,\cdots,p}\{\overline{\varepsilon}_{i}\}+\frac{1}{\alpha}\sum\limits_{i=1}^{p}\overline{\varepsilon}_{i}$,

(i)若$\widehat{x}$是$(\rm SOP)$的$(\epsilon,\overline{\epsilon})$ -严格拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(ii)若$\widehat{x}$是$(\rm SOP) $的$(\epsilon,\overline{\epsilon})$ -拟近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似弱有效解.

证 (i)设$\widehat{x}$是$(\rm SOP)$的$(\epsilon,\overline{\epsilon})$ -严格拟近似最优解, 故对任意的$x\in X,x\neq\widehat{x}$, 有

$\mathop {\max }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {f_i}(\hat x) - {r_i}\} + \frac{1}{\alpha }(\sum\limits_{i = 1}^p {({f_i}(} \hat x) - {r_i})){\rm{ }} < \mathop {\max }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {f_i}(x) - {r_i}\} + \frac{1}{\alpha }(\sum\limits_{i = 1}^p {({f_i}(} x) - {r_i})) + ||x - \hat x|| + \bar{\epsilon }.$

若$\widehat{x}$不是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解, 则存在$ x\in X,$使得$f(\widehat{x})\geq f(x)+\varepsilon||x-\widehat{x}||+\overline{\varepsilon}.$从而有

$\begin{array}{l} \mathop {\max }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {f_i}(\hat x) - {r_i}\} + \frac{1}{\alpha }(\sum\limits_{i = 1}^p {({f_i}(} \hat x) - {r_i}))\\ > \mathop {\max }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {f_i}(x) + {\varepsilon _i}||x - \hat x|| + {{\bar \varepsilon }_i} - {r_i}\} + \frac{1}{\alpha }(\sum\limits_{i = 1}^p {({f_i}(} x) + {\varepsilon _i}||x - \hat x|| + {{\bar \varepsilon }_i} - {r_i})\\ \ge \mathop {\max }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {f_i}(x) - {r_i}\} + \mathop {\min }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {\varepsilon _i}||x - \hat x||\} + \mathop {\min }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {{\bar \varepsilon }_i}\} + \frac{1}{\alpha }(\sum\limits_{i = 1}^p {({f_i}(} x) - {r_i})\\ + \frac{1}{\alpha }\sum\limits_{i = 1}^p {{\varepsilon _i}} ||x - \hat x|| + \frac{1}{\alpha }\sum\limits_{i = 1}^p {{{\bar \varepsilon }_i}} \\ \ge \mathop {\max }\limits_{i = 1, \cdots ,p} {\mkern 1mu} \{ {f_i}(x) - {r_i}\} + \frac{1}{\alpha }(\sum\limits_{i = 1}^p {({f_i}(} x) - {r_i})) + ||x - \hat x|| + \bar{\epsilon }. \end{array}$

这与$\widehat{x}$是$(\rm SOP)$的$(\epsilon,\overline{\epsilon})$ -严格拟近似最优解矛盾.所以$\widehat{x}$是$(\rm MOP)$的$(\varepsilon,\overline{\varepsilon})$ -拟近似有效解.

(ii)的证明与(i)的证明类似.

推论4.2 设$\varepsilon\in R^{p}_{+}, \epsilon\leq\min\limits_{i=1,\cdots,p}\{\varepsilon_{i}\}+\frac{1}{\alpha}\sum\limits_{i=1}^{p}\varepsilon_{i},$

(i)若$\widehat{x}$是$(\rm SOP)$的$\epsilon$ -严格近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$\varepsilon$ -近似有效解.

(ii)若$\widehat{x}$是$(\rm SOP)$的$\epsilon$ -近似最优解, 则$\widehat{x}$是$(\rm MOP)$的$\varepsilon$ -近似弱有效解.

注4.3 定理4.1的逆命题不一定成立.参见下面的例子.

例4.4 在问题(MOP)中, 设$X=\{(x_{1},x_{2}):x_{1}+x_{2}\geqq0, 0\leqq x_{1}\leqq1\}\cup\{(x_{1},x_{2}):x_{1}^{2}+(x_{2}-1)^{2}\leqq1\}\subset R^{2}, f(x)=(f_{1}(x),f_{2}(x)), f_{1}(x)=x_{1}, f_{2}(x)=x_{2}.$取理想点$r=(-2,-2), \widehat{x}=(0,0).$当$ \varepsilon=\overline{\varepsilon}=0,\epsilon=\overline{\epsilon}=0 $时, $\widehat{x}$是$(\rm MOP)$的有效解, 但$\widehat{x}$不是$(\rm MOP)$的真有效解.故由文献[5]中的定理3.4.10可知$\widehat{x}$不是$(\rm SOP)$的最优解.所以即使对于精确解定理4.1的逆命题都不一定成立.