Characterizations of Approximate Optimality Conditions for Fractional Semi-Infinite Optimization Problems With Uncertainty
-
摘要:
该文研究了一类带不确定参数的多目标分式半无限优化问题。首先借助鲁棒优化方法，引入该不确定多目标分式优化问题的鲁棒对应优化模型，并借助Dinkelbach方法，将该鲁棒对应优化模型转化为一般的多目标优化问题。随后借助一种标量化方法，建立了该优化问题的标量化问题，并刻画了它们的解之间的关系。最后借助一类鲁棒型次微分约束规格，建立了该不确定多目标分式优化问题拟近似有效解的鲁棒最优性条件。
Abstract:A class of multi-objective fractional semi-infinite optimization problems with uncertain data were investigated. Firstly, a robust optimization model corresponding to the uncertain multi-objective optimization problem was introduced. Then the optimization model was converted to a multi-objective optimization problem with the Dinkelbach method. In turn, by means of the scalarization method, the corresponding scalarization optimization problem was built, and the relationship between robust solutions to the multi-objective optimization problem and its corresponding scalarization optimization problem was described. Finally, through a robust-type sub-differential constraint qualification, the robust optimality condition for approximate quasi-efficient solutions to the multi-objective fractional optimization problem was established.
-
引 言
作为非线性优化问题的一个重要模型，分式优化在资源分配、投资组合和生产计划等问题中具有广泛的应用，因此在过去几十年里引起众多学者的广泛关注，如文献[1-5]。在研究分式优化问题时，近似解的最优性条件和对偶理论是其研究的重点内容，近些年取得了丰硕成果，如文献[6-9]。
值得注意的是，上述文献在研究近似解的最优性和对偶性时，常常需要假定所考虑优化问题模型的数据是精确的。然而由于实际应用中测量或制作误差、以及不精确信息的存在等诸多原因，许多优化问题都会涉及到不确定数据。这些不确定数据对问题求解有着不同程度的影响。因此，带不确定参数的优化问题引起了广泛关注。譬如，Li等[10]建立了带不确定参数的凸优化问题与其不确定共轭对偶问题之间的鲁棒强对偶关系，并应用到数据分类问题中；Sun等[11]借助一种标量化方法，刻画了不确定多目标优化问题的鲁棒近似弱有效解的最优性条件和对偶理论；通过引入一类新的广义凸性概念，Fakhar等[12]刻画了鲁棒弱有效解的充分最优性条件与对偶理论，并应用到投资组合优化问题中；借助一类次微分约束规格，Sun等[13]刻画了一类不确定半无限优化问题拟近似最优解的最优性条件和混合型对偶理论；赵丹和孙祥凯[14]研究了一类目标函数和约束函数均带不确定参数的多目标优化问题的鲁棒拟近似有效解的最优性条件；Lee等[15]讨论了不确定分式优化问题鲁棒近似最优性条件和对偶理论；借助一类鲁棒型约束规格，Zeng等[16]刻画了带不确定参数的半无限分式优化问题的鲁棒近似最优性条件和混合型对偶理论。
受上述文献启发，本文考虑如下多目标分式半无限优化问题：
$$ \left( {{\text{UMFP}}} \right)\quad \mathop {{\text{min}}}\limits_{{\boldsymbol{x}} \in C} \left\{ {\left.\left( {\frac{{{f_1}\left( {\boldsymbol{x}} \right)}}{{{g_1}\left( {\boldsymbol{x}} \right)}}, \cdot \cdot \cdot ,\frac{{{f_k}\left( {\boldsymbol{x}} \right)}}{{{g_k}\left( {\boldsymbol{x}} \right)}}} \right){\text{ }}\right| {{\text{ }}{h_t}\left( {{\boldsymbol{x}},{v_t}} \right) \leqslant 0} ,\;t \in T} \right\} , $$ 其中
$ C \subseteq {R^n} $ 是非空子集；${f_i}:{{{R}}^n} \to \bf{R}$ ，$ {g_i}:{R^n} \to {\bf{R}} $ ，$i = 1, 2,\cdot \cdot \cdot ,k$ 和${h_t}:{R^n} \times {R^s} \to \bf R$ ，$ t \in T $ 均为实值函数；$ {v_t} \in {V_t} $ 为不确定参数，$ {V_t} \subseteq {R^s} $ 为不确定集合。借助经典的鲁棒优化方法[17]，引入问题
$ \left( {{\text{UMFP}}} \right) $ 的鲁棒对应模型：$$ \left( {{\text{RUMFP}}} \right)\quad \mathop {{\text{min}}}\limits_{{\boldsymbol{x}} \in C} \left\{ {\left.\left( {\frac{{{f_1}\left( {\boldsymbol{x}} \right)}}{{{g_1}\left( {\boldsymbol{x}} \right)}}, \cdot \cdot \cdot ,\frac{{{f_k}\left( {\boldsymbol{x}} \right)}}{{{g_k}\left( {\boldsymbol{x}} \right)}}} \right){\text{ }}\right| {{\text{ }}{h_t}\left( {{\boldsymbol{x}},{v_t}} \right) \leqslant 0,{\text{ }}\forall {v_t} \in {V_t}} ,{\text{ }}t \in T} \right\} {\text{.}} $$ 记
$ \left( {{\text{RUMFP}}} \right) $ 的可行集为$$ \varOmega : = \left\{ {{\boldsymbol{x}} \in C:{h_t}\left( {{\boldsymbol{x}},{v_t}} \right) \leqslant 0,{\text{ }}\forall {v_t} \in {V_t},{\text{ }}t \in T} \right\} {\text{.}} $$ 不失一般性，对任意
${\boldsymbol{x}} \in C$ ，本文假设${f_i}\left( {\boldsymbol{x}} \right) \geqslant 0$ ，${g_i}\left( {\boldsymbol{x}} \right) \gt 0$ ,$i = 1, 2,\cdot \cdot \cdot ,k$ 。本文将刻画
$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟近似有效解的最优性条件， 大致框架如下：首先，借助Dinkelbach方法[1]， 将$ \left( {{\text{UMFP}}} \right) $ 的鲁棒对应模型$ \left( {{\text{RUMFP}}} \right) $ 转化为一般的多目标优化问题； 随后，借助标量化方法， 引入该多目标优化问题的标量化问题， 并讨论该多目标优化问题的拟近似有效解与其对应的标量化问题的拟近似最优解之间的关系；最后，借助一类鲁棒型次微分约束规格， 建立$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟近似有效解的必要最优性条件。1. 预 备 知 识
本节给出本文所要用到的定义和结论。假设
$ {R^n} $ 为赋予欧式范数$ \left\| \cdot \right\| $ 的$ n $ 维向量空间，$ {{ B}^*} $ 为$ {R^n} $ 的闭单位球。对任意$ {\boldsymbol{x}},{\boldsymbol{y}} \in {R^n} $ ，定义$ {R^n} $ 的内积$ \left\langle {{\boldsymbol{x}},{\boldsymbol{y}}} \right\rangle : = {{\boldsymbol{x}}^{\rm{T}}}{\boldsymbol{y}} $ 。设T为一个非空无限指标集，$ {R^{\left( T \right)}} $ 定义为${R}^{\left(T\right)}:=\{{\boldsymbol{\mu}} ={\left({\mu }_{t}\right)}_{t\in T}: \text{ }{\mu }_{t}=0, \text{ }t\in T,\text{ }仅有有限个{\mu }_{t}\ne 0\} {\text{.}}$ $ {R^{\left( T \right)}} $ 的非负锥$ R_ + ^{\left( T \right)} $ 定义为$R_ + ^{\left( T \right)}: = \{ {{\boldsymbol{\mu}} \in R_{}^{\left( T \right)}:{\text{ }}{\mu _t} \geqslant 0,{\text{ }}\forall t \in T} \} {\text{.}}$ 定义1[18] 设
$ \varphi :{R^n} \to \bf R $ 为实值函数以及${\boldsymbol{ x}} \in {R^n} $ 。若存在正数L以及$ {\boldsymbol{x}} $ 的邻域$ N({\boldsymbol{x}}) $ ，使得$$ \left| {\varphi \left( {{{\boldsymbol{x}}_1}} \right) - \varphi \left( {{{\boldsymbol{x}}_2}} \right)} \right| \leqslant L\left\| {{{\boldsymbol{x}}_1} - {{\boldsymbol{x}}_2}} \right\| , \qquad \forall {{\boldsymbol{x}}_1},{{\boldsymbol{x}}_2}\in N({\boldsymbol{x}}) , $$ 则称函数
$ \varphi $ 在${\boldsymbol{x}}$ 处为Lipschitz连续。定义2[18] 设
$ \varphi :{R^n} \to \bf R $ 在$ {\boldsymbol{x}} $ 处为Lipschitz连续。(i)
$ \varphi $ 在$ {\boldsymbol{x}} \in {R^n} $ 的Clarke方向导数定义为$$ {\varphi ^{\rm{c}}}\left( {{\boldsymbol{x}},{\boldsymbol{d}}} \right): = \mathop {\lim {\text{ sup}}}\limits_{{\boldsymbol{y}} \to {\boldsymbol{x}},{\text{ }}t \downarrow 0} \;\frac{{\varphi \left( {{\boldsymbol{y}} + t{\boldsymbol{d}}} \right) - \varphi \left( {\boldsymbol{y}} \right)}}{t}{\text{.}} $$ (ii)
$ \varphi $ 在$ {\boldsymbol{x}} \in {R^n} $ 的Clarke次微分定义为$$ {\partial ^{\rm{c}}}\varphi \left( {\boldsymbol{x}} \right): = \left\{ {{\boldsymbol{\xi}} \in {R^n}:{\varphi ^{\rm{c}}}\left( {{\boldsymbol{x}},{\boldsymbol{d}}} \right) \geqslant \left\langle {{\boldsymbol{\xi}} ,{\boldsymbol{d}}} \right\rangle ,{\text{ }}\forall {\boldsymbol{d}} \in {R^n}} \right\} {\text{.}} $$ 注1[18] (i) 若
$ { \varphi }$ 为凸函数，则$ { \varphi }$ 在$ { \bar{\boldsymbol{ x}} \in {R^n} }$ 处的Clarke次微分退化为$$ { \partial \varphi \left( {\bar {\boldsymbol{x}}} \right): = \left\{ {{{\boldsymbol{x}}^*} \in {R^n}:\varphi \left( {\boldsymbol{x}} \right) \geqslant \varphi \left( {\bar {\boldsymbol{x}}} \right) + \left\langle {{{\boldsymbol{x}}^*},{\boldsymbol{x}} - \bar{\boldsymbol{ x}}} \right\rangle ,{\text{ }}\forall {\boldsymbol{x}} \in {R^n}} \right\} {\text{.}}} $$ (ii) 显然，若
$ { \varphi :{R^n} \to \bf R }$ 在$ { {\boldsymbol{x}} \in {R^n} }$ 处为Lipschitz连续，则$$ {{\partial ^{\rm{c}}}\left( {s\varphi } \right)\left( {\boldsymbol{x}} \right) = s{\partial ^{\rm{c}}}\varphi \left( {\boldsymbol{x}} \right) ,\qquad \forall s \in {\bf R }{\text{.}} }$$ (1) 定义3[18] 设
$ D \subseteq {R^n} $ 是非空子集，$ {\boldsymbol{x}} \in D $ 。则集合$ D $ 在$ {\boldsymbol{x}} $ 处的Clarke法锥定义为$$ {N^{\rm{c}}}\left( {D,{\boldsymbol{x}}} \right): = \left\{ {{\boldsymbol{\xi}} \in {R^n}:\left\langle {{\boldsymbol{\xi}} ,{\boldsymbol{u}}} \right\rangle \leqslant 0,{\text{ }}\forall {\boldsymbol{u}} \in {T_D}\left( {\boldsymbol{x}} \right)} \right\} , $$ 其中，
${T_D}\left( {\boldsymbol{x}} \right) = \{ {{\boldsymbol{u}} \in {R^n}:d_D^0\left( {{\boldsymbol{x}},{\boldsymbol{u}}} \right) = 0} \}$ 为$ D $ 在$ {\boldsymbol{x}} $ 处的Clarke切锥，$ d_D^0 $ 为$ D $ 的距离函数。引理1[18] 设
$ D \subseteq {R^n} $ 是非空子集。若函数$ \varphi :{R^n} \to \bf R $ 在$ {\boldsymbol{x}} \in D $ 处是Lipschitz连续，且$ \varphi $ 在$ {\boldsymbol{x}} $ 处取得最小值。则$$ {\boldsymbol{0}} \in {\partial ^{\rm{c}}}\varphi \left( {\boldsymbol{x}} \right) + {N^{\rm{c}}}\left( {D,{\boldsymbol{x}}} \right) {\text{.}} $$ 引理2[18] 设
$ {\varphi _i}:{R^n} \to \bf R $ ,$ i = 1, 2,\cdot \cdot \cdot ,k $ ，在$ {\boldsymbol{x}} \in {R^n} $ 处是Lipschitz连续。则$$ {\partial ^{\rm{c}}}\left( {{\varphi _1} + \cdot \cdot \cdot + {\varphi _k}} \right)\left( {\boldsymbol{x}} \right) \subseteq {\partial ^{\rm{c}}}{\varphi _1}\left( {\boldsymbol{x}} \right) + \cdot \cdot \cdot + {\partial ^{\rm{c}}}{\varphi _k}\left( {\boldsymbol{x}} \right) {\text{.}} $$ 引理3[18] 设
$ {\varphi _1} $ ，$ {\varphi _2}:{R^n} \to \bf R $ ，在$ {\boldsymbol{x}} \in {R^n} $ 处是Lipschitz连续。则$ {\varphi _1}{\varphi _2} $ 在${\boldsymbol{ x}} $ 处是Lipschitz连续, 且$$ {\partial ^{\rm{c}}}\left( {{\varphi _1}{\varphi _2}} \right)\left( {\boldsymbol{x}} \right) \subseteq {\varphi _2}\left( {\boldsymbol{x}} \right){\partial ^{\rm{c}}}{\varphi _1}\left( {\boldsymbol{x}} \right) + {\varphi _1}\left({\boldsymbol{ x}} \right){\partial ^{\rm{c}}}{\varphi _2}\left( {\boldsymbol{x}} \right) {\text{.}} $$ 2. 鲁棒近似最优性条件刻画
本节将鲁棒优化方法、Dinkelbach方法以及标量化方法相结合，通过借助一类鲁棒型约束规格, 刻画
$ \left( {{\text{UMFP}}} \right) $ 鲁棒拟近似有效解的必要最优性条件。定义4 设
${\boldsymbol{ \varepsilon}} : = \left( {{\varepsilon _1},{\varepsilon _2}, \cdot \cdot \cdot ,{\varepsilon _k}} \right) \in R_ + ^k\backslash \left\{ 0 \right\} $ 和$ \bar {\boldsymbol{x}} \in \varOmega $ 。若不存在$ {\boldsymbol{x}} \in \varOmega $ ，使得对于任意的$ i = 1,2, \cdot \cdot \cdot ,k $ ，有$$ \frac{{{f_i}\left( {\boldsymbol{x}} \right)}}{{{g_i}\left({\boldsymbol{ x}} \right)}} \leqslant \frac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| {{\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| , $$ 以及存在
${i_0} \in \left\{ {1,2, \cdot \cdot \cdot ,k} \right\}$ ，使得$$ \frac{{{f_{{i_0}}}\left( {\boldsymbol{x}} \right)}}{{{g_{{i_0}}}\left( {\boldsymbol{x}} \right)}} \lt \frac{{{f_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _{{i_0}}}\left\| {{\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| , $$ 则称
$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解。记
${\boldsymbol{ \phi}} = \left( {{\phi _1},{\phi _2}, \cdot \cdot \cdot ,{\phi _k}} \right) $ ，其中$ {\phi _i}:{R^n} \to {\bf R_ + }: = \left[ {0, + \infty } \right) $ ,$ i = 1,2, \cdot \cdot \cdot ,k $ 。借助Dinkelbach方法[1]，将$ \left( {{\text{RUMFP}}} \right) $ 转化为如下优化问题：$$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} }\quad \mathop {{\text{min}}}\limits_{{\boldsymbol{x}} \in C} \{ {\left( {{f_1}\left( {\boldsymbol{x}} \right) - {\phi _1}\left( {\boldsymbol{x}} \right){g_1}\left( {\boldsymbol{x}} \right), \cdot \cdot \cdot ,{f_k}\left( {\boldsymbol{x}} \right) - {\phi _k}\left( {\boldsymbol{x}} \right){g_k}\left( {\boldsymbol{x}} \right)} \right){\text{ }}| {{\text{ }}{h_t}\left( {{\boldsymbol{x}},{v_t}} \right) \leqslant 0,{\text{ }}\forall {v_t} \in {V_t}} ,{\text{ }}t \in T} \} {\text{.}} $$ 注2 设
$ {{\boldsymbol{ \varepsilon }}: = \left( {{\varepsilon _1},{\varepsilon _2}, \cdot \cdot \cdot ,{\varepsilon _k}} \right) \in R_ + ^k\backslash \left\{ 0 \right\} }$ 和$ { {\boldsymbol{\phi}} = \left( {{\phi _1},{\phi _2}, \cdot \cdot \cdot ,{\phi _k}} \right) }$ 。类似于定义4，我们可以给出$ { {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } }$ 的拟$ { {\boldsymbol{\varepsilon}} }$ -有效解的概念。如无特殊说明，本文总是假设
$ {\phi _i}:{R^n} \to {\bf R_ + }: = \left[ {0, + \infty } \right) $ ,$ i = 1, 2,\cdot \cdot \cdot ,k $ 。下述命题刻画了$ \left( {{\text{UMFP}}} \right) $ 和$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 之间拟近似有效解的关系。命题1 设
$ {\boldsymbol{\varepsilon}} \in R_ + ^k\backslash \left\{ 0 \right\} $ ,$\bar {\boldsymbol{x}} \in \varOmega$ 以及${\phi _i} = \dfrac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| { \cdot - \bar {\boldsymbol{x}}} \right\|$ ，$ i = 1,2, \cdot \cdot \cdot ,k $ 。若$ \bar{\boldsymbol{ x}} $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟${\boldsymbol{ \varepsilon}} $ -有效解, 则$ \bar {\boldsymbol{x}} $ 是$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的拟${\boldsymbol{ \varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) $ -有效解，其中${\boldsymbol{ \varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right): = \left( {{\varepsilon _1}{g_1}\left( {\bar {\boldsymbol{x}}} \right), \cdot \cdot \cdot ,{\varepsilon _k}{g_k}\left( {\bar {\boldsymbol{x}}} \right)} \right)$ 。证明 假设
$ \bar {\boldsymbol{x}} $ 不是$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的拟${\boldsymbol{ \varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) $ -有效解。则存在$ \hat {\boldsymbol{x}} \in \varOmega $ ，使得对于任意的$ i = 1, 2,\cdot \cdot \cdot ,k $ ，有$$ {f_i}\left( {\hat {\boldsymbol{x}}} \right) - {\phi _i}\left( {\hat {\boldsymbol{x}}} \right){g_i}\left( {\hat {\boldsymbol{x}}} \right) \leqslant {f_i}\left( {\bar{\boldsymbol{ x}}} \right) - {\phi _i}\left( {\bar {\boldsymbol{x}}} \right){g_i}\left( {\bar {\boldsymbol{x}}} \right) - {\varepsilon _i}{g_i}\left( {\bar {\boldsymbol{x}}} \right)\left\| {\hat {\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| , $$ (2) 以及存在
$ {i_0} \in \left\{ {1, 2,\cdot \cdot \cdot ,k} \right\} $ ，使得$$ {f_{{i_0}}}\left( {\hat {\boldsymbol{x}}} \right) - {\phi _{{i_0}}}\left( {\hat {\boldsymbol{x}}} \right){g_{{i_0}}}\left( {\hat {\boldsymbol{x}}} \right) \lt {f_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right) - {\phi _{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right){g_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right) - {\varepsilon _{{i_0}}}{g_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right)\left\| {\hat {\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| {\text{.}} $$ (3) 由式(2)和
${\phi _i} = \dfrac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| { \cdot - \bar {\boldsymbol{x}}} \right\|$ ，可得对于任意的$i = 1,2, \cdot \cdot \cdot ,k$ ，有$$ {f_i}\left( {\hat {\boldsymbol{x}}} \right) - \left( {\frac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| {\hat {\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\|} \right){g_i}\left( {\hat {\boldsymbol{x}}} \right) \leqslant {f_i}\left( {\bar {\boldsymbol{x}}} \right) - \left( {\frac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| {\bar {\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\|} \right){g_i}\left( {\bar {\boldsymbol{x}}} \right) - {\varepsilon _i}{g_i}\left( {\bar {\boldsymbol{x}}} \right)\left\| {\hat {\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| \leqslant 0{\text{.}} $$ 又因为
${g_i}\left( {\bar {\boldsymbol{x}}} \right) \gt 0$ , 从而由上式可得$$ {f_i}\left( {\hat {\boldsymbol{x}}} \right){g_i}\left( {\bar {\boldsymbol{x}}} \right) - {f_i}\left( {\bar {\boldsymbol{x}}} \right){g_i}\left( {\hat {\boldsymbol{x}}} \right) + {\varepsilon _i}\left\| {\hat {\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\|{g_i}\left( {\hat {\boldsymbol{x}}} \right){g_i}\left( {\bar {\boldsymbol{x}}} \right) \leqslant 0 , \qquad \forall i = 1,2, \cdot \cdot \cdot ,k {\text{.}} $$ 故
$$ \frac{{{f_i}\left( {\hat {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\hat {\boldsymbol{x}}} \right)}} \leqslant \frac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| {{\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| , \qquad \forall i = 1,2, \cdot \cdot \cdot ,k {\text{.}} $$ (4) 同理，由式(3)可得
$$ \frac{{{f_{{i_0}}}\left( {\hat {\boldsymbol{x}}} \right)}}{{{g_{{i_0}}}\left( {\hat {\boldsymbol{x}}} \right)}} \lt \frac{{{f_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_{{i_0}}}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _{{i_0}}}\left\| {{\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| , \qquad \exists {i_0} \in \left\{ {1, 2,\cdot \cdot \cdot ,k} \right\} {\text{.}} $$ (5) 显然，式(4)和式(5)的矛盾在于
$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解。故，$ \bar {\boldsymbol{x}} $ 是$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的拟$ {\boldsymbol{\varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) $ -有效解。证毕。注3 不同于文献[15]的引理2.1，命题1中
$ { \left( {{\text{UMFP}}} \right) }$ 和$ { {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } }$ 的拟近似有效解之间不是等价关系，即若$ { \bar {\boldsymbol{x}} }$ 是$ { {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } }$ 的拟$ {{\boldsymbol{ \varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) }$ -有效解，则无法保证$ { \bar{\boldsymbol{ x}} }$ 是$ { \left( {{\text{UMFP}}} \right) }$ 的鲁棒拟$ { {\boldsymbol{\varepsilon}} }$ -有效解。下例解释了注3，即
$ \bar {\boldsymbol{x}} $ 是$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的拟$ {\boldsymbol{\varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) $ -有效解，而不是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解。例1 设
$ x \in C: =\bf R $ ,$ {v_t} \in {V_t}: = \left[ { - t + 2,t + 2} \right] $ ，其中$ t \in T: = \left[ {0,1} \right] $ 。考虑多目标分式优化问题：$$ \left( {{\text{UMFP}}} \right)\quad \mathop {{\text{min}}}\limits_{x \in C} \left\{ {\left.\left( {\frac{{{x^3} + 1}}{{x + 4}},\frac{{{x^3} + 2}}{{x + 4}}} \right){\text{ }}\right| {{\text{ }}t{x^2} - 2{v_t}x \leqslant 0,{\text{ }}t \in T} } \right\} {\text{.}} $$ 则
$ \left( {{\text{UMFP}}} \right) $ 的鲁棒对应模型为$$ \left( {{\text{RUMFP}}} \right)\quad \mathop {{\text{min}}}\limits_{x \in C} \left\{ {\left.\left( {\frac{{{x^3} + 1}}{{x + 4}},\frac{{{x^3} + 2}}{{x + 4}}} \right){\text{ }}\right| {{\text{ }}t{x^2} - 2{v_t}x \leqslant 0,{\text{ }}\forall {v_t} \in {V_t},{\text{ }}t \in T} } \right\} {\text{.}} $$ 显然，可行集
$ \varOmega = \left[ {0,2} \right] $ 。令
$ \bar x: = 0 \in \varOmega $ 和$ {\boldsymbol{\varepsilon}} : = \left( {\dfrac{1}{{32}},\dfrac{1}{{16}}} \right) $ 。显然，存在$ \hat x = \dfrac{1}{{10}} \in \varOmega $ ，使得对于任意的$ i = 1,2 $ ，均有$$ \frac{{{f_i}\left( {\hat x} \right)}}{{{g_i}\left( {\hat x} \right)}} \lt \frac{{{f_i}\left( {\bar x} \right)}}{{{g_i}\left( {\bar x} \right)}} - {\varepsilon _i}\left\| {\hat x - \bar x} \right\| {\text{.}} $$ 因此，
$ \bar x $ 不是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解。另一方面，由于
$ {\phi _1}(x) = \dfrac{1}{4} - {\varepsilon _1}\left| x \right| $ ，$ {\phi _2}(x) = \dfrac{1}{2} - {\varepsilon _2}\left| x \right| $ ，所以$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 为$$ \mathop {{\text{min}}}\limits_{{\boldsymbol{x}} \in C} \left\{ {\left.\left( {{x^3} + 1 - \left( {\frac{1}{4} - {\varepsilon _1}\left| x \right|} \right)\left( {x + 4} \right),{x^3} + 2 - \left( {\frac{1}{2} - {\varepsilon _2}\left| x \right|} \right)\left( {x + 4} \right)} \right){\text{ }}\right| {{\text{ }}t{x^2} - 2{v_t}x \leqslant 0,{\text{ }}\forall {v_t} \in {V_t}} ,{\text{ }}t \in T} \right\} {\text{.}} $$ 显然，
$ {\boldsymbol{\varepsilon}}{\boldsymbol{ g}}\left( {\bar {\boldsymbol{x}}} \right) = \left( {\dfrac{1}{8},\dfrac{1}{4}} \right) $ 。不难计算,$ \bar x $ 是$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi }}} $ 的鲁棒拟${\boldsymbol{ \varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) $ -有效解。下面借助标量化方法，刻画
$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的拟$ {\boldsymbol{\varepsilon g}}\left( {\bar {\boldsymbol{x}}} \right) $ -有效解。为此，首先引入$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的标量化问题：$$ \left( {{\text{SRUMFP}}} \right)_{\boldsymbol{\phi}} ^{}\quad \mathop {\min }\limits_{x \in C} \left\{ {\left.\sum\limits_{i = 1}^k {\left( {{f_i}\left( {\boldsymbol{x}} \right) - {\phi _i}\left( {\boldsymbol{x}} \right){g_i}\left( {\boldsymbol{x}} \right)} \right)} {\text{ }}\right| {{\text{ }}{h_t}\left( {{\boldsymbol{x}},{v_t}} \right) \leqslant 0,{\text{ }}\forall {v_t} \in {V_t},{\text{ }}t \in T} } \right\} {\text{.}} $$ 定义5 设
$\epsilon \geqslant 0$ 以及${\boldsymbol{ \phi}} = \left( {{\phi _1}, {\phi _2},\cdot \cdot \cdot ,{\phi _k}} \right) $ 。若存在$ \bar {\boldsymbol{x}} \in \varOmega $ ，使得$$ {\displaystyle \sum _{i=1}^{k}\left({f}_{i}\left(\overline{{\boldsymbol{x}}}\right)-{\phi }_{i}\left(\overline{{\boldsymbol{x}}}\right){g}_{i}\left(\overline{{\boldsymbol{x}}}\right)\right)\le {\displaystyle \sum _{i=1}^{k}\left({f}_{i}\left({\boldsymbol{x}}\right)-{\phi }_{i}\left({\boldsymbol{x}}\right){g}_{i}\left({\boldsymbol{x}}\right)\right)} + \epsilon \Vert {\boldsymbol{x}}-\overline{{\boldsymbol{x}}}\Vert } , \qquad \forall {\boldsymbol{x}} \in \varOmega , $$ 则称
$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{SRUMFP}}} \right)_{\boldsymbol{\phi}} ^{} $ 的$\epsilon$ -最优解。借助文献[19]中引理3.2的证明方法，易得
$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 和$ \left( {{\text{SRUMFP}}} \right)_{\boldsymbol{\phi}} ^{} $ 之间近似解的关系。命题2 设
$ {\boldsymbol{\varepsilon}} \in R_ + ^k\backslash \left\{ 0 \right\} $ ,$ \bar {\boldsymbol{x}} \in \varOmega $ 以及${\boldsymbol{ \phi}} = \left( {{\phi _1},{\phi _2}, \cdot \cdot \cdot ,{\phi _k}} \right) $ 。则$ \bar {\boldsymbol{x}} $ 是$ {\left( {{\text{RUMFP}}} \right)_{\boldsymbol{\phi}} } $ 的拟${\boldsymbol{ \varepsilon}} $ -有效解当且仅当$ \bar{\boldsymbol{ x}} $ 是$ \left( {{\text{SRUMFP}}} \right)_{\boldsymbol{\phi}} ^{} $ 的$\displaystyle \sum\nolimits_{i = 1}^k {{\varepsilon _i}}$ -最优解。众所周知，约束规格在刻画
$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟近似有效解的必要最优性条件中起着至关重要的作用。为此，类似于文献[13, 20]中所引入的约束规格，本文引入如下鲁棒型次微分约束规格。为方便起见，记${\boldsymbol{v}}: = {\left( {{v_t}} \right)_{t \in T}} \in {V_T}$ ，其中${V_T}: = \displaystyle \mathop \prod \nolimits_{t \in T} {V_t}$ 。定义6[20] 考虑
$ \left( {{\text{UMFP}}} \right) $ 。设$ \bar {\boldsymbol{x}} \in \varOmega $ ，记$T\left( {\bar {\boldsymbol{x}}} \right): = \{ {{\boldsymbol{\mu}} \in R_ + ^{\left( T \right)}:{\mu _t}{h_t}\left( {\bar {\boldsymbol{x}},{v_t}} \right) = 0,{\text{ }}\forall {v_t} \in {V_t},{\text{ }}t \in T} \}$ 。若$$ {N^{\rm{c}}}\left( {\varOmega ,\bar {\boldsymbol{x}}} \right) \subseteq \mathop \bigcup \limits_{{\boldsymbol{\mu}} \in T\left( {\bar {\boldsymbol{x}}} \right),{\boldsymbol{v}} \in {V_T}} \left[ {\sum\limits_{t \in T}^{} {{\mu _t}\partial _x^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{v_t}} \right)} } \right] + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) , $$ 则称鲁棒型次微分约束规格
$ \left( {{\text{RSCQ}}} \right) $ 在$\bar {\boldsymbol{x}}$ 处成立。注4 若
$ { C = {R^n} }$ ，则定义6退化为文献[13]中的定义3.2。 若$ { {V_t} }$ ，$ { t \in T }$ ， 为单点集，则定义6退化为文献[20]中第294页的约束规格$ { {\left( {{\rm{CQ}}} \right)_{{x_0}}} }$ 。接下来，借助鲁棒型次微分约束规格
$ \left( {{\text{RSCQ}}} \right) $ ，刻画$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解的必要最优性条件。定理1 设
$ {\boldsymbol{\varepsilon}} \in R_ + ^k\backslash \left\{ 0 \right\} $ ,$ \bar {\boldsymbol{x}} \in \varOmega $ 以及${\phi _i} = {{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}/{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}} - {\varepsilon _i}\left\| { \cdot - \bar {\boldsymbol{x}}} \right\|$ ，$ i = 1, 2,\cdot \cdot \cdot ,k $ 。设$ {f_i} $ 和$ {g_i} $ ，$ i = 1,2, \cdot \cdot \cdot ,k $ ，在$ \bar {\boldsymbol{x}} $ 处为Lipschitz连续，以及对于任意的$ {v_t} \in {V_t} $ ，$ {h_t}\left( { \cdot ,{v_t}} \right) $ ，$ t \in T $ ，在$ \bar {\boldsymbol{x}} $ 处为Lipschitz连续。假设$ \left( {{\text{RSCQ}}} \right) $ 在$ \bar {\boldsymbol{x}} $ 处成立。若$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解，则存在$ \bar {\boldsymbol{\mu}} \in R_ + ^{\left( T \right)} $ 以及$ {\bar v_t} \in {V_t} $ ，$ t \in T $ ，使得$$ {\boldsymbol{0}} \in \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}\left( {\bar {\boldsymbol{x}}} \right) - \sum\limits_{i = 1}^k {{\phi _i}\left( {\bar {\boldsymbol{x}}} \right){\partial ^{\rm{c}}}{g_i}\left( {\bar {\boldsymbol{x}}} \right)} } + \sum\limits_{t \in T}^{} {{{\bar \mu }_t}\partial _{\boldsymbol{x}}^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right)} + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) + 2\sum\limits_{i = 1}^k {{\varepsilon _i}{g_i}\left( {\bar {\boldsymbol{x}}} \right)} {{ B}^*} , $$ (6) $$ {\bar \mu _t}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right) = 0 , \qquad \forall t \in T {\text{.}} $$ (7) 证明 设
$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解。则由命题1及命题2知，$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{SRUMFP}}} \right)_{\boldsymbol{\phi}} ^{} $ 的$\gamma$ -最优解，其中$\gamma = \displaystyle \sum\nolimits_{i = 1}^k {{\varepsilon _i}{g_i}\left( {\bar {\boldsymbol{x}}} \right)}$ 。从而对于任意的$ {\boldsymbol{x}} \in \varOmega $ ，$$ \sum\limits_{i = 1}^k {\left( {{f_i}\left( {\boldsymbol{x}} \right) - {\phi _i}\left( {\boldsymbol{x}} \right){g_i}\left( {\boldsymbol{x}} \right)} \right)} \geqslant \sum\limits_{i = 1}^k {\left( {{f_i}\left( {\bar {\boldsymbol{x}}} \right) - {\phi _i}\left( {\bar {\boldsymbol{x}}} \right){g_i}\left( {\bar {\boldsymbol{x}}} \right)} \right)} - \gamma \left\| {{\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| {\text{.}} $$ 令
$$ {\varGamma _{\boldsymbol{\phi}} }\left( x \right) = \sum\limits_{i = 1}^k {\left( {{f_i}\left( {\boldsymbol{x}} \right) - {\phi _i}\left( {\boldsymbol{x}} \right){g_i}\left( {\boldsymbol{x}} \right)} \right)} - \sum\limits_{i = 1}^k {\left( {{f_i}\left( {\bar {\boldsymbol{x}}} \right) - {\phi _i}\left( {\bar {\boldsymbol{x}}} \right){g_i}\left( {\bar {\boldsymbol{x}}} \right)} \right)} + \gamma \left\| {{\boldsymbol{x}} - \bar {\boldsymbol{x}}} \right\| , $$ 则对任意
$ {\boldsymbol{x}} \in \varOmega $ ，${\varGamma _{\boldsymbol{\phi}} }\left( {\bar {\boldsymbol{x}}} \right) \leqslant {\varGamma _{\boldsymbol{\phi}} }\left( {\boldsymbol{x}} \right)$ 。从而对任意$ {\boldsymbol{x}} \in \varOmega $ ，$ {\varGamma _{\boldsymbol{\phi}} } $ 在$ \bar {\boldsymbol{x}} $ 处取得最小值，且函数$ {\varGamma _{\boldsymbol{\phi}} } $ 在$ \bar {\boldsymbol{x}} \in \varOmega $ 处是Lipschitz连续。因此，由引理1可得$$ {\boldsymbol{0}} \in {\partial ^{\rm{c}}}{\varGamma _{\boldsymbol{\phi}} }\left( {\bar {\boldsymbol{x}}} \right) + {N^{\rm{c}}}\left( {\varOmega ,\bar {\boldsymbol{x}}} \right) {\text{.}} $$ 又由引理2和
${\partial ^{\rm{c}}}\left( {\left\| { \cdot - \bar {\boldsymbol{x}}} \right\|} \right)\left( {\bar {\boldsymbol{x}}} \right) = {{ B}^*}$ ，易知$$ {\boldsymbol{0}} \in {\partial ^{\rm{c}}}\left[ {\sum\limits_{i = 1}^k {\left( {{f_i} - {\phi _i}{g_i}} \right)} } \right]\left( {\bar {\boldsymbol{x}}} \right) + {N^{\rm{c}}}\left( {\varOmega ,\bar {\boldsymbol{x}}} \right) + \gamma {{ B}^*} \subseteq \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}\left( {{f_i} - {\phi _i}{g_i}} \right)\left( {\bar {\boldsymbol{x}}} \right)} + {N^{\rm{c}}}\left( {\varOmega ,\bar {\boldsymbol{x}}} \right) + \gamma {{ B}^*} {\text{.}} $$ 进一步，又因为
$ \left( {{\text{RSCQ}}} \right) $ 在$\bar {\boldsymbol{x}}$ 处成立，所以$$ {\boldsymbol{0}} \in \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}\left( {{f_i} - {\phi _i}{g_i}} \right)\left( {\bar {\boldsymbol{x}}} \right)} + \mathop \bigcup \limits_{{\boldsymbol{\mu}} \in T\left( {\bar {\boldsymbol{x}}} \right),{\boldsymbol{v}} \in {V_T}} \left[ {\sum\limits_{t \in T}^{} {{\mu _t}\partial _x^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{v_t}} \right)} } \right] + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) + \gamma {{ B}^*} , $$ 其中
$$ T\left( {\bar {\boldsymbol{x}}} \right) = \{ {{\boldsymbol{\mu}} \in R_ + ^{\left( T \right)}:{\mu _t}{h_t}\left( {\bar {\boldsymbol{x}},{v_t}} \right) = 0,{\text{ }}\forall {v_t} \in {V_t},{\text{ }}t \in T} \}{\text{.}}$$ 从而存在
$\bar {\boldsymbol{\mu}} \in R_ + ^{\left( T \right)}$ 和$ {\bar v_t} \in {V_t} $ ，$ t \in T $ ，使得$$ {\boldsymbol{0}} \in \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}\left( {{f_i} - {\phi _i}{g_i}} \right)\left( {\bar {\boldsymbol{x}}} \right)} + \sum\limits_{t \in T}^{} {{{\bar \mu }_t}\partial _x^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right)} + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) + \gamma {{ B}^*} , $$ (8) $$ {\bar \mu _t}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right) = 0 ,\qquad \forall t \in T {\text{.}} $$ (9) 又由式(1)、引理2和引理3，可知
$$ \begin{split}& \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}\left( {{f_i} - {\phi _i}{g_i}} \right)\left( {\bar {\boldsymbol{x}}} \right)} \subseteq \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}\left( {\bar {\boldsymbol{x}}} \right) + \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}\left( {{\phi _i}\left( { - {g_i}} \right)} \right)\left( {\bar {\boldsymbol{x}}} \right)} }\subseteq \hfill \\& \qquad \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}\left( {\bar {\boldsymbol{x}}} \right) + \sum\limits_{i = 1}^k {{\phi _i}(\bar {\boldsymbol{x}}){\partial ^{\rm{c}}}\left( { - {g_i}} \right)\left( {\bar {\boldsymbol{x}}} \right) + } } \sum\limits_{i = 1}^k {\left( { - {g_i}} \right)\left( {\bar {\boldsymbol{x}}} \right){\partial ^{\rm{c}}}{\phi _i}\left( {\bar {\boldsymbol{x}}} \right)}= \hfill \\& \qquad \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}\left( {\bar {\boldsymbol{x}}} \right) - \sum\limits_{i = 1}^k {{\phi _i}\left( {\bar {\boldsymbol{x}}} \right){\partial ^{\rm{c}}}{g_i}\left( {\bar {\boldsymbol{x}}} \right) + \sum\limits_{i = 1}^k {{\varepsilon _i}{g_i}\left( {\bar {\boldsymbol{x}}} \right)} {{ B}^*}} } {\text{.}} \hfill \\ \end{split} $$ 从而，由式(8)可知
$$ {\boldsymbol{0}} \in \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}\left( {\bar {\boldsymbol{x}}} \right) - \sum\limits_{i = 1}^k {{\phi _i}\left( {\bar {\boldsymbol{x}}} \right){\partial ^{\rm{c}}}{g_i}\left( {\bar {\boldsymbol{x}}} \right)} } + \sum\limits_{t \in T}^{} {{{\bar \mu }_t}\partial _{\boldsymbol{x}}^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right)} + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) + 2\sum\limits_{i = 1}^k {{\varepsilon _i}{g_i}\left( {\bar {\boldsymbol{x}}} \right)} {{ B}^*} {\text{.}} $$ 故，结合式(9)可知，存在
$ \bar {\boldsymbol{\mu}} \in R_ + ^{\left( T \right)} $ 以及$ {\bar v_t} \in {V_t} $ ,$ t \in T $ ，使得式(6)和式(7)成立。证毕。下例说明了定理1中约束规格
$ \left( {{\text{RSCQ}}} \right) $ 是必不或缺的。例2 设
$ x \in C: = \left[ {0,1} \right] $ ，$ {v_t} \in {V_t}: = \left[ { - t + 2,t + 2} \right] $ ，其中$ t \in T: = \left[ {0,1} \right] $ 。考虑多目标分式优化问题：$$ \left( {{\text{UMFP}}} \right)\quad \mathop {{\text{min}}}\limits_{x \in C} \left\{ {\left.\left( {\frac{{ - {x^2} - 2x + 3}}{{x + 4}},\frac{{ - {x^2} - x + 2}}{{x + 2}}} \right){\text{ }}\right| {{\text{ }}\left( {t + 2{v_t}} \right){x^2} \leqslant 0,{\text{ }}t \in T} } \right\} , $$ 则
$ {f_1}\left( x \right) = - {x^2} - 2x + 3 $ ，$ {f_2}\left( x \right) = - {x^2} - x + 2 $ ，$ {g_1}\left( x \right) = x + 4 $ ，$ {g_2}\left( x \right) = x + 2 $ ，$ {h_t}\left( {x,{v_t}} \right) = \left( {t + 2{v_t}} \right){x^2} $ 。另一方面，易得$ \varOmega = \left\{ 0 \right\} $ 。令
$ \bar x: = 0 \in \varOmega $ 和$ {\boldsymbol{\varepsilon}} : = \left( {\dfrac{1}{8},\dfrac{1}{8}} \right) $ 。显然，$ \bar x $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解。另一方面，易得$ \partial _x^{\rm{c}}{h_t}\left( {\bar x,{v_t}} \right) = \left\{ 0 \right\} $ 。因此$$ \mathop \bigcup \limits_{{\boldsymbol{\mu}} \in T\left( {\bar x} \right),{\boldsymbol{v}} \in {V_T}} \left[ {\sum\limits_{t \in T}^{} {{\mu _t}\partial _{{x}}^{\rm{c}}{h_t}\left( {\bar x,{v_t}} \right)} } \right] = \left\{ 0 \right\} {\text{.}} $$ 而
${N^{\rm{c}}}\left( {\varOmega ,\bar x} \right) = \bf R$ 和$ {N^{\rm{c}}}\left( {C,\bar x} \right) = \left( { - \infty ,0} \right] $ 。显然$$ {N}^{{\rm{c}}}\left(\varOmega ,\overline{x}\right) \nsubseteq \underset{{\boldsymbol{\mu}} \in T\left(\overline{{{x}}}\right),{\boldsymbol{v}}\in {V}_{T}}{\bigcup }\left[{\displaystyle \sum _{t\in T}^{}{\mu }_{t}{\partial }_{{{x}}}^{{\rm{c}}}{h}_{t}\left(\overline{x},{v}_{t}\right)}\right] + {N}^{{\rm{c}}}\left(C,\overline{x}\right) {\text{.}} $$ 因此，
$ \left( {{\text{RSCQ}}} \right) $ 在$ \bar x $ 处是不成立的。另一方面，对于
$ \bar x = 0 $ 和$ {\boldsymbol{\varepsilon}} : = \left( {\dfrac{1}{8},\dfrac{1}{8}} \right) $ 。易知$ {\partial ^{\rm{c}}}{f_1}\left( {\bar x} \right) = - 2 $ ,$ {\partial ^{\rm{c}}}{f_2}\left( {\bar x} \right) = - 1 $ ,$ {\partial ^{\rm{c}}}{g_1}\left( {\bar x} \right) = {\partial ^{\rm{c}}}{g_2}\left( {\bar x} \right) = 1 $ ,$ {\phi _1}\left( {\bar x} \right) = \dfrac{3}{4} $ ,${\phi _2}\left( {\bar x} \right) = 1$ 和$\displaystyle \sum\nolimits_{i = 1}^2 {{\varepsilon _i}{g_i}\left( {\bar x} \right)} = \dfrac{3}{4}$ 。显然，对任意${\boldsymbol{\mu}} \in R_ + ^{\left( T \right)}$ 以及$ {v_t} \in {V_t} $ ，$ t \in T $ ，有$$ 0 \notin \left( { - \infty , - \frac{{13}}{4}} \right] = \sum\limits_{i = 1}^2 {{\partial ^{\rm{c}}}{f_i}\left( {\bar x} \right) - \sum\limits_{i = 1}^2 {{\phi _i}\left( {\bar x} \right){\partial ^{\rm{c}}}{g_i}\left( {\bar x} \right)} } + \sum\limits_{t \in T}^{} {{\mu _t}\partial _x^{\rm{c}}{h_t}\left( {\bar x,{v_t}} \right)} + {N^{\rm{c}}}\left( {C,\bar {{x}}} \right) + 2\sum\limits_{i = 1}^2 {{\varepsilon _i}{g_i}\left( {\bar x} \right)} {{ B}^*} {\text{.}} $$ 故，定理1不成立。
若
$ {\boldsymbol{\varepsilon}} : = \left( {0, \cdot \cdot \cdot ,0} \right) \in R_ + ^k $ ，则有如下关于问题$ \left( {{\text{UMFP}}} \right) $ 鲁棒有效解的必要最优性条件。推论1 考虑
$ \left( {{\text{UMFP}}} \right) $ 。假定$ \bar{\boldsymbol{ x}} \in \varOmega $ 。设$ {f_i} $ 和$ {g_i} $ ，$ i = 1, 2,\cdot \cdot \cdot ,k $ ，在$ \bar {\boldsymbol{x}} $ 处为Lipschitz连续，以及对于任意的$ {v_t} \in {V_t} $ ，$ {h_t}\left( { \cdot ,{v_t}} \right) $ ，$ t \in T $ ，在$ \bar {\boldsymbol{x}} $ 处为Lipschitz连续。假设$ \left( {{\text{RSCQ}}} \right) $ 在$ \bar {\boldsymbol{x}} $ 处成立。若$ \bar {\boldsymbol{x}} \in \varOmega $ 是$ \left( {{\text{UMFP}}} \right) $ 的鲁棒有效解，则存在$ \bar {\boldsymbol{\mu}} \in R_ + ^{\left( T \right)} $ 以及$ {\bar v_t} \in {V_t} $ ，$ t \in T $ ，使得$${\boldsymbol{ 0}} \in \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}\left( {\bar {\boldsymbol{x}}} \right) - \sum\limits_{i = 1}^k {\frac{{{f_i}\left( {\bar {\boldsymbol{x}}} \right)}}{{{g_i}\left( {\bar {\boldsymbol{x}}} \right)}}{\partial ^{\rm{c}}}{g_i}\left( {\bar {\boldsymbol{x}}} \right)} } + \sum\limits_{t \in T}^{} {{{\bar \mu }_t}\partial _{\boldsymbol{x}}^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right)} + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) , $$ $$ {\bar \mu _t}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right) = 0 , \qquad \forall t \in T {\text{.}} $$ 若问题
$ \left( {{\text{UMFP}}} \right) $ 中的${g_i}\left( {\boldsymbol{x}} \right) \equiv 1$ ,$i = 1,2,\cdot \cdot \cdot ,k$ , 则易得如下结论。推论2 考虑如下多目标优化问题：
$$ \left( {{\text{UMP}}} \right) \quad \mathop {{\text{min}}}\limits_{{\boldsymbol{x}} \in C} \{ {\left( {{f_1}\left( {\boldsymbol{x}} \right), {f_2}\left( {\boldsymbol{x}} \right),\cdot \cdot \cdot ,{f_k}\left( {\boldsymbol{x}} \right)} \right){\text{ }}| {{\text{ }}{h_t}\left( {{\boldsymbol{x}},{v_t}} \right) \leqslant 0} ,{\text{ }}t \in T} \} {\text{.}} $$ 设
$ {\boldsymbol{\varepsilon}} \in R_ + ^k\backslash \left\{ 0 \right\} $ ，$ \bar {\boldsymbol{x}} \in \varOmega $ 以及$ {\tau _i} = {f_i}\left( {\bar{\boldsymbol{ x}}} \right) - {\varepsilon _i}\left\| { \cdot - \bar {\boldsymbol{x}}} \right\| $ ,$ i = 1,2, \cdot \cdot \cdot ,k $ 。设$ {f_i} $ ，$ i = 1,2, \cdot \cdot \cdot ,k $ ，在$ \bar {\boldsymbol{x}} $ 处为Lipschitz连续，以及对于任意的$ {v_t} \in {V_t} $ ,$ {h_t}\left( { \cdot ,{v_t}} \right) $ ，$ t \in T $ ，在$ \bar {\boldsymbol{x}} $ 处为Lipschitz连续。设$ \left( {{\text{RSCQ}}} \right) $ 在$ \bar {\boldsymbol{x}} $ 处成立。若$ \bar {\boldsymbol{x}} $ 是$ \left( {{\text{UMP}}} \right) $ 的鲁棒拟$ {\boldsymbol{\varepsilon}} $ -有效解，则存在$ \bar {\boldsymbol{\mu}} \in R_ + ^{\left( T \right)} $ 以及$ {\bar v_t} \in {V_t} $ ，$ t \in T $ ，使得$$ {\boldsymbol{0}} \in \sum\limits_{i = 1}^k {{\partial ^{\rm{c}}}{f_i}} \left( {\bar {\boldsymbol{x}}} \right) + \sum\limits_{t \in T}^{} {{{\bar \mu }_t}\partial _{\boldsymbol{x}}^{\rm{c}}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right)} + {N^{\rm{c}}}\left( {C,\bar {\boldsymbol{x}}} \right) + \sum\limits_{i = 1}^k {{\varepsilon _i}} {{ B}^*} , $$ $$ {\bar \mu _t}{h_t}\left( {\bar {\boldsymbol{x}},{{\bar v}_t}} \right) = 0 ,\qquad \forall t \in T {\text{.}} $$ 注5 若
$ { \left( {{\text{UMP}}} \right) }$ 中的$ { i = 1 }$ 以及不确定集$ { {V_t} }$ ，$ { t \in T }$ ，为单点集，则$ { \left( {{\text{UMP}}} \right) }$ 退化为经典的单目标半无限优化问题，文献[20]详细刻画了这类问题的最优性条件。若$ { \left( {{\text{UMP}}} \right) }$ 中的不确定集$ { {V_t} }$ ，$ { t \in T }$ ，为单点集，则$ { \left( {{\text{UMP}}} \right) }$ 退化为经典的多目标半无限优化问题，这类问题的最优性条件和对偶问题也得到详细刻画，如文献[21]详细刻画了其对偶问题。3. 总 结
本文主要对一类带不确定参数的多目标分式半无限优化问题进行了研究。首先结合鲁棒方法和Dinkelbach方法，将该问题的鲁棒对应模型转化为一般的多目标优化问题。再借助标量化方法，建立了该多目标优化问题的标量化问题，得到了它们拟近似解之间的关系。最后，借助一类鲁棒型次微分约束规格，建立了该不确定多目标分式半无限优化问题拟近似有效解的鲁棒最优性条件。本文推广了文献[16，19]的相关结果。另一方面，对偶理论也是最优化理论研究的重点内容，因此如何用本文的方法刻画带有不确定参数的多目标分式半无限优化问题的鲁棒对偶理论，这将是我们进一步要研究的课题。
致谢 本文作者衷心感谢重庆工商大学科研团队项目(ZDPTTD201908)以及重庆工商大学研究生“创新型科研项目”(yjscxx2021-112-58)对本文的资助。
-
[1] DINKELBACH W. On nonlinear fractional programming[J]. Management Science, 1967, 13(7): 492-498. doi: 10.1287/mnsc.13.7.492 [2] YANG X M, TEO K L, YANG X Q. Symmetric duality for a class of nonlinear fractional programming problems[J]. Journal of Mathematical Analysis and Applications, 2002, 271(1): 7-15. doi: 10.1016/S0022-247X(02)00042-2 [3] LONG X J, HUANG N J, LIU Z B. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs[J]. Journal of Industrial and Management Optimization, 2008, 4(2): 287-298. doi: 10.3934/jimo.2008.4.287 [4] SUN X K, CHAI Y, ZENG J. Farkas-type results for constrained fractional programming with DC functions[J]. Optimization Letters, 2014, 8: 2299-2313. doi: 10.1007/s11590-014-0737-7 [5] ZHOU Z A, CHEN W. Optimality conditions and duality of the set-valued fractional programming[J]. Pacific Journal of Optimization, 2019, 15(4): 639-651. [6] LIU J C, YOKOYAMA K. $ \varepsilon $ -optimality and duality for multiobjective fractional programming[J]. Computers and Mathematics With Applications, 1999, 37(8): 119-128. doi: 10.1016/S0898-1221(99)00105-4[7] GUPTA P, SHIRAISHI S, YOKOYAMA K. $ \varepsilon $ -optimality without constraint qualification for multiobjective fractional problem[J]. Journal of Nonlinear and Convex Analysis, 2005, 6(2): 347-357.[8] VERMA R U. Weak $ \varepsilon $ -efficiency conditions for multiobjective fractional programming[J]. Applied Mathematics and Computation, 2013, 219(12): 6819-6827. doi: 10.1016/j.amc.2012.12.087[9] KIM M H, KIM G S, LEE G M. On $ \varepsilon $ -optimality conditions for multiobjective fractional optimization problems[J]. Fixed Point Theory and Applications, 2011, 2011: 6. doi: 10.1186/1687-1812-2011-6[10] LI G Y, JEYAKUMAR V, LEE G M. Robust conjugate duality for convex optimization under uncertainty with application to data classification[J]. Nonlinear Analysis, 2011, 74(6): 2327-2341. doi: 10.1016/j.na.2010.11.036 [11] SUN X K, LI X B, LONG X J, et al. On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization[J]. Pacific Journal of Optimization, 2017, 13(4): 621-643. [12] FAKHAR M, MAHYARINIA M, ZAFARANI J. On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization[J]. European Journal of Operational Research, 2018, 265(1): 39-48. doi: 10.1016/j.ejor.2017.08.003 [13] SUN X K, TEO K L, ZENG J, et al. Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty[J]. Optimization, 2020, 69(9): 2109-2129. doi: 10.1080/02331934.2020.1763990 [14] 赵丹, 孙祥凯. 非凸多目标优化模型的一类鲁棒逼近最优性条件[J]. 应用数学和力学, 2019, 40(6): 694-700. (ZHANG Dan, SUN Xiangkai. Some robust approximate optimality conditions for nonconvex multi-objective optimization problems[J]. Applied Mathematics and Mechanics, 2019, 40(6): 694-700.(in Chinese)ZHANG Dan, SUN Xiangkai. Some robust approximate optimality conditions for nonconvex multi-objective optimization problems[J]. Applied Mathematics and Mechanics, 2019, 40(6): 694-700. (in Chinese)) [15] LEE J H, LEE G M. On $ \varepsilon $ -solutions for robust fractional optimization problems[J]. Journal of Inequalities and Applications, 2014, 2014: 501. doi: 10.1186/1029-242X-2014-501[16] ZENG J, XU P, FU H Y. On robust approximate optimal solutions for fractional semi-infinite optimization with uncertainty data[J]. Journal of Inequalities and Applications, 2019, 2019: 45. doi: 10.1186/s13660-019-1997-7 [17] BEN-TAL A, GHAOUI L E, NEMIROVSKI A. Robust Optimization[M]. Princeton: Princeton University Press, 2009. [18] CLARKE F H. Optimization and Nonsmooth Analysis[M]. New York: John Wiley and Sons Inc, 1983. [19] ANTCZAK T. Parametric approach for approximate efficiency of robust multiobjective fractional programming problems[J]. Mathematical Methods in Applied Sciences, 2021, 44(14): 11211-11230. doi: 10.1002/mma.7482 [20] LONG X J, XIAO Y B, HUANG N J. Optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems[J]. Journal of the Operations Research Society of China, 2018, 6(2): 289-299. doi: 10.1007/s40305-017-0167-1 [21] 刘娟, 龙宪军. 非光滑多目标半无限规划问题的混合型对偶[J]. 应用数学和力学, 2021, 42(6): 595-601. (LIU Juan, LONG Xianjun. Mixed type duality for nonsmooth multiobjective semi-infinite programming problems[J]. Applied Mathematics and Mechanics, 2021, 42(6): 595-601.(in Chinese)LIU Juan, LONG Xianjun. Mixed type duality for nonsmooth multiobjective semi-infinite programming problems[J]. Applied Mathematics and Mechanics, 2021, 42(6): 595-601. (in Chinese) 期刊类型引用(3)
1. 曹琪，冯敏. 非光滑半无限多目标优化的高阶KKT最优性充分条件. 应用数学和力学. 2024(04): 502-508 . 
本站查看2. 吴浩，李向有. 非光滑多目标优化鲁棒近似解的最优性和鞍点定理. 延安大学学报(自然科学版). 2024(04): 63-68 . 百度学术3. 钟维宇，戴俨炯，蔡敢为，柳林燕. 基于认知不确定可靠性的产品功能优化算法研究. 机械制造与自动化. 2023(06): 6-9+27 . 百度学术其他类型引用(1)
-
计量
- 文章访问数: 746
- HTML全文浏览量: 263
- PDF下载量: 68
- 被引次数: 4
下载:
渝公网安备50010802005915号