西南大学学报 (自然科学版)  2019, Vol. 41 Issue (9): 77-86.  DOI: 10.13718/j.cnki.xdzk.2019.09.010
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  • E-拟α-预不变型凸函数与最优化    [PDF全文]
    王子元1,2, 王泾晶1, 彭再云1, 邵重阳1, 周大琼1     
    1. 重庆交通大学 数学与统计学院, 重庆 400074;
    2. 英属哥伦比亚大学 数学系, 加拿大 英属哥伦比亚省 基隆拿 V1V1V7
    摘要:研究了E-拟α-预不变型凸函数的性质与应用.首先,给出了E-拟α-预不变凸函数的定义,用例子说明了其存在性,并给出了在条件A与条件B下(半)严格E-拟α-预不变凸函数的等价刻画.其次,提出了E-拟α-预不变凸条件下的一类约束优化问题(NP1),证明了问题(NP1)可行解集、最优解集的E-α-不变凸性,并给出了问题(NP1)局部最优解的性质.最后,讨论了E-α-预不变凸函数的性质,给出了该类函数的等价刻画,获得了不等式约束下E-α-预不变凸多目标规划问题(MOP1)的最优性结果,并举例验证了所得结论的正确性.
    关键词E-拟α-预不变凸函数    非线性规划问题    最优性条件    E-α-预不变凸函数    

    凸性与广义凸性在最优化理论的研究中起着重要的作用.近年来,许多学者对凸函数进行了推广,得到了一系列广义凸函数[1-9].

    文献[10]研究了α-预不变凸函数的性质.文献[11]把文献[10]的结论推广到了拟α-预不变凸函数,并在一定假设下给出了拟α-预不变凸函数、(半)严格拟α-预不变凸函数的充要条件.

    本文将文献[11]的结果进一步推广,研究了E-拟α-预不变凸性与E-α-预不变凸性和它们在最优化问题中的应用.

    1 预备知识

    设ℝnn维欧几里得空间,K是ℝn的一个非空子集,f: K→ℝ与α: K×K→ℝ\{0}为实值函数,η: K×K→ℝn为向量值函数.

    定义1[6]  如果对于∀xyK,∀λ∈[0, 1],存在向量值映射η: K×K→ℝn使得y+λα(xy)η(xy)∈K,则称K是关于αηα-不变凸集.

    定义2[6]  设K是关于αηα-不变凸集.若∀xyK,∀λ∈[0, 1],满足

    $ f(\mathit{\boldsymbol{y}} + \lambda \alpha (\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}})\eta (\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}})) \le \max \{ f(\mathit{\boldsymbol{x}}),f(\mathit{\boldsymbol{y}})\} $

    则称f是关于αη的拟α-预不变凸函数.

    设存在映射E: ℝn→ℝn.下面给出E-α-不变凸集的定义.

    定义3  如果∀xyK,∀λ∈[0, 1],满足E(y)+λα(E(x),E(y))η(E(x),E(y))∈K,则称K是关于αηE-α-不变凸集.

    例1  设K=[-1, 0],对∀xKE(x)=|x|-1.对∀xyK,令

    $ \alpha (x,y) = \left\{ {\begin{array}{*{20}{l}} {xy}&{若\;x \ne 0\;且\;y \ne 0}\\ 1&{若\;x,y\;至少一个为零} \end{array}} \right. $
    $ \eta (x,y) = \left\{ \begin{array}{l} - \frac{1}{{2x}}\;\;\;\;若\;x \ne 0\;且\;y \ne 0\\ 0\;\;\;\;\;\;\;\;\;若\;x,y\;至少一个为零 \end{array} \right. $

    分析  1)当x≠-1且y≠-1时,对于∀xyK,∀λ∈[0, 1],有E(y)+λα(E(x),E(y))η(E(x),E(y))=$\left( {\frac{\lambda }{2} - 1} \right)$(1+y)∈K.

    2) 当xy至少一个为-1时,对于∀xyK,∀λ∈[0, 1],有E(y)+λα(E(x),E(y))η(E(x),E(y))=-y-1∈K,故K是关于αηE-α-不变凸集.

    定义4  设K是关于αηE-α-不变凸集.若对∀xyK,∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) \le \lambda f(E(\mathit{\boldsymbol{x}})) + (1 - \lambda )f(E(\mathit{\boldsymbol{y}})) $

    则称f是关于αηE-α-预不变凸函数.

    定义5  设K是关于αηE-α-不变凸集.若对∀xyK,∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) \le \max \{ f(E(\mathit{\boldsymbol{x}})),f(E(\mathit{\boldsymbol{y}}))\} $

    则称f是关于αηE-拟α-预不变凸函数.

    注1  由定义4与定义5可知,E-拟α-预不变凸函数是E-α-预不变凸函数的真推广.但反之,E-拟α-预不变凸函数不一定是E-α-预不变凸函数.

    下例说明E-拟α-预不变凸函数的存在性.

    例2  设K=(0,1],对∀xKE(x)=x2.对∀xyK,令α(xy)=xy$\eta \left( {x, y} \right) = \frac{{x - y}}{{xy}}$.定义f: K→ℝ为f(x)=x2.

    分析  容易证明K是关于αηE-α-不变凸集.对于∀xyK,∀λ∈[0, 1],有

    $ \begin{array}{*{20}{l}} {f(E(y) + \lambda \alpha (E(x),E(y))\eta (E(x),E(y))) = f\left( {\lambda {x^2} + (1 - \lambda ){y^2}} \right) = }\\ {{\lambda^2}{x^4} + {{(1 - \lambda )}^2}{y^4} + 2\lambda (1 - \lambda ){x^2}{y^2} \le {\lambda ^2}{x^4} + {{(1 - \lambda )}^2}{y^4} + \lambda (1 - \lambda )\left( {{x^4} + {y^4}} \right) = }\\ {(1 - \lambda ){y^4} + \lambda {x^4} = (1 - \lambda )f(E(y)) + \lambda f(E(x)) \le \max \{ f(E(x)),f(E(y))\} } \end{array} $

    f是关于αηE-拟α-预不变凸函数.

    下例说明E-拟α-预不变凸函数不一定是E-α-预不变凸函数.

    例3  设K=[-1, 1]×[-1, 1],对∀(xy)∈KE(xy)=(x2y2).对∀(x1y1),(x2y2)∈K,令α((x1y1),(x2y2))=x1y2+2,η((x1y1),(x2y2))=$\left( {\frac{{{x_1} - {x_2}}}{{{x_1}{y_2} + 2}}, \frac{{{y_1} - {y_2}}}{{{x_1}{y_2} + 2}}} \right)$.定义g: K→ℝ为g(xy)=y2-x3(图 12).

    图 1 g(xy)=y2-x3
    图 2 g(xy)=y2-x3

    分析  根据图 12及定义5,易知g是关于αηE-拟α-预不变凸函数.取K上两点ω=(0,1)与ν=(1,1),令λ=$\frac{1}{2}$,有

    $ g(E(\mathit{\boldsymbol{v}}) + \lambda \alpha (E(\mathit{\boldsymbol{\omega }}),E(\mathit{\boldsymbol{v}}))\eta (E(\mathit{\boldsymbol{\omega }}),E(\mathit{\boldsymbol{v}}))) = g\left( {\frac{1}{2},1} \right) = 0.875 > \frac{1}{2}g(0,1) + \frac{1}{2}g(1,1) = 0.5 $

    g不是关于αηE-α-预不变凸函数.

    定义6[12]  设K⊂ℝn是非空凸集,f是定义在K上的函数.

    1) 如果对于∀xyK,∀λ∈[0, 1],有

    $ f(\lambda \mathit{\boldsymbol{x}} + (1 - \lambda )\mathit{\boldsymbol{y}}) \le \max \{ f(\mathit{\boldsymbol{x}}),f(\mathit{\boldsymbol{y}})\} $

    则称fK上的拟凸函数;

    2) 如果对于∀xyKxy,∀λ∈[0, 1],有

    $ f(\lambda \mathit{\boldsymbol{x}} + (1 - \lambda )\mathit{\boldsymbol{y}}) < \max \{ f(\mathit{\boldsymbol{x}}),f(\mathit{\boldsymbol{y}})\} $

    则称fK上的严格拟凸函数;

    3) 如果对于∀xyKf(x)≠f(y),∀λ∈[0, 1],有

    $ f(\lambda \mathit{\boldsymbol{x}} + (1 - \lambda )\mathit{\boldsymbol{y}}) < \max \{ f(\mathit{\boldsymbol{x}}),f(\mathit{\boldsymbol{y}})\} $

    则称fK上的半严格拟凸函数.

    将严格和半严格拟凸函数进行推广,可分别得到严格与半严格E-拟α-预不变凸函数的定义.

    定义7  设K是关于αηE-α-不变凸集,f是定义在K上的函数.

    1) 若对∀xyKE(x)≠E(y),∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) < \max \{ f(E(\mathit{\boldsymbol{x}})),f(E(\mathit{\boldsymbol{y}}))\} $

    则称f是关于αη的严格E-拟α-预不变凸函数;

    2) 若对∀xyKf(E(x))≠f(E(y)),∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) < \max \{ f(E(\mathit{\boldsymbol{x}})),f(E(\mathit{\boldsymbol{y}}))\} $

    则称f是关于αη的半严格E-拟α-预不变凸函数.

    2 E-拟α-预不变凸性与约束优化问题

    本节将讨论(半)严格E-拟α-预不变凸函数的充要条件,及E-拟α-预不变凸型约束优化问题的最优性结果.下面给出后面将用到的关于映射αη的一个重要引理.

    引理1  设K是关于映射αηE-α-不变凸集,且E(·)是满射.若对∀xyK,∀λ∈[0, 1],有

    $ \eta (E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) = - \lambda \eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) $
    $ \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) = \alpha (E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) $

    则对∀λ1λ2∈(0,1],有

    $ \begin{array}{*{20}{l}} {\eta \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {\left( {{\lambda _1} - {\lambda _2}} \right)\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \end{array} $

    $ \begin{array}{*{20}{l}} {\alpha \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right. = }\\ {\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \end{array} $

      根据假设,有

    $ \begin{array}{l} \begin{array}{*{20}{l}} {\eta \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha [E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {\eta \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha [E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) + } \right.} \end{array}\\ \begin{array}{*{20}{l}} {\left. {\left( {{\lambda _2} - {\lambda _1}} \right)\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {\eta \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right. + } \end{array}\\ \left( {\frac{{{\lambda _1} - {\lambda _2}}}{{{\lambda _1}}}} \right)\alpha \left( {E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right)\eta \left( {E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right.\\ \begin{array}{*{20}{l}} {\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) = - \frac{{{\lambda _1} - {\lambda _2}}}{{{\lambda _1}}}\eta (E(\mathit{\boldsymbol{y}}),}\\ {\left. {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right)\left( {{\lambda _1} - {\lambda _2}} \right)\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \end{array} \end{array} $

    $ \begin{array}{l} \begin{array}{*{20}{l}} {\alpha \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha [E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {\alpha \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha [E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha [E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) + } \right.} \end{array}\\ \frac{{{\lambda _1} - {\lambda _2}}}{{{\lambda _1}}}\alpha \left( {E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right)\eta (E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + \\ \begin{array}{*{20}{l}} {\left. {{\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {\alpha \left( {E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \end{array} \end{array} $

    我们给出条件A与条件B的定义.

    条件A  设K是关于映射αηE-α-不变凸集.称函数f满足条件A,如果对∀xyK,有

    $ f(E(\mathit{\boldsymbol{y}}) + \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) \le f(E(\mathit{\boldsymbol{x}})) $

    条件B  设K是关于映射αηE-α-不变凸集.称αη满足条件B,如果对∀xyK,∀λ∈[0, 1],有

    $ \eta (E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) = - \lambda \eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) $
    $ \eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) = (1 - \lambda )\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) $

    借助条件A与条件B,我们给出严格与半严格E-拟α-预不变凸函数的等价刻画.

    定理1  设K是关于αηE-α-不变凸集,映射E(·)是满射,且f满足条件A,η满足条件B.若对∀xyKE(x)≠E(y),∀λ∈[0, 1],有

    $ \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) \ne 0,\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) \ne 0 $

    $ \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) = \alpha (E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) $

    成立,则fK上是关于映射αη的严格E-拟α-预不变凸函数,当且仅当对∀xyK,∀λ∈(0,1],g(λ)=f(E(y)+λα(E(x),E(y))η(E(x),E(y)))是严格拟凸的.

      先证定理的必要性.设g(λ)=f(E(y)+λα(E(x),E(y))η(E(x),E(y)))是严格拟凸的.根据定义,对∀xyKE(x)≠E(y),∀λ∈[0, 1],

    $ \begin{array}{*{20}{l}} {f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) = g(\lambda ) = g(1 \cdot \lambda + 0 \cdot (1 - \lambda )) < }\\ {\max \{ g(1),g(0)\} = \max \{ f(E(\mathit{\boldsymbol{y}}) + \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))),f(E(\mathit{\boldsymbol{y}}))\} } \end{array} $

    根据条件A,可知

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) < \max \{ f(E(\mathit{\boldsymbol{x}})),f(E(\mathit{\boldsymbol{y}}))\} $

    f是关于映射αη的严格E-拟α-预不变凸函数.

    再证定理的充分性.设f是关于映射αη的严格E-拟α-预不变凸函数.根据定义,对∀xyKE(x)≠E(y),∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) < \max \{ f(E(\mathit{\boldsymbol{x}})),f(E(\mathit{\boldsymbol{y}}))\} $

    由条件可知E(x)≠E(y)时有α(E(x),E(y))≠0,η(E(x),E(y))≠0成立,则对于∀λ1λ2β∈[0, 1],λ1λ2(不失一般性,假设λ2λ1),有

    $ \begin{array}{*{20}{l}} {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) \ne }\\ {E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),\quad \forall E(\mathit{\boldsymbol{x}}) \ne E(\mathit{\boldsymbol{y}})} \end{array} $

    根据引理1,下列不等式成立

    $ \begin{array}{*{20}{l}} {g\left( {\beta {\lambda _1} + (1 - \beta ){\lambda _2}} \right) = f\left( {E(\mathit{\boldsymbol{y}}) + \left( {\beta {\lambda _1} + (1 - \beta ){\lambda _2}} \right)\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {f\left( {E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) + \beta \left( {{\lambda _1} - {\lambda _2}} \right)\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) = }\\ {f\left( {E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) + \beta \alpha \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right.} \right.} \end{array} $
    $ \begin{array}{l} E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta \left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})),E(\mathit{\boldsymbol{y}})} \right),E(\mathit{\boldsymbol{y}}) + \\ \begin{array}{*{20}{l}} {\left. {{\lambda _2}\alpha (E(\mathit{\boldsymbol{x}})E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right) < }\\ {\max \left\{ {f\left( {E(\mathit{\boldsymbol{y}}) + {\lambda _1}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right),f\left( {E(\mathit{\boldsymbol{y}}) + {\lambda _2}\alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))} \right)} \right\}} \end{array}\\ \max \left\{ {g\left( {{\lambda _1}} \right),g\left( {{\lambda _2}} \right)} \right\} \end{array} $

    g(λ)为严格拟凸函数,证毕.

    使用类似方法,可以得到关于半严格E-拟α-预不变凸函数的如下刻画.

    定理2  设K是关于αηE-α-不变凸集,映射E(·)是满射,且f满足条件A,η满足条件B.若对∀xyKf(E(x))≠f(E(y)),∀λ∈[0, 1],有α(E(x),E(y))=α(E(y),E(y)+λα(E(x),E(y))η(E(x),E(y)))成立,则fK上是关于映射αη的半严格E-拟α-预不变凸函数,当且仅当对∀xyK,∀λ∈[0, 1],g(λ)=f(E(y)+λα(E(x),E(y))η(E(x),E(y)))是半严格拟凸的.

    考虑如下非线性规划问题(NP1)

    $ \min f(E(\mathit{\boldsymbol{x}})) $
    $ {\rm{s}}.\;{\rm{t}}.\;\;\;\;\;{g_i}(E(\mathit{\boldsymbol{x}})) \le {b_i}\quad i = 1,2,3, \cdots ,n $
    $ \mathit{\boldsymbol{x}} \in K $

    其中: K是关于映射αηE-α-不变凸集;函数fgi(i=1,2,3,…,n)为关于映射αηE-拟α-预不变凸函数.

    规定

    $ I = \{ 1,2,3, \cdots ,n\} ,{X_i} = \left\{ {\mathit{\boldsymbol{x}}|{g_i}(E(\mathit{\boldsymbol{x}})) \le {b_i},\mathit{\boldsymbol{x}} \in K} \right\} $
    $ X = \left\{ {\mathit{\boldsymbol{x}}|{g_i}(E(\mathit{\boldsymbol{x}})) \le {b_i},i = 1,2,3, \cdots ,n,\mathit{\boldsymbol{x}} \in K} \right\} $

    使用与文献[5]在引理2中类似的证明方法,可以得到引理2.

    引理2  若Ki(iI)皆为关于同一αηE-α-不变凸集,则集合$\bigcap\limits_{i \in I} {{K_i}} $仍然是关于同一αηE-α-不变凸集.

    下面给出问题(NP1)的3个最优性结果.

    定理3  非线性规划问题(NP1)的可行解集是关于映射αηE-α-不变凸集.

      设xy是问题(NP1)的可行解,则对∀xyXi,∀λ∈[0, 1],有

    $ E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) \in K $

    gi(x)的E-拟α-预不变凸性可知,对∀xyXi,∀λ∈[0, 1]有

    $ {g_i}(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) \le \max \left\{ {{g_i}(E(\mathit{\boldsymbol{x}})),{g_i}(E(\mathit{\boldsymbol{y}}))} \right\} \le {b_i} $

    $ E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) \in {X_i} $

    Xi是关于映射αηE-α-不变凸集.

    因为$X = \bigcap\limits_{i \in I} {{K_i}} $,根据引理2可知X是关于映射αηE-α-不变凸集,证毕.

    定理4  非线性规划问题(NP1)的最优解集S是关于映射αηE-α-不变凸集.

      设x1*x2*是问题(NP1)的最优解,

    $ {f^*} = \min f(E(\mathit{\boldsymbol{x}})) $

    则有

    $ \mathit{\boldsymbol{x}}_1^*,\mathit{\boldsymbol{x}}_2^* \in S,f\left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right)} \right) = f\left( {E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right) = {f^*} $

    由定理3得可行解集X是关于映射αηE-α-不变凸集,故

    $ E\left( {\mathit{\boldsymbol{x}}_2^*} \right) + \lambda \alpha \left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right),E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right)\eta \left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right),E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right) \in X,\forall \lambda \in [0,1] $

    对∀x1*x2*∈S,∀λ∈[0, 1],有

    $ f\left( {E\left( {\mathit{\boldsymbol{x}}_2^*} \right) + \lambda \alpha \left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right),E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right)\eta \left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right),E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right)} \right) \le \max \left\{ {f\left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right)} \right),f\left( {E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right)} \right\} = {f^ * } $ (1)

    根据(1)式可知对∀x1*x2*∈S,∀λ∈[0, 1],有

    $ E\left( {\mathit{\boldsymbol{x}}_2^*} \right) + \lambda \alpha \left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right),E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right)\eta \left( {E\left( {\mathit{\boldsymbol{x}}_1^*} \right),E\left( {\mathit{\boldsymbol{x}}_2^*} \right)} \right) \in S $

    S是关于映射αηE-α-不变凸集,证毕.

    定理5  如果x*是非线性规划问题(NP1)的局部最优解,则x*是(NP1)的全局最优解.

      设x*是(NP1)的局部最优解,则存在δ>0,使得

    $ f\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) < f(E(\mathit{\boldsymbol{x}})),\forall \mathit{\boldsymbol{x}} \in X \cap B\left( {{\mathit{\boldsymbol{x}}^*};\delta } \right) $

    其中B(x*δ)={x|0<‖x-x*‖≤δxK}.若x*不是问题(NP1)的全局最优解,则必存在xX(xx*),使得f(E(x))<f(E(x*)).

    由于f是关于映射αηE-拟α-预不变凸函数,对∀λ∈[0, 1],有

    $ \begin{array}{*{20}{l}} {f\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right) + \lambda \alpha \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)\eta \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \right) \le }\\ {\max \left\{ {f(E(\mathit{\boldsymbol{\bar x}})),f\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \right\} = f\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \end{array} $

    $ \bar \lambda = \min \left\{ {1,\frac{\delta }{{\left\| {\alpha \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)\eta \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \right\|}}} \right\} $

    显然有λ∈(0,1].对∀λ∈[0,λ],令

    $ \mathit{\boldsymbol{x}} = E\left( {{\mathit{\boldsymbol{x}}^*}} \right) + \lambda \alpha \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)\eta \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) $

    $ \left\| {\mathit{\boldsymbol{x}} - {\mathit{\boldsymbol{x}}^*}} \right\| = \lambda \left\| {\alpha \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)\eta \left( {E(\mathit{\boldsymbol{\bar x}}),E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \right\| \le \delta $

    由定理3可知xX,则xXB(x*δ),且f(E(x))≤f(E(x*)),这与x*是问题(NP1)的局部最优解矛盾.故x*是问题(NP1)的全局最优解,证毕.

    例4  考虑下面的非线性规划问题(NP2)

    $ \min f(E(x)) $
    $ {\rm{s}}.{\rm{t}}.\;\;\;\;\;g(E(x)) \le 2 $
    $ x \in K $

    其中集合$K = \left( {0, \frac{1}{2}} \right]$$E\left( x \right) = {x^2} + \frac{1}{4}$.定义g(x)=log2(1-x),f(x)=x2-x+1.对∀xyK,令α(xy)=xy$\eta \left( {x, y} \right) = \frac{{x - y}}{{xy}}$.由定义3和定义4易知集合A是关于αηE-α-不变凸集,f(x)与g(x)是关于αηE-拟α-预不变凸函数.

    (NP2) 的可行解集为$X = \left\{ {x\left| {0 < x \le \frac{1}{2}} \right.} \right\}$.易知$x = \frac{1}{2}$为问题(NP2)的局部最优解,且是问题(NP2)的全局最优解.该结果验证了定理5.

    3 E-α-预不变凸性与多目标规划

    本节主要讨论E-α-预不变凸性以及其在一类多目标规划问题中的应用.首先给出关于αηE-α-预不变凸函数的两个性质.

    定理6  设fK上关于αηE-α-预不变凸函数,且f可微.若▽f≥(≤)0,且对于∀xyKαη满足α(E(x),E(y))η(E(x),E(y))≥(≤)E(x)-E(y),则下列不等式成立:

    $ f(E(\mathit{\boldsymbol{x}})) - f(E(\mathit{\boldsymbol{y}})) \ge \nabla f{(E(\mathit{\boldsymbol{y}}))^{\rm{T}}}(E(\mathit{\boldsymbol{x}}) - E(\mathit{\boldsymbol{y}})) $

      根据▽f≥(≤)0,α(E(x),E(y))η(E(x),E(y))≥(≤)E(x)-E(y)成立,对∀xyK,∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda (E(\mathit{\boldsymbol{x}}) - E(\mathit{\boldsymbol{y}}))) \le f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) $ (2)

    E-α-预不变凸函数的定义可知,对∀xyK,∀λ∈[0, 1],有

    $ f(E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) \le \lambda f(E(\mathit{\boldsymbol{x}})) + (1 - \lambda )f(E(\mathit{\boldsymbol{y}})) $ (3)

    由(2),(3)式可知,下列不等式成立

    $ \begin{array}{*{20}{c}} {f(E(\mathit{\boldsymbol{y}}) + \lambda (E(\mathit{\boldsymbol{x}}) - E(\mathit{\boldsymbol{y}}))) \le \lambda f(E(\mathit{\boldsymbol{x}})) + (1 - \lambda )f(E(\mathit{\boldsymbol{y}}))}\\ {f(E(\mathit{\boldsymbol{y}}) + \lambda (E(\mathit{\boldsymbol{x}}) - E(\mathit{\boldsymbol{y}}))) \le f(E(\mathit{\boldsymbol{y}})) + \lambda (f(E(\mathit{\boldsymbol{x}})) - f(E(\mathit{\boldsymbol{y}})))}\\ {f(E(\mathit{\boldsymbol{y}}) + \lambda (E(\mathit{\boldsymbol{x}}) - E(\mathit{\boldsymbol{y}}))) - f(E(\mathit{\boldsymbol{y}})) \le \lambda \left( {f(E(\mathit{\boldsymbol{x}})) - f(E(\mathit{\boldsymbol{y}}))} \right)} \end{array} $ (4)

    λ=0或λ=1时,(4)式恒成立.现考虑λ∈(0,1)时的情况,显然有

    $ f(E(\mathit{\boldsymbol{x}})) - f(E(\mathit{\boldsymbol{y}})) \ge \frac{{f(E(\mathit{\boldsymbol{y}}) + \lambda (E(\mathit{\boldsymbol{x}}) - E(\mathit{\boldsymbol{y}}))) - f(E(\mathit{\boldsymbol{y}}))}}{\lambda } $

    λ→0+时,得到不等式f(E(x))-f(E(y))≥▽f(E(y))T(E(x)-E(y)),证毕.

    定理7  设K是关于映射αηE-α-不变凸集,E(·)是满射.函数f满足条件A,映射η满足条件B.若对∀xyK,∀λ∈[0, 1],有

    $ \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}})) = \alpha (E(\mathit{\boldsymbol{y}}),E(\mathit{\boldsymbol{y}}) + \lambda \alpha (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))\eta (E(\mathit{\boldsymbol{x}}),E(\mathit{\boldsymbol{y}}))) $

    成立,则函数f是关于αηE-α-预不变凸函数当且仅当对∀xyK,∀λ∈[0, 1],g(λ)=f(E(y)+λα(E(x),E(y))η(E(x),E(y)))为凸函数.

      利用定理1的方法可类似证明.

    考虑下列多目标规划问题(MOP1)

    $ \min f(E(\mathit{\boldsymbol{x}})) = \left( {{f_1}(E(\mathit{\boldsymbol{x}})),{f_2}(E(\mathit{\boldsymbol{x}})), \cdots ,{f_m}(E(\mathit{\boldsymbol{x}}))} \right) $
    $ {\rm{s}}.\;{\rm{t}}.\;\;\;g(E(\mathit{\boldsymbol{x}})) = \left( {{g_1}(E(\mathit{\boldsymbol{x}})),{g_2}(E(\mathit{\boldsymbol{x}})), \cdots ,{g_n}(E(\mathit{\boldsymbol{x}}))} \right) \le 0 $
    $ h(E(\mathit{\boldsymbol{x}})) = \left( {{h_1}(E(\mathit{\boldsymbol{x}})),{h_2}(E(\mathit{\boldsymbol{x}})), \cdots ,{h_p}(E(\mathit{\boldsymbol{x}}))} \right) = 0 $
    $ \mathit{\boldsymbol{x}} \in K $

    其中K⊂ℝn是关于αηE-α-不变凸集,fi(i=1,2,3,…,m),gj(j=1,2,3,…,n)与hk(k=1,2,3,…,p)是K上关于同一αηE-α-预不变凸函数,设多目标规划问题(MOP1)的可行域为D.

    先引入以下几个符号:设x=(x1x2,…,xn),y=(y1y2,…,yn).

    $ \mathit{\boldsymbol{x}} = \mathit{\boldsymbol{y}} \Leftrightarrow {x_i} = {\mathit{\boldsymbol{y}}_i}(i = 1,2,3, \cdots ,n);x < \mathit{\boldsymbol{y}} \Leftrightarrow {x_i} < {\mathit{\boldsymbol{y}}_i}(i = 1,2,3, \cdots ,n) $
    $ \mathit{\boldsymbol{x}} \le q\mathit{\boldsymbol{y}} \Leftrightarrow {x_i} \le {\mathit{\boldsymbol{y}}_i}(i = 1,2,3, \cdots ,n);\mathit{\boldsymbol{x}} \le \mathit{\boldsymbol{y}} \Leftrightarrow \mathit{\boldsymbol{x}} \le q\mathit{\boldsymbol{y}},\mathit{\boldsymbol{x}} \ne \mathit{\boldsymbol{y}} $
    $ \mathbb{R}_ + ^n = \left\{ {\left( {{x_1},{x_2}, \cdots ,{x_n}} \right)|{x_i} \geqslant 0,i = 1,2,3, \cdots ,n} \right\} $
    $ \mathbb{R}_{ + + }^n = \left\{ {\left( {{x_1},{x_2}, \cdots ,{x_n}} \right)|{x_i} > 0,i = 1,2,3, \cdots ,n} \right\} $

    定义8  设x*是多目标规划问题(MOP1)的可行解,若不存在另一可行解x,使f(E(x))≤f(E(x*))(或f(E(x))<f(E(x*)))成立,则称x*为该问题的有效解(或弱有效解).

    定理8  设x*是多目标规划问题(MOP1)的可行解,figjhk具有一阶连续偏导数,$\sum\limits_{k = 1}^p {{v_k}{h_k}} $v=(v1v2,…,vp)∈ℝp是关于αηE-α-预不变凸函数,并且定理6中的条件皆成立.若存在λ∈ℝ++mμ∈ℝ+nv∈ℝp,使得

    $ \sum\limits_{i = 1}^m {{\lambda _i}} \nabla {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \sum\limits_{j = 1}^n {{\mu _j}} \nabla {g_j}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \sum\limits_{k = 1}^p {{v_k}} \nabla {h_k}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) = 0 $ (5)
    $ \sum\limits_{j = 1}^n {{\mu _j}} {g_j}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) = 0 $ (6)

    x*是多目标规划问题(MOP1)的有效解(弱有效解).

      反证.假设x*不是(MOP1)的有效解(弱有效解),则存在xD,使得f(E(x))≤(<)f(E(x*)).由于λ∈ℝ++m,下列不等式成立

    $ \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}(E(\mathit{\boldsymbol{x}})) < \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) $ (7)

    由于figj$\sum\limits_{k = 1}^p {{v_k}{h_k}} $E-α-预不变凸函数,且满足定理6的假设,则以下不等式成立

    $ {f_i}(E(\mathit{\boldsymbol{x}})) - {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) \geqslant \nabla {f_i}{\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)^{\text{T}}}\left( {E(\mathit{\boldsymbol{x}}) - E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)\quad i = 1,2,3, \cdots ,m $ (8)
    $ {g_j}(E(\mathit{\boldsymbol{x}})) - {g_j}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) \geqslant \nabla {g_j}{\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)^{\text{T}}}\left( {E(\mathit{\boldsymbol{x}}) - E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)\quad j = 1,2,3, \cdots ,n $ (9)
    $ \sum\limits_{k = 1}^p {{v_k}} {h_k}(E(\mathit{\boldsymbol{x}})) - \sum\limits_{k = 1}^p {{v_k}} {h_k}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) \geqslant {\left( {\nabla \sum\limits_{k = 1}^p {{v_k}} {h_k}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \right)^{\text{T}}}\left( {E(\mathit{\boldsymbol{x}}) - E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) $ (10)

    将(8)式中第i式乘以λi,(9)中第j式乘以μj,并与(10)式相加得到

    $ \begin{array}{l} \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}(E(\mathit{\boldsymbol{x}})) - \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \sum\limits_{j = 1}^n {{\mu _j}} {g_j}(E(\mathit{\boldsymbol{x}})) - \sum\limits_{j = 1}^n {{\mu _j}} {g_j}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \\ \sum\limits_{k = 1}^p {{v_k}} {h_k}(E(\mathit{\boldsymbol{x}})) - \sum\limits_{k = 1}^p {{v_k}} {h_k}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) \ge \\ {\left( {\sum\limits_{i = 1}^m {{\lambda _i}} \nabla {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \sum\limits_{j = 1}^n {{\mu _j}} \nabla {g_j}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \nabla \sum\limits_{k = 1}^p {{v_k}} {h_k}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right)} \right)^{\rm{T}}}\left( {E(\mathit{\boldsymbol{x}}) - E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) = 0 \end{array} $

    则有

    $ \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}(E(\mathit{\boldsymbol{x}})) \ge \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) - \sum\limits_{j = 1}^n {{\mu _j}} {g_j}(E(\mathit{\boldsymbol{x}})) + \sum\limits_{j = 1}^n {{\mu _j}} {g_j}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) \ge \sum\limits_{i = 1}^m {{\lambda _i}} {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) $ (11)

    式(11)与(7)矛盾,故x*为有效解(弱有效解),证毕.

    下面给出例5来证明以上结果的正确性.

    例5  考虑多目标规划问题(MOP2)

    $ \min \left( {{f_1}(E(x,y)),{f_2}(E(x,y))} \right) $
    $ {\rm{s}}.\;{\rm{t}}.\;\;\;\;g(E(x,y)) \le 0 $
    $ h(E(x,y)) = 0 $
    $ (x,y) \in K $

    其中K=[0, 1]×[0, 1].对∀(xy)∈KE(xy)=(x2y2),f1(xy)=x2f2(xy)=x3-1,g(xy)=${y^{\frac{3}{2}}} - 1$h(xy)=0.对∀(x1y1),(x2y2)∈K,令

    $ \alpha \left( {\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)} \right) = {x_1} + {y_2} + 1,\eta \left( {\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)} \right) = \left( {\frac{{{x_1} - {x_2}}}{{{x_1} + {y_2} + 1}},\frac{{{y_1} - {y_2}}}{{{x_1} + {y_2} + 1}}} \right) $

    分析  容易证明K是关于αηE-α-不变凸集,f1f2gh是关于αηE-α-预不变凸函数且满足定理6的条件,x*=(0,0)是(MOP2)的一个可行解.任取λ1λ2>0,v∈ℝ,μ=0有

    $ \sum\limits_{i = 1}^2 {{\lambda _i}} \nabla {f_i}\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + \mu \nabla g\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) + v\nabla h\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) = 0 $
    $ \mu g\left( {E\left( {{\mathit{\boldsymbol{x}}^*}} \right)} \right) = 0 $

    因此x*=(0,0)是(MOP2)的有效解.

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    E-α-prequasiinvex-Type Functions and Their Optimization
    WANG Zi-yuan1,2, WANG Jing-jing1, PENG Zai-yun1, SHAO Chong-yang1, ZHOU Da-qiong1     
    1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China;
    2. Department of Mathematics, University of British Columbia, Kelowna, BC V1V1V7, Canada
    Abstract: In this paper, we mainly study the concept of E-α-prequasiinvex-type functions and their applications. First, the definition of E-α-prequasiinvex functions is given, with examples demonstrating their existence. The necessary and sufficient conditions of strictly (semi-strictly) E-α-prequasiinvex functions under the assumption of Condition A and Condition B are discussed. Then, the nonlinear programming problem (NP1) with inequality constraints under the E-α-prequasiinvex condition is proposed. The E-α-invexity of the feasible solution set and optimal solution set of (NP1) is proved, and the property of the local optimal solution of NP1 is given. Finally, the concepts of E-α-preinvex functions are discussed, the equivalent characterization of E-α-preinvex functions is given, and the optimal result of the multi-objective programming problem under E-α-preinvex condition is obtained. Examples are given to verify the conclusions reached.
    Key words: E-α-prequasiinvex function    nonlinear programming problem    optimal criterion    E-α-preinvex function    
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