Demyanov difference is an important notion in non-smooth analysis and optimization, especially in quasidifferentiable analysis and optimization.Basic operation rules have been established, including addition, positive scalar multiplication, cancellation law for addition and so on. However, some rules are represented in inclusion relation, which is inconvenient in practice. Two new rules represented in equality for Demyanov difference: the Demyanov difference of summations of two pairs of convex compact complementary sets;the Demyanov difference of a set, which is the convex hull of finite number of convex compact sets, and a convex compact set complementary to these sets,are given. These two rules are useful to compute Demyanov difference of the sub-differential and the super-differential for the summation function and the maximal (minimal) function. Therefore, they are helpful to characterize the optimality conditions for quasidifferentiable optimization with equality and inequality constraints.%Demyanov差是非光滑分析与优化,尤其是拟可微分析与优化中的一个重要概念.基本的运算法则已经形成,包括加法运算、数乘运算、加法的消去律等.但是,其中部分公式是用包含关系表达的,这给使用造成了很大的不便.为此给出了两个关于Demyanov差的新的用等式表述的运算法则:两个彼此互补的凸紧集对的和的Demyanov差,有限个凸紧集的凸包和与这些凸紧集正交互补的凸紧集的Demyanov差.这两个法则可以用于计算和函数和极大值函数的次微分与超微分的Demyanov差,从而有助于表述既含等式约束,又含不等式约束的拟可微优化的最优性条件.
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