特征函数$1_{\left(0,+\infty \right)}\left(z\right)$的一个光滑D.C.近似
A Smooth D.C. Approximation of the Characteristic Function $1_{\left(0,+\infty \right)}\left(z\right)$

特征函数1(0,+∞)(z)已经广泛地使用在实践中，许多重要的实际问题，如概率约束优化问题，可以表示为与特征函数相关的问题，该问题通常是非凸和非光滑的，因而在数值计算上有困难。本文构造了特征函数1(0,+∞)(z)的一个光滑D.C.近似函数。讨论了函数φ(z,t)的一些性质。在一定条件下，当所涉及的参数足够小时，所提出的光滑近似函数收敛到1(0,+∞)(z)。基于该函数，构建了概率约束优化问题的一个等价的近似问题。
Characteristic function1(0,+∞)(z)have been widely used in practice, and many important practical problems, such as probabilistic constrained optimization problems, can be expressed as the problems associated with the characteristic function, which are usually non-convex and non-smooth, and thus difficult to calculate numerically. In this paper, we construct a smooth D.C. approximation functionφ(z,t)of the characteristic function. Some properties of the functionφ(z,t)are discussed. Under certain conditions, the proposed smooth approximation function converges to1(0,+∞)(z)when the involved parameters are sufficiently small. Based on this function, an equivalent approximation of the constrained optimization problem is constructed.