In this paper, we derive deterministic inner approximations for single and joint independent or dependent probabilistic constraints based on classical inequalities from probability theory such as the one-sided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality (see Pinter, 1989). The dependent case has been modelled via copulas. New assumptions under which the bounds based approximations are convex allowing to solve the problem efficiently are derived. When the convexity condition can not hold, an efficient sequential convex approximation approach is further proposed to solve the approximated problem. Piecewise linear and tangent approximations are also provided for Chernoff and Hoeffding inequalities allowing to reduce the computational complexity of the associated optimization problem. Extensive numerical results on a blend planning problem under uncertainty are finally provided allowing to compare the proposed bounds with the Second Order Cone (SOCP) formulation and Sample Average Approximation (SAA).

在本文中，我们根据概率论中的经典不等式，如单边切比雪夫不等式、伯恩斯坦不等式、切尔诺夫不等式和霍夫丁不等式（参见 Pinter，1989），推导出单个和联合独立或相关概率约束的确定性内近似。依赖案例已通过 copula 建模。推导出基于边界的近似是凸的允许有效地解决问题的新假设。当凸性条件不成立时，进一步提出了一种有效的顺序凸逼近方法来解决逼近问题。还为 Chernoff 和 Hoeffding 不等式提供了分段线性和切线近似，以减少相关优化问题的计算复杂性。