Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers
Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers
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DOI：
10.1007/s10957-016-0943-9
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发表时间：
2016-08-01
影响因子：
1.9
通讯作者：
Branda, Martin
中科院分区：
文献类型：
--
作者：
Adam, Lukas;Branda, Martin
讨论了具有离散分布的非线性随机函数可微的机会约束问题。我们允许非凸函数的约束和目标。我们将问题转化为一个混合整数非线性规划，并将整数变量松弛为连续变量。我们将松弛问题转化为具有互补约束的数学问题，并通过扩大可行解集来正则化它。对于所有考虑的问题，我们推导出必要的最优性条件的基础上Fr，切特对象对应于强平稳性。我们讨论了不动点和极小值之间的关系。我们提出了两个迭代算法找到一个稳定点的原始问题。第一个是基于放松的重新制定，而第二个采用其正则化的版本。在约束条件的有效性下，我们证明了正则化问题的稳定点收敛到松弛重构的稳定点，并且在附加条件下，它甚至是原问题的稳定点。最后通过一个数值例子对本文进行了总结。
We deal with chance constrained problems with differentiable nonlinear random functions and discrete distribution. We allow nonconvex functions both in the constraints and in the objective. We reformulate the problem as a mixed-integer nonlinear program and relax the integer variables into continuous ones. We approach the relaxed problem as a mathematical problem with complementarity constraints and regularize it by enlarging the set of feasible solutions. For all considered problems, we derive necessary optimality conditions based on Fr,chet objects corresponding to strong stationarity. We discuss relations between stationary points and minima. We propose two iterative algorithms for finding a stationary point of the original problem. The first is based on the relaxed reformulation, while the second one employs its regularized version. Under validity of a constraint qualification, we show that the stationary points of the regularized problem converge to a stationary point of the relaxed reformulation and under additional condition it is even a stationary point of the original problem. We conclude the paper by a numerical example.