Stochastic programming problems with generalized integrated chance constraints
Stochastic programming problems with generalized integrated chance constraints
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DOI：
10.1080/02331934.2011.587007
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发表时间：
2012-01-01
期刊：
影响因子：
2.2
通讯作者：
Branda, Martin
中科院分区：
文献类型：
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作者：
Branda, Martin
如果优化问题中的约束依赖于随机参数，我们希望确保它们以高水平的可靠性得到满足。最自然的方法是使用机会约束。然而，由此产生的问题很难解决。我们提出了一种使用罚函数的随机规划的替代公式。惩罚的预期可以作为约束，导致广义的综合机会约束，或者作为惩罚条款纳入目标。我们证明了惩罚问题在相当温和的条件下是渐近等价的。讨论了样本逼近技术在具有广义积分机会约束问题中的应用，并给出了可行解集的收敛速率。我们将把注意力集中在可行解集合是有限的情况下，这种情况可能出现在整数规划中。然后将结果推广到具有连续变量的有界集合。附加二进制变量是解决样本近似机会约束问题所必需的，这导致了一个大型的混合整数非线性规划。另一方面，带有惩罚的问题可以在不添加二元变量的情况下解决；只有连续变量才能模拟惩罚。将所介绍的方法应用于混合问题，得到了比较可靠的解。
If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.