BAYESIAN SOLUTION ESTIMATORS IN STOCHASTIC OPTIMIZATION
BAYESIAN SOLUTION ESTIMATORS IN STOCHASTIC OPTIMIZATION
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随机优化中的贝叶斯解估计器
DOI：
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发表时间：
2017
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通讯作者：
D. Davarnia
中科院分区：
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作者：
D. Davarnia
我们研究了一类随机规划，其中目标函数中的某些元素是随机的，其概率分布具有未知参数。目标是利用随机元素分布的抽样数据，为随机规划的最优解找到一个好的估计。我们研究了两个评价解估计器质量的自然标准，一个基于目标值的差异，另一个基于解之间的欧几里德距离。我们使用风险作为这些标准在样本空间上的期望值。在贝叶斯框架下，假设未知参数为先验分布，出现了两种自然的估计-优化策略。一个单独的方案首先找到未知参数的估计器，然后在优化问题中使用该估计器。联合方案通过直接调整随机规划中的分布来结合估计和优化步骤。我们研究了几类随机规划从这两种方案得到的解之间的风险差异，同时提供了解决这些问题的计算努力的洞察力。特别地，(I)我们确定了两种格式的解估计相等的条件，(Ii)对于一般问题，我们证明了两种格式之间的风险差可以是任意大的，(Iii)对于随机分段线性规划，我们得到了风险差的显式界，以及(Iv)对于随机几何规划，我们讨论了两种格式的计算复杂性的差异，并给出了计算实验。
We study a class of stochastic programs where some of the elements in the objective function are random, and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two natural criteria for evaluating the quality of a solution estimator, one based on the difference in objective values, and the other based on the Euclidean distance between solutions. We use risk as the expected value of such criteria over the sample space. Under a Bayesian framework, where a prior distribution is assumed for the unknown parameters, two natural estimation-optimization strategies arise. A separate scheme first finds an estimator for the unknown parameters, and then uses this estimator in the optimization problem. A joint scheme combines the estimation and optimization steps by directly adjusting the distribution in the stochastic program. We study the risk difference between the solutions obtained from these two schemes for several classes of stochastic programs, while providing insight on the computational effort to solve these problems. In particular, (i) we identify conditions under which the solution estimators of both schemes are equal, (ii) for general problems, we show that the risk difference between the two schemes can be arbitrarily large, (iii) for stochastic piecewise linear programs, we derive explicit bounds on risk differences, and (iv) for stochastic geometric programs, we discuss the difference in computational complexity of the two schemes and provide computational experiments.