This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We present a computable stochastic approximation type algorithm, namely the stochastic linearized proximal method of multipliers, to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits $O(K^{-1/2})$ expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where $K$ denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has $O(\log(K)K^{-1/2})$ constraint violation bound and $O(\log^{3/2}(K)K^{-1/2})$ objective bound. Some preliminary numerical results demonstrate the performance of the proposed algorithm.
翻译:本文考虑了以一系列不平等等同期望限制来最大限度地减少等离子期望函数的问题。 我们提出了一个可计算到的随机近似类型算法, 即随机线性直线性准乘法, 以解决这个等离子线性优化问题。 这个算法大致上可以被看成是随机近似和传统的近似乘数方法的混合体。 在温和的条件下, 我们显示这个算法显示如果正确选择了算法参数, 则客观削减和约束性违反的预期趋同率为$O( K ⁇ -1/2}) 美元, 其中美元表示迭代数。 此外, 我们还表明, 极有可能, 算法有美元( log( K) K ⁇ -1/2 } 约束值约束值和 $O( log3/2}( K) K ⁇ -1/2} 目标约束值。 一些初步的数字结果显示提议的算法的性能 。