有风险控制的log-最优投资组合问题的一个黎曼几何随机算法

针对有风险控制的log-最优投资组合问题，提出了一个自适应的随机算法．该算法通过引进松弛变量，把对风险控制的不等式约束化为等式约束；再通过引进罚参数，运用罚函数法对风险控制的等式约束进行处理，从而将原来的问题化为一系列新的随机优化问题，再利用黎曼流形上的随机优化算法对其进行自适应求解．最后，使用该算法对上海证券交易所的实际数据进行了模拟计算，得到了很好的计算效果．

A Riemannian Geometry Underlying Stochastic Algorithm for Log-Optimal Portfolio Problem with Risk Control

This paper proposed an adaptively stochastic algorithm for solving a log-optimal portfolio problem with risk control. By introducing some slack variables, the original problem with inequality constraints is transformed into a new one with equality constraints, and the latter problem is further changed into a series of unconstrained stochastic optimization problems after the disposal of equality constraints (involving the risk-constraint) by virtue of the penalty function method. These new problems are then adaptively solved by some stochastic algorithms on Riemannian manifolds. Finally, the algorithm is applied to solve the risk-constrained log-optimal portfolio problem with the real data from the Institute of Shanghai Security. The numerical results are satisfactory.