In this article, we address a class of non convex, integer, non linear mathematical programs using dynamic programming. The mathematical program considered, whose properties are studied in this article, may be used to model the optimal liquidation problem of a single asset portfolio, held in a very large quantity, in a low volatility and perfect memory market, with few market participants. In this context, the Portfolio Manager's selling actions convey information to market participants, which in turn lower bid prices and further penalize the liquidation proceeds we attempt to maximize. We show the problem can be solved exactly using Dynamic Programming (DP) in polynomial time. However, exact resolution is only efficient for small instances. For medium size and large instances, we introduce dedicated heuristics which provide thin admissible solutions, hence tight lower bounds for the initial problem. We also benchmark them against a commercial solver, such as LocalSolver [7]. We are also interested in the continuously relaxed problem, which is non convex. Firstly, we use continuous solutions, obtained by free solver NLopt [26] and transform them into thin admissible solutions of the discrete problem. Secondly, we provide, under some convexity assumptions, an upper bound for the continuous relaxation, and hence for the initial (integer) problem. Numerical experiments confirm the quality of proposed heuristics (lower bounds), which often reach the optimal, or prove very tight, for small and medium size instances, with a very fast CPU time. Our upper bound, however, is not tight.
翻译:在文章中,我们用动态编程处理一组非混凝、整数、非线性数学程序。考虑的数学程序,其性质在本条中研究,可以用来模拟单一资产组合的最佳清算问题,一个资产组合在低波动和完善的记忆市场中大量持有,规模小,规模小,存储范围小,市场参与者很少。在这方面,投资组合管理人的出售行动向市场参与者传递信息,而市场参与者则反过来降低出价价格,进一步惩罚我们试图最大化的清算收益。我们用动态编程(DP)来表明问题完全可以解决。然而,精确的解析只对小案例有效。对于中等规模和大案例,我们引入了专门的超重资产组合清算问题,提供稀薄的可接受解决方案,从而降低初始问题的门槛。我们还根据当地Solver(7)等商业清算者对持续放松的问题感兴趣。我们使用持续解析的问题,即自由解算器(DP) [26] 和将问题转化为不易碎的可接受的解决方案,但对于小型案例来说,精确的解析度有限。 其次,我们提供某种快速的深度的深度的缩度假设,因此,在一定的深度的深度的深度的试测测测度之下, 。(我们提供了某种深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度的深度, 。