Random projections map a set of points in a high dimensional space to a lower dimensional one while approximately preserving all pairwise Euclidean distances. Although random projections are usually applied to numerical data, we show in this paper that they can be successfully applied to quadratic programming formulations over a set of linear inequality constraints. Instead of solving the higher-dimensional original problem, we solve the projected problem more efficiently. This yields a feasible solution of the original problem. We prove lower and upper bounds of this feasible solution w.r.t. the optimal objective function value of the original problem. We then discuss some computational results on randomly generated instances, as well as a variant of Markowitz’ portfolio problem. It turns out that our method can find good feasible solutions of very large instances.

中文翻译：

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二次规划的随机投影

随机投影将高维空间中的一组点映射到低维空间，同时近似保留所有成对欧几里得距离。尽管随机投影通常应用于数值数据，但我们在本文中表明它们可以成功应用于一组线性不等式约束上的二次规划公式。我们不是解决更高维的原始问题，而是更有效地解决投影问题。这产生了原始问题的可行解。我们证明了这个可行解的上下界和原问题的最优目标函数值。然后我们讨论随机生成的实例的一些计算结果，以及 Markowitz 投资组合问题的一个变体。