In this paper we explore convex reformulation strategies for non-convex quadratically constrained optimization problems (QCQPs). First we investigate such reformulations using Pataki’s rank theorem iteratively. We show that the result can be used in conjunction with conic optimization duality in order to obtain a geometric condition for the S-procedure to be exact. Based upon known results on the S-procedure, this approach allows for some insight into the geometry of the joint numerical range of the quadratic forms. Then we investigate a reformulation strategy introduced in recent literature for bilinear optimization problems which is based on adjustable robust optimization theory. We show that, via a similar strategy, one can leverage exact reformulation results of QCQPs in order to derive lower bounds for more complicated quadratic optimization problems. Finally, we investigate the use of reformulation strategies in order to derive characterizations of set-copositive matrix cones. Empirical evidence based upon first numerical experiments shows encouraging results.

在本文中，我们探索了非凸二次约束优化问题（QCQP）的凸重构策略。首先，我们使用Pataki的秩定理迭代地研究这种重构。我们表明，该结果可以与圆锥优化对偶一起使用，以便获得精确的S程序的几何条件。根据有关S程序的已知结果，此方法可以使您深入了解二次形式的联合数值范围的几何形状。然后，我们研究了基于可调鲁棒优化理论的近期文献中针对双线性优化问题引入的一种重构策略。我们证明，通过类似的策略，可以利用QCQP的精确重新编制结果来为更复杂的二次优化问题得出下界。最后，我们研究了重新制定策略的使用，以得出集合正矩阵锥的特征。基于第一个数值实验的经验证据显示出令人鼓舞的结果。