Characterizing Convexity of Images for Quadratic-Linear Mappings with Applications in Nonconvex Quadratic Optimization
研究了向量映射（部分二次、部分线性）图像凸性的刻画，发现全图像凸性可简化为低维图像凸性，并分析了含一或两个二次分量的情形，给出了凸性的几何充要条件，修正了已有结果，并应用于S引理和强对偶性刻画。
Various characterizations of convexity for images of a vector mapping where some of its components are quadratic and the remaining ones are linear are established. In a certain sense, one might conclude that convexity of the full image is reduced to the convexity of an image in a lower dimension by deleting the linear components. The latter may be considered as the analogue to the reduction of the number of constraints once the dual is associated. The cases of having one or two quadratic components while the other are linear are particularly analyzed. This allows us to formulate some (geometric) sufficient and necessary conditions for convexity. As a byproduct, a result obtained in [Xia, Wang, and Sheu, Math. Program. Ser. A, 156 (2016), pp. 513--547] is corrected. Finally, as some applications, we obtain an S-lemma (with equality and on an affine subspace) and a characterization of strong duality in terms of convexity of some image set associated to the minimization problem under consideration.