Uterine sarcoma is significantly rarer than leiomyoma and has poor prognosis. Moreover, the diagnosis of leiomyosarcoma is difficult because its symptoms, including pelvic pain, uterine mass, and/or uterine bleeding, are very similar to those of leiomyoma. There are a few cases of leiomyosarcoma wherein leiomyoma was treated with uterine artery embolization (UAE); these reports revealed that the symptoms of hypermenorrhea or/and pelvic pain persisted even after UAE. Symptoms persisting even after UAE treatment for leiomyomas, especially multiple leiomyomas, should be investigated to rule out leiomyosarcoma. Therefore, long-term follow-up is needed. Here, we describe a case of a 39-year-old woman diagnosed with leiomyosarcoma 3 years after undergoing UAE for multiple leiomyomas.

We aim to provide better relaxation for generalized completely positive (copositive) programming. We first develop an inner-approximation hierarchy for the generalized copositive cone over a symmetric cone. Exploiting this hierarchy as well as the existing hierarchy proposed by Zuluaga et al. (SIAM J Optim 16(4):1076-1091, 2006), we then propose two (NN and ZVP) generalized doubly nonnegative (GDNN) cones. They are (if defined) always tractable, in contrast to the existing (BD) GDNN cone proposed by Burer and Dong (Oper Res Lett 40(3): [203][204][205][206] 2012). We focus our investigation on the inclusion relationship between the three GDNN cones over a direct product of a nonnegative orthant and second-order cones or semidefinite cones. We find that the NN GDNN cone is included in the ZVP one theoretically and in the BD one numerically. Although there is no inclusion relationship between the ZVP and BD GDNN cones theoretically, the result of solving GDNN programming relaxation problems of mixed 0-1 second-order cone programming shows that the proposed GDNN cones provide a tighter bound than the existing one in most cases. To sum up, the proposed GDNN cones have theoretical and numerical superiority over the existing one.

We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (Structured semidefinite programs and semialgebraic geometry methods in Robustness and optimization, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a symmetric cone using the usual COP cone. By replacing the usual COP cone appearing in this characterization with the inner- or outer-approximation hierarchy provided by de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) or Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), we obtain an inner- or outer-approximation hierarchy described by semidefinite but not by SOS constraints for the COP matrix cone over the direct product of a nonnegative orthant and a second-order cone. We then compare them with the existing hierarchies provided by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, https://doi.org/10.1137/03060151X, 2006) and Lasserre (Math Program 144:265–276, https://doi.org/10.1007/s10107-013-0632-5, 2014). Theoretical and numerical examinations imply that we can numerically increase a depth parameter, which determines an approximation accuracy, in the approximation hierarchies derived from de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) and Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), particularly when the nonnegative orthant is small. In such a case, the approximation hierarchy derived from Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012) can yield nearly optimal values numerically. Combining the proposed approximation hierarchies with existing ones, we can evaluate the optimal value of COP programming problems more accurately and efficiently.