We consider the extended trust-region subproblem with two linear inequalities. In the "nonintersecting" case of this problem, Burer and Yang(2015) have proved that its semi-definite programming relaxation with second-order-cone reformulation(SDPR-SOCR) is a tight relaxation. In the more complicated "intersecting" case, which is discussed in this paper, so far there is no result except for a counterexample for the SDPR-SOCR. We present a necessary and sufficient condition for the SDPR-SOCR to be a tight relaxation in both the "nonintersecting" and "intersecting" cases. As an application of this condition, it is verified easily that the "nonintersecting" SDPR-SOCR is a tight relaxation indeed. Furthermore, as another application of the condition, we prove that there exist at least three regions among the four regions in the trust-region ball divided by the two intersecting linear cuts, on which the SDPR-SOCR must be a tight relaxation. Finally, the results of numerical experiments show that the SDPR-SOCR can work efficiently in decreasing or even eliminating the duality gap of the nonconvex extended trust-region subproblem with two intersecting linear inequalities indeed.
Service-oriented architecture is becoming a major software framework for complex application and it can be dynamically and flexibly composed by integrating existing component web services provided by different providers with standard protocols. The rapid introduction of new web services into a dynamic business environment can adversely affect the service quality and user satisfaction. Therefore, how to leverage, aggregate and make use of individual component’s quality of service (QoS) information to derive the optimal QoS of the composite service which meets the needs of users is still an ongoing hot research problem. This study aims at reviewing the advance of the current state-of-the-art in technologies and inspiring the possible new ideas for web service selection and composition, especially with nature-inspired computing approaches. Firstly, the background knowledge of web services is presented. Secondly, various nature-inspired web selection and composition approaches are systematically reviewed and analysed for QoS-aware web services. Finally, challenges, remarks and discussions about QoS-aware web service composition are presented.
