Special topics in cone complementarity

Gao, Yingchao (2024). Special topics in cone complementarity. University of Birmingham. Ph.D.

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Abstract

In this thesis, the concept of Monotone Extended Second Order Cone (MESOC), which is a new generalisation of second order cone, has been present. We discussed the fundamental properties of MESOC and demonstrated the positive operator, the Lyapunov-like transformation, as well as the reducibility of this cone. The value Lyapunov rank has also been provided. We also investigated the isotonicity property of MESOC, and we showed the cylinder is the only isotonic projection set with respect to MESOC in the ambient space. Then we present the mixed complementarity problem on a general close, and convex cone can be solved by using an iterative method based on the isotonicity property of MESOC. Meanwhile, a numerical example has been illustrated to show the applicability of MESOC. We also investigated the formulas to show how to project onto MESOC. In the most general case, the formula we obtained is dependent on an equation for one real variable. The linear complementarity problem on the MESOC has also been studied. We have demonstrated that the linear complementarity problem on the MESOC can be converted to a mixed complementarity problem on the nonnegative orthant. The algorithms are discussed and numerical examples are also present. Moreover, we present an application of the MESOC, which is a portfolio optimisation problem with an analytical solution. At last, we studied the gradient projection method on the sphere. We showed that this method could be used in discussing the solvability of the complementarity problem and checking the copositivity of an operator with respect to cones. The numerical experiments which illustrate the copositivity of operators are also provided.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):