How to project onto extended second order cones

The extended second order cones were introduced by S. Z. N\'emeth and G. Zhang in [S. Z. N\'emeth and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. N\'emeth and G. Zhang in [S.Z. N\'emeth and G. Zhang. Positive operators of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods

Linear complementarity problems on extended second order cones

In this paper, we study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We state necessary and sufficient conditions for a point to be a solution of the converted problem. We also present solution strategies for this problem, such as the Newton method and Levenberg-Marquardt algorithm. Finally, we present some numerical examples

Complementarity problems, variational inequalities and extended lorentz cones

In this thesis, we introduced the concept of extended Lorentz cones. We discussed the solvability of variational inequalities and complementarity problems associated with an unrelated closed convex cone. This cone does not have to be an isotone projection cone. We showed that the solution of variational inequalities and complementarity problems can be reached as a limit of a sequence defined in an ordered space which is ordered by extended Lorentz cone. Moreover, we applied our results in game theory and conic optimization problems. We also discussed the positive operators. We showed necessary and sufficient conditions under which a linear operator is a positive operator of extended Lorentz cone. We also showed sufficient and necessary conditions under which a linear operator in a specific form is a positive operator

Complementarity and related problems

In this thesis, we present results related to complementarity problems. We study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We present algorithms for this problem, and exemplify it by a numerical example. Following this result, we explore the stochastic version of this linear complementarity problem. Finally, we apply complementarity problems on extended second order cones in a portfolio optimisation problem. In this application, we exploit our theoretical results to find an analytical solution to a new portfolio optimisation model. We also study the spherical quasi-convexity of quadratic functions on spherically self-dual convex sets. We start this study by exploring the characterisations and conditions for the spherical positive orthant. We present several conditions characterising the spherical quasi-convexity of quadratic functions. Then we generalise the conditions to the spherical quasi-convexity on spherically self-dual convex sets. In particular, we highlight the case of spherical second order cones