  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.
