Lattice-like operations and isotone projection sets

By using some lattice-like operations which constitute extensions of ones introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E. A. Ok.Comment: Proofs of Theorem 1 and Corollary 4 have been corrected. arXiv admin note: substantial text overlap with arXiv:1210.232

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

Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

© 2017, Springer Science+Business Media New York.In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others