To what extent are second-order cone and positive semidefinite cone alike?

[[abstract]]The cone of positive semidefinite matrices and second-order cone are both self-dual and special cases of symmetric cones. Each of them play an important role in semidefinite programming (SDP) and second-order cone programming (SOCP), respectively. It is known that an SOCP problem can be viewed as an SDP problem via certain relation between positive semidefinite cone and second-order cone. Nonetheless, most analysis used for dealing SDP can not carried over to SOCP due to some difference, for instance, the matrix multiplication is associative for positive semidefinite cone whereas the Jordan product is not for second-order cone. In this paper, we try to have a thorough study on the similarity and difference between these two cones which provide theoretical for further investigation of SDP and SOCP.

Differentiability v.s. convexity for complementarity functions

The J.-S. Chen's work is supported by Ministry of Science and Technology, Taiwan. The work of J. E. Martinez-Legaz has been supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, and under Australian Research Council's Discovery Projects funding scheme (project number DP140103213). He is affiliated to MOVE (Markets, Organizations and Votes in Economics).It is known that complementarity functions play an important role in dealing with complementarity problems. The most well known complementarity problem is the symmetric cone complementarity problem (SCCP) which includes nonlinear complementarity problem (NCP), semidefinite complementarity problem (SDCP), and second-order cone complementarity problem (SOCCP) as special cases. Moreover, there is also so-called generalized complementarity problem (GCP) in infinite dimensional space. Among the existing NCP-functions, it was observed that there are no differentiable and convex NCP-functions. In particular, Miri and Effati (J Optim Theory Appl 164:723-730, 2015) show that convexity and differentiability cannot hold simultaneously for an NCP-function. In this paper, we further establish that such result also holds for general complementarity functions associated with the GCP

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

A regularized smoothing Newton method for symmetric cone complementarity problems

This paper extends the regularized smoothing Newton method in vector complementarity problems to symmetric cone complementarity problems (SCCP), which includes the nonlinear complementarity problem, the second-order cone complementarity problem, and the semidefinite complementarity problem as special cases. In particular, we study strong semismoothness and Jacobian nonsingularity of the total natural residual function for SCCP. We also derive the uniform approximation property and the Jacobian consistency of the Chen–Mangasarian smoothing function of the natural residual. Based on these properties, global and quadratical convergence of the proposed algorithm is established