Three Transformations in the Semidefinite Linear Complementarity Problem

(Abstract)

Let L be a linear transformation from Sⁿ (the space of symmetric, real matrices of order n) to Sⁿ. The semidefinite lineary complementarity problem (SDLCP) is:

Given Q Є Sⁿ find if possible an X Є Sⁿ₊ (the space of positive semidefinite matrices) such that L(x) + Q Є Sⁿ₊ with trace(X(L(X)+Q))=0.

It is not hard to see that SDLCP is a generalization of the linear complementarity problem (LCP).

In this talk we introduce three linear transformations, namely Lyapunov, Stein and Double Multiplicative transformation and examine if they have SDLCP solutions for every Q Є Sⁿ. The main difficulty in SDLCP is Sⁿ₊ is not polyhedral. Also matrix multiplication is not commutative. A key objective here is to see how far the results from LCP can be carried over to the SDLCP problem.