The condition number

  The condition number of an SDP is defined in [7] and its significance is discussed there. Briefly, the condition number of an SQLP is defined as the condition number of the Jacobian of the function to which Newton's method is applied in defining the XZ+ZX search direction, evaluated at the solution. This quantity is if and only if the solution of the SQLP fails to satisfy the strict complementarity or primal or dual nondegeneracy conditions. The routine is

sqlcond.m

Given the data of an SQLP and the solutions and , this routine verifies the optimality conditions and computes a lower bound (in the 1-norm) of the condition number of an SQLP. The calling sequence is:

In exact arithmetic, the routine would return the value ( ) if and only if strict complementarity or primal or dual nondegeneracy are violated. A large value of cndsql is a strong indication of that at least one of these three conditions fails to hold. This routine takes a long time to execute compared to pcond.m and dcond.m, and can therefore be used only for small problems, but its advantage is that no tolerance is required to determine ranks.

In order to use sqlcond on a diagonally constrained SDP or a Lovász problem, the user must ensure that , and have been constructed, by first calling the appropriate script (dsdp.m or lsdp.m) with and .