Strengthened existence and uniqueness conditions for search directions in semidefinite programming Original Research Article

Primal–dual interior-point (p–d i-p) methods for Semidefinite Programming (SDP) are generally based on solving a system of matrix equations for a Newton type search direction for a symmetrization of the optimality conditions. These search directions followed the derivation of similar p–d i-p methods for linear programming (LP). Among these, a computationally interesting search direction is the AHO direction. However, in contrast to the LP case, existence and uniqueness of the AHO search direction is not guaranteed under the standard nondegeneracy assumptions. Two different sufficient conditions are known that guarantee the existence and uniqueness independent of the specific linear constraints. The first is given by Shida–Shindoh–Kojima and is based on the semidefiniteness of the symmetrization of the product SX at the current iterate. The second is a centrality condition given first by Monteiro–Zanjácomo and then improved by Monteiro–Todd.