The limits of a class of primal and dual solution trajectories associated with the Sequential Unconstrained Minimization Technique (SUMT) are investigated for convex programming problems with non-unique optima. Logarithmic barrier terms are assumed. For linear programming problems, such limits of both primal and dual trajectories - are strongly optimal, strictly complementary, and can be characterized as analytic centers of, loosely speaking, optimality regions. Examples are given, which show that those results do not hold in general for convex programming problems. If the latter are weakly analytic (Bank et al. ), primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability. That class of programming problems contains faithfully convex linear, and convex quadratic programming problems as strict subsets. In the differential case, dual trajectory limits can be characterized similarly, abeit under different conditions, one of which suffices for strict complementarity.
analytic center, barrier function, convex programming, degeneracy, interior point method, optimization, rank-integrity, SUMT, trajectory
and Witzgall, C.
Logarithmic SUMT Limits in Convex Programming, Mathematical Programming
(Accessed June 9, 2023)