Generalized affine scaling trajectory analysis for linearly constrained convex programming

Abstract

In this paper, we propose and analyze a continuous trajectory, which is the solution of an ordinary differential equation (ODE) system for solving linearly constrained convex programming. The ODE system is formulated based on a first-order interior point method in [Math. Program., 127, 399–424 (2011)] which combines and extends a first-order affine scaling method and the replicator dynamics method for quadratic programming. The solution of the corresponding ODE system is called the generalized affine scaling trajectory. By only assuming the existence of a finite optimal solution, we show that, starting from any interior feasible point, (i) the continuous trajectory is convergent; and (ii) the limit point is indeed an optimal solution of the original problem.