Analysis of the primal affine scaling continuous trajectory for convex programming

A weighted first-order primal affine scaling continuous trajectory for linearly constrained smooth convex programming is studied in this paper. By assuming the existence of an optimal solution in the linear case or the boundness of the optimal solution set in the general case, we show that starting from any interior feasible point, (i) every accumulation point is indeed an optimal solution; and (ii) if the objective function is analytic, the primal affine scaling continuous trajectory converges to a point which is actually in the relative interior of the optimal solution set. As we know, this result is the first one to obtain the convergence of the primal affine scaling continuous trajectory in the nonlinear case for linearly constrained convex programming.