In this paper, we consider the projected gradient algorithms for solving the quadratic program with bound constraints and a single linear equality constraint (SLBQP). We establish the relationship between the Lagrangian multiplier in the projection subproblem and the Lagrangian multiplier in the original optimization problem. Then we give an improved initial estimate of the Lagrangian multiplier in the subproblem based on this relationship. It appears that this initial estimate is very close to the optimal Lagrangian multiplier after several iterations of the outer loop. This will reduce at most 40% of the computing time in the projection subproblem. This initial guess can also be used in all kinds of projected gradient methods for solving the SLBQP problem. The numerical results show that it brings much more improvement in monotone algorithms than in nonmonotone algorithms. We also apply the adaptive steepest descent step-size and the Dai-Yuan step-size which are two monotone step-sizes to the projected gradient method of this SLBQP problem. Our numerical experiments showed that their performance can be better than some other monotone projected gradient methods.
On the Convergence of Projected-Gradient Methods with Low-Rank Projections for Smooth Convex Minimization over Trace-Norm Balls and Related Problems
