A simple polynomial-time algorithm for convex quadratic programming

In this note we propose a polynomial-time algorithm for convex quadratic programming. This algorithm augments the objective by a logarithmic penalty function and then solves a sequence of quadratic approximations of this program. This algorithm has a complexity of O(ml/2-L) iterations and O(m 3-5 .L) arithmetic operations, where m is the number of variables and L is the size of the problem encoding in binary. The novel feature of this algorithm is that it admits a very simple proof of its complexity, which makes it valuable both as a teaching and as a research tool. The proof uses a new Lyapunov function to measure the duality gap, which has itself interesting properties that can be used in a line search procedure to accelerate convergence. If the cost is separable, the line search is particularly simple to implement and, if the cost is linear, the line search stepsize is obtainable in a closed form. This algorithm maintains both primal and dual feasibility at all iterations.