Gradient Projection for Solving Quadratic Programs with Standard Simplex Constraints

Abstract

An important method to optimize a function on standard simplex is the active set algorithm, which requires the gradient of the function to be projected onto a hyperplane, with sign constraints on the variables that lie in the boundary of the simplex. We propose a new algorithm to efficiently project the gradient for this purpose. Furthermore, we apply the proposed gradient projection method to quadratic programs (QP) with standard simplex constraints, where gradient projection is used to explore the feasible region and, when we believe the optimal active set is identified, we switch to constrained conjugate gradient to accelerate convergence. Specifically, two different directions of gradient projection are used to explore the simplex, namely, the projected gradient and the reduced gradient. We choose one of the two directions according to the angle between the directions. Moreover, we propose two conditions for guessing the optimal active set heuristically. The first condition is that the working set remains unchanged for many iterations, and the second condition is that the angle between the projected gradient and the reduced gradient is small enough. Based on these strategies, a new active set algorithm for solving quadratic programs on standard simplex is proposed.