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

Gradient Projection for Solving Quadratic Programs with Standard Simplex Constraints