A conditional gradient homotopy method with applications to Semidefinite Programming
This work addresses computational efficiency for researchers and practitioners in optimization, particularly for semidefinite programming applications, though it appears incremental as it builds on existing conditional gradient and homotopy methods.
The authors tackled the problem of solving convex optimization with many simple conic constraints, such as in semidefinite programming relaxations of combinatorial problems, by proposing a new homotopy-based conditional gradient method that achieves competitive theoretical iteration complexity with cheap projection-free subroutines.
We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.