  Seeking tighter relaxations of combinatorial optimization problems,
semidefinite programming is a generalization of linear programming that offers
better bounds and is still polynomially solvable. Yet, in practice, a
semidefinite program is still significantly harder to solve than a similar-size
Linear Program (LP). It is well-known that a semidefinite program can be
written as an LP with infinitely-many cuts that could be solved by repeated
separation in a Cutting-Planes scheme; this approach is likely to end up in
failure. We proposed in [Projective Cutting-Planes, Daniel Porumbel, Siam
Journal on Optimization, 2020] the Projective Cutting-Planes method that
upgrades t he well-known separation sub-problem to the projection sub-problem:
given a feasible $y$ inside a polytope $P$ and a direction $d$, find the
maximum $t^*$ so that $y+t^*d\in P$. Using this new sub-problem, one can
generate a sequence of both inner and outer solutions that converge to the
optimum over $P$. This paper shows that the projection sub-problem can be
solved very efficiently in a semidefinite programming context, enabling the
resulting method to compete very well with state-of-the-art semidefinite
optimization software (refined over decades). Results suggest it may the
fastest method for matrix sizes larger than $2000\times 2000$.

Optimization and Control;Mathematical Software;Numerical Analysis;Numerical Analysis