A geodesic interior-point method for linear optimization over symmetric cones

We develop a new interior-point method for symmetric-cone optimization, a common generalization of linear, second-order-cone, and semidefinite programming. Our key idea is updating iterates with a geodesic of the cone instead of the kernel of the linear constraints. This approach yields a primal-dual-symmetric, scale-invariant, and line-search-free algorithm that uses just half the variables of a … Read more

DIRECT SEARCH ALGORITHMS OVER RIEMANNIAN MANIFOLDS

We generalize the Nelder-Mead simplex and LTMADS algorithms and, the frame based methods for function minimization to Riemannian manifolds. Examples are given for functions defined on the special orthogonal Lie group $\mathcal{SO}(n)$ and the Grassmann manifold $\mathcal{G}(n,k)$. Our main examples are applying the generalized LTMADS algorithm to equality constrained optimization problems and, to the Whitney … Read more

EQUALITY CONSTRAINTS, RIEMANNIAN MANIFOLDS AND DIRECT SEARCH METHODS

We present a general procedure for handling equality constraints in optimization problems that is of particular use in direct search methods. First we will provide the necessary background in differential geometry. In particular, we will see what a Riemannian manifold is, what a tangent space is, how to move over a manifold and how to … Read more

On the Riemannian Geometry Defined by Self-Concordant Barriers and Interior-Point Methods

We consider the Riemannian geometry defined on a convex set by the Hessian of a self-concordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the … Read more