Interior Point Methods with a Gradient Oracle

We provide an interior point method based on quasi-Newton iterations, which only requires first-order access to a strongly self-concordant barrier function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC '07] to maintain a preconditioner, while using only first-order information. We measure the quality of this preconditioner in terms of its relative excentricity to the unknown Hessian matrix, and we generalize these techniques to convex functions with a slowly-changing Hessian. We combine this with an interior point method to show that, given first-order access to an appropriate barrier function for a convex set K, we can solve well-conditioned linear optimization problems over K to epsilon precision in time (O) over tilde (T +n(2)) root log (1/epsilon)), where.. is the self-concordance parameter of the barrier function, and T is the time required to make a gradient query. As a consequence we show that: Linear optimization over..-dimensional convex sets can be solved in time (O) over tilde (T +n(2)) . This parallels the running time achieved by state of the art algorithms for cutting plane methods, when replacing separation oracles with first-order oracles for an appropriate barrier function. We can solve semidefinite programs involving matrices in R-nxn in time (O) over tilde (mn(4)+m(1.25)n(3.5) log(1/epsilon)) , improving over the state of the art algorithms, in the case where m = Omega(3.5/n omega-1.25) Along the way we develop a host of tools allowing us to control the evolution of our potential functions, using techniques from matrix analysis and Schur convexity.