Hessian barrier algorithms for linearly constrained optimization problems

Immanuel M. Bomze, Panayotis Mertikopoulos, Werner Schachinger, Mathias Staudigl

In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent (MD), and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a non-degeneracy condition, the algorithm converges to the problem's set of critical points; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is $\mathcal{O}(1/k^\rho)$ for some $\rho\in(0,1]$ that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard non-convex test functions and large-scale traffic assignment problems.

Department of Statistics and Operations Research, Research Platform Data Science @ Uni Vienna
SIAM Journal on Optimization
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Accepted/In press
Peer reviewed
Austrian Fields of Science 2012
101015 Operations research
math.OC, cs.LG, Primary: 90C51, 90C30, secondary: 90C25, 90C26
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