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Publication# Convergence of the Exponentiated Gradient Method with Armijo Line Search

Abstract

Consider the problem of minimizing a convex differentiable function on the probability simplex, spectrahedron, or set of quantum density matrices. We prove that the expo-nentiated gradient method with Armijo line search always converges to the optimum, if the sequence of the iterates possesses a strictly positive limit point (element-wise for the vector case, and with respect to the Löwner partial ordering for the matrix case). To the best of our knowledge, this is the first convergence result for a mirror descent-type method that only requires differentiability. The proof exploits self-concordant likeness of the l og-partition function, which is of independent interest.

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Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalentl

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Gradient method

In optimization, a gradient method is an algorithm to solve problems of the form
:\min_{x\in\mathbb R^n}; f(x)
with the search directions defined by the gradient of the function at th