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The minimum volume enclosing ellipsoid (MVEE) problem is an optimizationproblem in the basis of many practical problems. This paper describes some newproperties of this model and proposes a first-order oracle algorithm, theAdjusted Coordinate Descent (ACD) algorithm, to address the MVEE problem. TheACD algorithm is globally linear convergent and has an overwhelming advantageover the other algorithms in cases where the dimension of the data is large.Moreover, as a byproduct of the convergence property of the ACD algorithm, weprove the global linear convergence of the Frank-Wolfe type algorithm(illustrated by the case of Wolfe-Atwood's algorithm), which supports theconjecture of Todd. Furthermore, we provide a new interpretation for the meansof choosing the coordinate axis of the Frank-Wolfe type algorithm from theperspective of the smoothness of the coordinate axis, i.e., the algorithmchooses the coordinate axis with the worst smoothness at each iteration. Thisfinding connects the first-order oracle algorithm and the linear optimizationoracle algorithm on the MVEE problem. The numerical tests support ourtheoretical results.

Global linear convergent algorithm to compute the minimum volume enclosing ellipsoid