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Assume that f is a strict convex function with a unique minimum in R^n. Wedivide the vector of n-variables to d groups of vector subvariables with d atleast two. We assume that we can find the partial minimum of f with respect toeach vector subvariable while other variables are fixed. We then describe analgorithm that partially minimizes each time on a specifically chosen vectorsubvariable. This algorithm converges geometrically to the unique minimum. Therate of convergence depends on the uniform bounds on the eigenvalues of theHessian of f in the compact sublevel set f whose values are at most f(x_0),where x_0 is the starting point of the algorithm. In the case where f is apolynomial of degree two, with positive definite quadratic term, and d=n ourmethod can be considered as a generalization of the classical conjugategradient method. The main result of this paper is the observation that thecelebrated Sinkhorn diagonal scaling algorithm for matrices, and thecorresponding diagonal scaling of tensors, can be viewed as partialminimization of certain logconvex functions.

Partial minimization of strict convex functions and tensor scaling