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S.: Iteration bounds for finding stationary points of structured nonconvex optimization. Working Paper
, 2014
"... In this paper we study proximal conditionalgradient (CG) and proximal gradientprojection type algorithms for a blockstructured constrained nonconvex optimization model, which arises naturally from tensor data analysis. First, we introduce a new notion of stationarity, which is suitable for the s ..."
Abstract

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In this paper we study proximal conditionalgradient (CG) and proximal gradientprojection type algorithms for a blockstructured constrained nonconvex optimization model, which arises naturally from tensor data analysis. First, we introduce a new notion of stationarity, which is suitable for the structured problem under consideration. We then propose two types of firstorder algorithms for the model based on the proximal conditionalgradient (CG) method and the proximal gradientprojection method respectively. If the nonconvex objective function is in the form of mathematical expectation, we then discuss how to incorporate randomized sampling to avoid computing the expectations exactly. For the general block optimization model, the proximal subroutines are performed for each block according to either the blockcoordinatedescent (BCD) or the maximumblockimprovement (MBI) updating rule. If the gradient of the nonconvex part of the objective f satisfies ‖∇f(x) − ∇f(y)‖q ≤ M‖x − y‖δp where δ = p/q with 1/p + 1/q = 1, then we prove that the new algorithms have an overall iteration complexity bound of O(1/q) in finding an stationary solution. If f is concave then the iteration complexity reduces to O(1/). Our numerical experiments for tensor approximation problems show promising performances of the new solution algorithms.