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In this paper, we introduce a new class of nonnegative tensors -strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T , there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)-i.e. they are approximately norm-preserving-the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to eliminate spurious local minima, so existing results can only hold for near-perfect RIP. In fact, counterexamples are ubiquitous: we prove that every x is the spurious local minimum of a rank-1 instance of matrix recovery that satisfies RIP. One specific counterexample has RIP constant δ = 1/2, but causes randomly initialized stochastic gradient descent (SGD) to fail 12% of the time. SGD is frequently able to avoid and escape spurious local minima, but this empirical result shows that it can occasionally be defeated by their existence. Hence, while exact recovery guarantees will likely require a proof of no spurious local minima, arguments based solely on norm preservation will only be applicable to a narrow set of nearly-isotropic instances.

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MSC: 90C33 90C56 65K10Keywords: Complementarity problem NCP-function Merit function Derivative free algorithm a b s t r a c tIn this paper, we propose a new family of NCP-functions and the corresponding merit functions, which are the generalization of some popular NCP-functions and the related merit functions. We show that the new NCP-functions and the corresponding merit functions possess a system of favorite properties. Specially, we show that the new NCPfunctions are strongly semismooth, Lipschitz continuous, and continuously differentiable; and that the corresponding merit functions have SC 1 property (i.e., they are continuously differentiable and their gradients are semismooth) and LC 1 property (i.e., they are continuously differentiable and their gradients are Lipschitz continuous) under suitable assumptions. Based on the new NCP-functions and the corresponding merit functions, we investigate a derivative free algorithm for the nonlinear complementarity problem and discuss its global convergence. Some preliminary numerical results are reported.

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A symmetric tensor is called copositive if it generates a multivariate form taking nonnegative values over the nonnegative orthant. Copositive tensors have found important applications in polynomial optimization and tensor complementarity problems. In this paper, we consider copositivity detection of tensors both from theoretical and computational points of view. After giving several necessary conditions for copositive tensors, we propose several new criteria for copositive tensors based on the representation of the multivariate form in barycentric coordinates with respect to the standard simplex and simplicial partitions. It is verified that, as the partition gets finer and finer, the concerned conditions eventually capture all strictly copositive tensors. Based on the obtained theoretical results with the help of simplicial partitions, we propose a numerical method to judge whether a tensor is copositive or not. The preliminary numerical results confirm our theoretical findings.