We consider global optimization of nonconvex problems containing mul- tilinear functions. It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP). When used as cutting planes, these facets can sig- nificantly enhance the quality of conventional relaxations in general-purpose global solvers. However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function. To cope with this growth, we propose a graph decomposition scheme that exploits the structure of a multilinear function to decom- pose it to lower-dimensional components, for which the aforementioned LP can be solved very efficiently by employing a customized simplex algorithm. We embed this cutting plane generation strategy at every node of the branch-and-reduce global solver BARON, and carry out an extensive computational study on quadratically constrained quadratic problems, multilinear problems, and polynomial optimization problems. Results show that the proposed multilinear cuts enable BARON to solve many more problems to global optimality and lead to an average 60 % CPU time reduction.

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Multi-waydataarisesinmanyapplicationssuchaselectroencephalography classification, face recognition, text mining and hyperspectral data analysis. Tensor decomposition has been commonly used to find the hidden factors and elicit the intrin- sic structures of the multi-way data. This paper considers sparse nonnegative Tucker decomposition (NTD), which is to decompose a given tensor into the product of a core tensor and several factor matrices with sparsity and nonnegativity constraints. An alternating proximal gradient method is applied to solve the problem. The algorithm is then modified to sparse NTD with missing values. Per-iteration cost of the algorithm is estimated scalable about the data size, and global convergence is established under fairly loose conditions. Numerical experiments on both synthetic and real world data demonstrate its superiority over a few state-of-the-art methods for (sparse) NTD from partial and/or full observations. The MATLAB code along with demos are accessible from the author’s homepage.

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Computational methods are considered for finding a point that satisfies the second-order necessary conditions for a general (possibly nonconvex) quadratic program (QP). The first part of the paper considers the formulation and analysis of an active-set method for a generic QP with both equality and inequality constraints. The method uses a search direction that is the solution of an equality-constrained subproblem involving a “working set” of linearly independent constraints. The method is a reformulation of a method for general QP first proposed by Fletcher, and modified subsequently by Gould. The reformulation facilitates a simpler analysis and has the benefit that the algorithm reduces to a variant of the simplex method when the QP is a linear program. The search direction is computed from a KKT system formed from the QP Hessian and the gradients of the working-set constraints. It is shown that, under certain circumstances, the solution of this KKT system may be updated using a simple recurrence relation, thereby giving a significant reduction in the number of KKT systems that need to be solved. The second part of the paper focuses on the solution of QP problems with constraints in so-called standard form. We describe how the constituent KKT systems are solved, and discuss how an initial basis is defined. Numerical results are presented for all QPs in the CUTEst test collection.

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