Invited Session Wed.1.H 0112

Wednesday, 10:30 - 12:00 h, Room: H 0112

Cluster 16: Nonlinear programming [...]

Solution methods for matrix, polynomial, and tensor optimization


Chair: Shuzhong Zhang



Wednesday, 10:30 - 10:55 h, Room: H 0112, Talk 1

Xin Liu
Beyond heuristics: Applying alternating direction method of multiplier method in solving matrix factorization problems

Coauthor: Yin Zhang


Alternating direction method of multiplier(ADMM) applies alternating technique on the KKT system of augmented Lagrangian function, which is a powerful algorithm for optimization problems with linear equality constraints and certain separable structures. However, its convergence has not been established except in two blocks, separable and convex cases.
In this talk, we will show ADMM also has excellent performances in solving some matrix factorization problems in which either separability or convexity does not apply. Furthermore we will present some preliminary results on the converegence of ADMM in these cases.



Wednesday, 11:00 - 11:25 h, Room: H 0112, Talk 2

Zhening Li
Maximum block improvement and polynomial optimization

Coauthors: Bilian Chen, Simai He, Shuzhong Zhang


We propose an efficient method for solving a large class of polynomial optimization problems, in particular, the spherically constrained homogeneous polynomial optimization. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we accept only a block update that achieves the maximum improvement, hence the name of our new search method: maximum block improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proved. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus we can maximize a homogeneous polynomial over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: For a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.


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