A Note on a Lower Bound on the Minimum Rank of a Positive Semidefinite Hankel Matrix Rank Minimization Problem

Yi Xu,Xiaorong Ren,Xihong Yan,He Chen

doi:10.1155/2021/2524016

Yi Xu, Xiaorong Ren + Show 2 more

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https://doi.org/10.1155/2021/2524016

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Abstract

This paper investigates the problem of approximating the global minimum of a positive semidefinite Hankel matrix minimization problem with linear constraints. We provide a lower bound on the objective of minimizing the rank of the Hankel matrix in the problem based on conclusions from nonnegative polynomials, semi-infinite programming, and the dual theorem. We prove that the lower bound is almost half of the number of linear constraints of the optimization problem.

Highlights

Several evolutionary algorithms were proposed by Cai, including an iterative hard thresholding algorithm based on the nuclear norm model [18], an alternating direction method of multipliers based on Vandermonde factorization of a Hankel matrix model [19], and a projected Wirtinger gradient algorithm [20]. e major drawbacks of these methods are that the solutions obtained are not guaranteed to be a good approximate global solution for the problem since there have been no Mathematical Problems in Engineering approximation bounds developed for these methods
In this paper, we provide a lower bound on the minimum rank of a positive semidefinite Hankel matrix minimization problem with linear constraints, which is modeled as the following program: min r(X), A(X) b, (2)
Note that the positive semidefinite Hankel matrix completion problem is a special case of problem (2)

Summary

Introduction

Is problem can be formulated as the following program: min r(X), (1) In view of improving performance of algorithms for solving the problem, it is important to find an approximation for the rank of the optimal Hankel matrix. In this paper, we provide a lower bound on the minimum rank of a positive semidefinite Hankel matrix minimization problem with linear constraints, which is modeled as the following program: min r(X), A(X) b, (2)

Results

Conclusion