Minimizing Polynomials via Sum of Squares over the Gradient Ideal

Sparse SOS Relaxations for Minimizing Functions that are Summations of Small Polynomials

This paper discusses how to find the global minimum of functions that are summations of small polynomials (``small'' means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these sparse relaxations. The proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems. Numerical experiments are presented, which show that the proposed…

Sensor placement minimizing the state estimation mean square error: Performance guarantees of greedy solutions

This paper studies selecting a subset of the system's output to minimize the state estimation mean square error (MSE). This results in the maximization problem of a set function defined on possible sensor selections subject to a cardinality constraint. We consider to solve it approximately by a greedy search. Since the MSE function is not submodular nor supermodular, the well-known performance guarantees for the greedy solutions do not hold in the present case. Thus, we use the quantities---the submodularity ratio and the curvature---to evaluate the degrees of submodularity and supermodularity…

关于最小化非加权平方和的DIGing最优参数设计

Optimal Parameter Design for DIGing on Minimizing Unweighted Sum of Squares

研究分布式优化算法DIGing在最小化非加权平方和时的最优参数设计，通过将迭代算法表示为动态线性系统并分解，利用劳斯稳定性判据推导最优收敛速率及参数公式，为DIGing算法最优参数设计迈出第一步。

Hermitian Sums of Squares Modulo Hermitian Ideals

In this work we study the problem of writing a Hermitian polynomial as a Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea of Putinar-Scheiderer to obtain necessary matrix positivity conditions for Hermitian polynomials to be Hermitian sums of squares modulo Hermitian ideals. We show that the conditions are sufficient for a class of examples making a connection to the operator-valued Riesz-Fejer theorem and block Toeplitz forms. The work fits into the larger themes of Hermitian versions of Hilbert's 17-th problem and characterizations of positivity.

Critical Point Finding with Newton-MR by Analogy to Computing Square Roots

Understanding of the behavior of algorithms for resolving the optimization problem (hereafter shortened to OP) of optimizing a differentiable loss function (OP1), is enhanced by knowledge of the critical points of that loss function, i.e. the points where the gradient is 0. Here, we describe a solution to the problem of finding critical points by proposing and solving three optimization problems: 1) minimizing the norm of the gradient (OP2), 2) minimizing the difference between the pre-conditioned update direction and the gradient (OP3), and 3) minimizing the norm of the gradient along the upd…

Splittings and symbolic powers of square-free monomial Ideals

We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism which resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung which states that the normalized $a$-invariants and the Castelnuovo-Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic…