High-order sum-of-squares structured tensors:theory and applications

Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience.An important class of tensor decomposition is sum-of-squares(SOS)tensor decomposition.SOS tensor decomposition has a close connection with SOS polynomials,and SOS polynomials are very important in polynomial theory and polynomial optimization.In this paper,we give a detailed survey on recent advances of high-order SOS tensors and their applications.It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case.Then,the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established.Further,a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided,and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified.Some potential research directions in the future are also listed in this paper.

Safety-critical Policy Iteration Algorithm for Control under Model Uncertainty

Safety is an important aim in designing safe-critical systems.To design such systems,many policy iterative algorithms are introduced to find safe optimal controllers.Due to the fact that in most practical systems,finding accurate information from the system is rather impossible,a new online training method is presented in this paper to perform an iterative reinforcement learning based algorithm using real data instead of identifying system dynamics.Also,in this paper the impact of model uncertainty is examined on control Lyapunov functions(CLF)and control barrier functions(CBF)dynamic limitations.The Sum of Square program is used to iteratively find an optimal safe control solution.The simulation results which are applied on a quarter car model show the efficiency of the proposed method in the fields of optimality and robustness.

On computing minimal H-eigenvalue of sign-structured tensors

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.

Proving total correctness and generating preconditions for loop programs via symbolic-numeric computation methods

We present a symbolic-numeric hybrid method, based on sum-of-squares （SOS） relaxation and rational vec- tor recovery, to compute inequality invariants and ranking functions for proving total correctness and generating pre- conditions for programs. The SOS relaxation method is used to compute approximate invariants and approximate rank- ing functions with floating point coefficients. Then Gauss- Newton refinement and rational vector recovery are applied to approximate polynomials to obtain candidate polynomials with rational coefficients, which exactly satisfy the conditions of invariants and ranking functions. In the end, several exam- ples are given to show the effectiveness of our method.