This note uses sum of squares (SOS) relaxation to solve stochastic reachability problems for jump-diffusion processes. The main result is a polynomial characterization of the infinitesimal generator for the solution of a jump-diffusion process’ boundary value problem, thereby enabling one to compute a bound on the probability of reaching a target set in finite time using SOS optimization.

where f(x) is a real multivariate polynomial in x ∈ Rn and S is a feasible set defined by polynomial equalities or inequalities. In this thesis, we do not have any convexity/concavity assumptions on f(x) or S. The goal is to find the global minimum and global minimizers if any. Polynomial optimization of form (1.1) is quite general in practical applications. Many NP-hard problems like max cut, ...

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that n+2d−1 2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjectu...

We investigate the relationships between various sum of squares (SOS) and semidefinite programming (SDP) relaxations for the sensor network localization problem. In particular, we show that Biswas and Ye’s SDP relaxation is equivalent to the degree one SOS relaxation of Kim et al. We also show that Nie’s sparse-SOS relaxation is stronger than the edge-based semidefinite programming (ESDP) relax...

Motivation [Warning: the following discussion is based on my very rudimentary understanding of this problem area... don’t place too much trust in it. Also, while the problem is practical our solution is most certainly not— we need to run the SOS algorithm on a large number of variables and degree k that is a large constant or sometimes even logarithmic. However, one can hope that the ideas behi...

This paper discusses how to find the global minimum and minimizers of functions that are given as the summation of small polynomials (“small” means involving a small number of variables). Some sparse sum of squares (SOS) relaxations are proposed. We compare the computational complexity and lower bound with prior SOS relaxations. The proposed methods are specially useful in solving nonlinear lea...

This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal POPs. A numerical example i...

This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal POPs. A numerical example i...

We consider the commonly used "Sum-of-Squares" (SoS) reconstruction method for phased-array magnetic resonance imaging with unknown coil sensitivities. We show that the signal-to-noise ratio (SNR) in the image produced by SoS is asymptotically (as the input SNR--> infinity ) equal to that of maximum-ratio combining, which is the best unbiased reconstruction method when the coil sensitivities ar...

There has been significant increase in penetration of renewable generation (RG) sources all over the world. Localized concentration of many such generators could initiate a cascade tripping sequence that might threaten the stability of the entire system. Understanding the impact of cascade tripping process would help the system planner identify trip sequences that must be blocked in order to in...