Jiawang Nie

Minimizing Polynomials via Sum of Squares over the Gradient Ideal

A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient ideal is radical or the polynomial is strictly positive on the real gradient variety. This opens up the possibility of solving previously intractable polynomial optimization problems. The related problem of constrained minimization is also considered, and numerical examples are discussed. Experiments show that our method using the gradient variety outperforms prior SOS methods.

Sum of squares method for sensor network localization

Abstract We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, and discuss its various properties. When distances are given exactly, this SOS relaxation often returns true sensor locations. At each step of interior point methods solving this SOS relaxation, the complexity is

Optimality conditions and finite convergence of Lasserre's hierarchy

\mathcal{O}(n^{3})

Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations

, where n is the number of sensors. When the distances have small perturbations, we show that the sensor locations given by this SOS relaxation are accurate within a constant factor of the perturbation error under some technical assumptions. The performance of this SOS relaxation is tested on some randomly generated problems.

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

Lasserre’s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of Lasserre’s hierarchy. Our main results are: (i) Lasserre’s hierarchy has finite convergence when the constraint qualification, strict complementarity and second order sufficiency conditions hold at every global minimizer, under the standard archimedean condition; the proof uses a result of Marshall on boundary hessian conditions. (ii) These optimality conditions are all satisfied at every local minimizer if a finite set of polynomials, which are in the coefficients of input polynomials, do not vanish at the input data (i.e., they hold in a Zariski open set). This implies that, under archimedeanness, Lasserre’s hierarchy has finite convergence generically.

The algebraic degree of semidefinite programming

This paper studies the so-called biquadratic optimization over unit spheres

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

\min_{x\in \mathbb{R}^n,y\in \mathbb{R}^m}\sum_{1\leq i,k\leq n,\,1\leq j,l \leq m}b_{ijkl}x_{i}y_{j}x_{k}{y}_l

All Real Eigenvalues of Symmetric Tensors

, subject to

An exact Jacobian SDP relaxation for polynomial optimization

\|x\| = 1

Addressing the needs of complex MEMS design

,