# Strong duality in Lasserre's hierarchy for polynomial optimization

arXiv: https://arxiv.org/abs/1405.7334 · HTML: https://mlsanity.com/paper/1405.7334/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 May 2014.

## Built on by (most cited first, 3 of 3)

- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization](https://mlsanity.com/paper/2105.14033/) (2021)
- [SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications](https://mlsanity.com/paper/2410.21194/) (2024)

## Abstract (arXiv)

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set K described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J.~B.~Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, when the POP feasibility set K is compact, we show that there is no duality gap between each primal and dual SDP problem in Lasserre's hierarchy, provided a redundant ball constraint is added to the description of set K. Our proof uses elementary results on SDP duality, and it does not assume that K has an interior point.