sum-of-squares polynomial

Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert

  • Ryan Cory-Wright
  • Cristina Cornelio
  • Sanjeeb Dash
  • Bachir El Khadir
  • Lior Horesh
    • Categories Basic Sciences Applications, Global Optimization Applications, Semi-definite Programming Tags , , ,

    The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in … Read more

    Exactness of Parrilo’s conic approximations for copositive matrices and associated low order bounds for the stability number of a graph

  • Monique Laurent
  • Luis Felipe Vargas
    • Categories Approximation Algorithms, Semi-definite Programming Tags , , , , ,

    De Klerk and Pasechnik (2002) introduced the bounds $\vartheta^{(r)}(G)$ ($r\in \mathbb{N}$) for the stability number $\alpha(G)$ of a graph $G$ and conjectured exactness at order $\alpha(G)-1$: $\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)$. These bounds rely on the conic approximations $\mathcal{K}_n^{(r)}$ by Parrilo (2000) for the copositive cone $\text{COP}_n$. A difficulty in the convergence analysis of $\vartheta^{(r)}$ is the bad behaviour … Read more

    Finite convergence of sum-of-squares hierarchies for the stability number of a graph

  • Monique Laurent
  • Luis Felipe Vargas
    • Categories Combinatorial Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , , , , ,

    We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp.875–892], who conjectured convergence to $\alpha(G)$ in $r=\alpha(G) -1$ steps. Even the weaker conjecture claiming finite convergence is still open. … Read more

    Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization

  • Etienne de Klerk
  • Monique Laurent
  • Roxana Hess
    • Categories Semi-definite Programming Tags , , , , ,

    We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\int_{[-1,1]^n} f(x)h(x)d\mu(x)$, where $d\mu(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It … Read more