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SDPB is an open-source, arbitrary-precision, parallelized semidefinite program solver, designed for the conformal bootstrap. It solves the following problem:

maximize:  b_0 + \sum_n b_n y_n over (y_1,...,y_N), such that: M_{0j}(x) + \sum_n y_n M_{nj}(x) is positive semidefinite for all x >= 0 and 1 <= j <= J, where each M_{nj}(x) is a polynomial matrix in x.

For more information, see A Semidefinite Program Solver for the Conformal Bootstrap and the manual.

Authors: David Simmons-Duffin (, Walter Landry (

On April 25, 2019, the main branch of this repository was updated to SDPB 2 which has very different performance characteristics and somewhat different usage instructions from version 1.0. Please see the changelog and other documentation for details.

Installation and Usage

The easiest way to run SDPB on a Windows or Mac machine is to follow the Docker instructions. For Linux and HPC centers, the Singularity instructions will probably work better. If you want to build it yourself, there are detailed instructions in

Usage instructions are detailed in

Two python wrappers for SDPB are available:

An unofficial Haskell wrapper is available:



If you use SDPB in work that results in publication, consider citing

  • D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 1506, 174 (2015) arXiv:1502.02033.
  • W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB arXiv:1909.09745.

Depending on how SDPB is used, the following other sources might be relevant:

The first use of semidefinite programming in the bootstrap:

  • D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 1205, 110 (2012) arXiv:1109.5176.

The generalization of semidefinite programming methods to arbitrary spacetime dimension:

  • F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) Vector Models, JHEP 1406, 091 (2014) arXiv:1307.6856.

The generalization of semidefinite programming methods to arbitrary systems of correlation functions:

  • F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 1411, 109 (2014) arXiv:1406.4858.


Derivation of linear and quadratic variations of the objective function, used in approx_objective:

  • M. Reehorst, S. Rychkov, D. Simmons-Duffin, B. Sirois, N. Su, B. van Rees, Navigator Function for the Conformal Bootstrap, arXiv:2104.09518.


Spectrum extraction was originally written for use in:

  • Z. Komargodski and D. Simmons-Duffin, The Random Bond Ising Model in 2.01 and 3 Dimensions, arXiv:1603.04444

An explanation of how it works appears in:

  • D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, arXiv:1612.08471


  • Version 2 of SDPB was made possible by the Simons Collaboration on the Nonperturbative Bootstrap.

  • The design of SDPB was partially based on the solvers SDPA and SDPA-GMP, which were essential sources of inspiration and examples.

  • Thanks to Filip Kos, David Poland, and Alessandro Vichi for collaboration in developing semidefinite programming methods for the conformal bootstrap and assistance testing SDPB.

  • Thanks to Amir Ali Ahmadi, Hande Benson, Pablo Parrilo, and Robert Vanderbei for advice and discussions about semidefinite programming.

  • Thanks also to Noah Stein, who first suggested the idea of semidefinite programming to me in this Math Overflow question.

Works Using SDPB

As of April 2019, SDPB has been used in approximately 70 works. Here is a list of papers citing SDPB.


A semidefinite program solver for the conformal bootstrap.







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