## SOSTOOLS: Sum of squares optimization toolbox for MATLAB (2004)

Citations: | 71 - 8 self |

### BibTeX

@MISC{Prajna04sostools:sum,

author = {Stephen Prajna and Antonis Papachristodoulou and Peter Seiler and Pablo A. Parrilo},

title = {SOSTOOLS: Sum of squares optimization toolbox for MATLAB},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

Version 2.00

### Citations

809 | Semidefinite programming
- Vandenverghe, Boyd
- 1996
(Show Context)
Citation Context ...ving sum of squares programs. The techniques behind it are based on the sum of squares decomposition for multivariate polynomials [4], which can be efficiently computed using semidefinite programming =-=[24]-=-. SOSTOOLS is developed as a consequence of the recent interest in sum of squares polynomials [12, 13, 20, 4, 18, 10, 9], partly due to the fact that these techniques provide convex relaxations for ma... |

802 | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software
- Sturm
- 1999
(Show Context)
Citation Context ...e 1.1: Diagram depicting relations between sum of squares program (SOSP), semidefinite program (SDP), SOSTOOLS, and SeDuMi or SDPT3. SOSP. At present, it uses other free MATLAB add-ons such as SeDuMi =-=[21]-=- or SDPT3 [23] as the SDP solver. This whole process is depicted in Figure 1.1. In the original release of SOSTOOLS, polynomials are implemented solely as symbolic objects, making full use of the capa... |

232 | SDPT3 — a Matlab software package for semidefinite programming
- Toh, Todd, et al.
- 1999
(Show Context)
Citation Context ... depicting relations between sum of squares program (SOSP), semidefinite program (SDP), SOSTOOLS, and SeDuMi or SDPT3. SOSP. At present, it uses other free MATLAB add-ons such as SeDuMi [21] or SDPT3 =-=[23]-=- as the SDP solver. This whole process is depicted in Figure 1.1. In the original release of SOSTOOLS, polynomials are implemented solely as symbolic objects, making full use of the capabilities of th... |

145 |
The K-moment problem for compact semi-algebraic sets
- Schmüdgen
- 1991
(Show Context)
Citation Context ... (x)gi1(x)gi2(x) + · · · , (3.7) then it follows that γ is a lower bound for the constrained optimization problem stated above. This specific kind of representation corresponds to Schmüdgen’s theorem =-=[19]-=-. By maximizing γ, we can obtain a lower bound that becomes increasingly tighter as the degree of the expression (3.7) is increased. As an example, consider the problem of minimizing x1 + x2, subject ... |

98 | Some concrete aspects of Hilbert’s 17th Problem
- Reznick
- 1996
(Show Context)
Citation Context ...on for multivariate polynomials [4], which can be efficiently computed using semidefinite programming [24]. SOSTOOLS is developed as a consequence of the recent interest in sum of squares polynomials =-=[12, 13, 20, 4, 18, 10, 9]-=-, partly due to the fact that these techniques provide convex relaxations for many hard problems such as global, constrained, and boolean optimization. Besides the optimization problems mentioned abov... |

45 |
Class of global minimum bounds of polynomial functions
- Shor
- 1987
(Show Context)
Citation Context ...on for multivariate polynomials [4], which can be efficiently computed using semidefinite programming [24]. SOSTOOLS is developed as a consequence of the recent interest in sum of squares polynomials =-=[12, 13, 20, 4, 18, 10, 9]-=-, partly due to the fact that these techniques provide convex relaxations for many hard problems such as global, constrained, and boolean optimization. Besides the optimization problems mentioned abov... |

33 | An algorithm for sums of squares of real polynomials
- Powers, Wörmann
- 1998
(Show Context)
Citation Context ... p(x) = Z T (x)QZ(x), (1.2) where Z(x) is some properly chosen vector of monomials. Expressing an SOS polynomial using a quadratic form as in (1.2) has also been referred to as the Gram matrix method =-=[4, 15]-=-. As hinted above, sums of squares techniques can be used to provide tractable relaxations for many hard optimization problems. A very general and powerful relaxation methodology, introduced in [12, 1... |

30 |
Extremal psd forms with few terms
- Reznick
- 1978
(Show Context)
Citation Context ...ide of the inequality is a high degree sparse polynomial (i.e., containing a few nonzero terms), it is computationally more efficient to impose the SOS condition using a reduced set of monomials (see =-=[17]-=-) in the Gram matrix form. This will result in a smaller size semidefinite program, which is easier to solve. By default, SOSTOOLS does not try to obtain this optimal reduced set of monomials, since t... |

20 |
Polynomial equations and convex polytopes
- Sturmfels
- 1998
(Show Context)
Citation Context ...ix case this word usually means that many coefficients are zero, in the polynomial case the specific vanishing pattern is also taken into account. This idea is formalized by using the Newton polytope =-=[22]-=-, defined as the convex hull of the set of exponents, considered as vectors in R n . It was shown by Reznick in [17] that Z(x) need only contain monomials whose squared degrees are contained in the co... |

17 | Semidefinite programming relaxations and algebraic optimization in control
- Parrilo, Lall
(Show Context)
Citation Context ...sum of squares decomposition can be reduced which results in a decrease of the size of the semidefinite program. Consider for example the polynomial p(x, y) = 4x 4 y 6 + x 2 − xy 2 + y 2 , taken from =-=[14]-=-. Its Newton polytope is a triangle, being the convex hull of the points (4, 6), (2, 0), (1, 2), (2, 0); see Figure 2.1. By the result mentioned above, we can always find a SOS decomposition that cont... |

3 | On convexity in stabilization of nonlinear systems
- Rantzer, Parrilo
- 2000
(Show Context)
Citation Context ...ms, such as: search for Lyapunov functions to prove stability of a dynamical system, computation of tight upper bounds for the structured singular value µ [12], and stabilization of nonlinear systems =-=[16]-=-. Some examples related to these problems, as well as several other optimization-related examples, are provided and solved in the demo files that are distributed with SOSTOOLS. In the next two section... |