### Abstract

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H<inf>∞</inf> control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

Original language | English (US) |
---|---|

Pages (from-to) | 2383-2417 |

Number of pages | 35 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 20 |

Issue number | 8 |

DOIs | |

State | Published - Oct 1 2015 |

### Fingerprint

### Keywords

- Convex optimization
- Handelman's theorem
- Lyapunov stability analysis
- Polya's theorem
- Polynomial optimization

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

**Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares.** / Kamyar, Reza; Peet, Matthew.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems - Series B*, vol. 20, no. 8, pp. 2383-2417. https://doi.org/10.3934/dcdsb.2015.20.2383

}

TY - JOUR

T1 - Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

AU - Kamyar, Reza

AU - Peet, Matthew

PY - 2015/10/1

Y1 - 2015/10/1

N2 - In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H∞ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

AB - In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H∞ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

KW - Convex optimization

KW - Handelman's theorem

KW - Lyapunov stability analysis

KW - Polya's theorem

KW - Polynomial optimization

UR - http://www.scopus.com/inward/record.url?scp=84939857089&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939857089&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2015.20.2383

DO - 10.3934/dcdsb.2015.20.2383

M3 - Article

VL - 20

SP - 2383

EP - 2417

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 8

ER -