## Complete solutions and extremality criteria to polynomial optimization problems (2006)

Venue: | Journal of Global Optimization |

Citations: | 10 - 3 self |

### BibTeX

@ARTICLE{Gao06completesolutions,

author = {David Yang Gao},

title = {Complete solutions and extremality criteria to polynomial optimization problems},

journal = {Journal of Global Optimization},

year = {2006},

volume = {35},

pages = {131--143}

}

### OpenURL

### Abstract

Abstract. This paper presents a set of complete solutions to a class of polynomial optimization problems. By using the so-called sequential canonical dual transformation developed in the author’s recent book [Gao, D.Y. (2000), Duality Principles in Nonconvex Systems: Theory, Method and Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, xviii + 454 pp], the nonconvex polynomials in R n can be converted into an one-dimensional canonical dual optimization problem, which can be solved completely. Therefore, a set of complete solutions to the original problem is obtained. Both global minimizer and local extrema of certain special polynomials can be indentified by Gao-Strang’s gap function and triality theory. For general nonconvex polynomial minimization problems, a sufficient condition is proposed to identify global minimizer. Applications are illustrated by several examples. Key words: critical point theory, duality, global optimization, nonlinear programming, NP-hard problem, polynomial minimization.

### Citations

330 | Global optimization with polynomials and the problem of moments
- Lasserre
- 2001
(Show Context)
Citation Context ...the identification of the global minima (see [13]). Therefore, many numerical methods and algorithms have been suggested recently for finding the lower bounds of polynomial optimization problems (see =-=[1,15,16]-=-).s132 D. YANG GAO The primary goal of this paper is to present a potentially useful canonical dual transformation method for solving a special polynomial minimization problem (P) where W is a so-call... |

184 |
Introduction to Global Optimization
- Horst, Pardalos, et al.
- 1996
(Show Context)
Citation Context ...lobal optimization problem. It is known that the application of traditional local optimization procedures for solving nonconvex problems can not guarantee the identification of the global minima (see =-=[13]-=-). Therefore, many numerical methods and algorithms have been suggested recently for finding the lower bounds of polynomial optimization problems (see [1,15,16]).s132 D. YANG GAO The primary goal of t... |

97 |
Squared functional systems and optimization problems,” in High Performance Optimization
- Nesterov
- 2000
(Show Context)
Citation Context ...ere x = (x1,x2,... ,xn) T ∈ R n is a real vector, f ∈ R n is a given vector, and W(x) is a polynomial of degree d. It is known that the polynomial minimization problem is NP-hard even when d = 4 (see =-=[14]-=-). Due to nonconvexity of the cost function P(x), the problem (1) may possess many local minimizers and it represents a global optimization problem. It is known that the application of traditional loc... |

46 | Minimizing polynomial functions
- Parrilo, Sturmfels
- 2003
(Show Context)
Citation Context ...the identification of the global minima (see [13]). Therefore, many numerical methods and algorithms have been suggested recently for finding the lower bounds of polynomial optimization problems (see =-=[1,15,16]-=-).s132 D. YANG GAO The primary goal of this paper is to present a potentially useful canonical dual transformation method for solving a special polynomial minimization problem (P) where W is a so-call... |

28 | Minimizing polynomials via sum of squares over the gradient ideal
- Sturmfels, Demmel, et al.
- 2006
(Show Context)
Citation Context ...the identification of the global minima (see [13]). Therefore, many numerical methods and algorithms have been suggested recently for finding the lower bounds of polynomial optimization problems (see =-=[1,15,16]-=-).s132 D. YANG GAO The primary goal of this paper is to present a potentially useful canonical dual transformation method for solving a special polynomial minimization problem (P) where W is a so-call... |

28 | G.Strang, Geometric Nonlinearity: Potential Energy, Complementary Energy and the Gap Function, Quartely
- Gao
- 1989
(Show Context)
Citation Context ...on P(x), while the vector ¯x2 = f/ ¯ς (2) is a local stationary point. REMARK. For p =1, the nonconvex function W(x) is a double-well function of |x|. By using the method introduced by Gao and Strang =-=[12]-=-, we let ξ1 = �1(x) = 1 2 |x|2 , then W(x) can be written as W(x) = W1(�1(x)), where W1(ξ1)= 1 2α1(ξ1 −λ1) 2 is the canonical function of ξ1 (see [5]). Its conjugate function can be easily obtained by... |

25 | Duality Principles in Nonconvex Systems: Theory, Methods, and Applications
- Gao
- 1999
(Show Context)
Citation Context ...r is to present a potentially useful canonical dual transformation method for solving a special polynomial minimization problem (P) where W is a so-called canonical polynomial of degree d = 2p+1 (see =-=[5]-=-), defined by W(x) = 1 2αp � 1 2αp−1 � � 1 ... 2α1 � 1 2 |x|2 �2 �2 �2 �2, − λ1 ... − λp−1 − λp (2) There αi,λi are given parameters. The nonconvex function W appears in many applications. In the simp... |

20 |
Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications
- Gao
- 1998
(Show Context)
Citation Context ...nima by using the sequential canonical dual transformation. This method has been successfully applied for solving a large class of nonconvex variational analysis and global optimization problems (see =-=[3,6,10,11]-=-).sCOMPLETE SOLUTIONS AND EXTREMALITY CRITERIA 133 2. Complete Solutions By the use of the sequential canonical dual transformation developed in [5], the perfect dual problem (with zero duality gap) (... |

17 | Dual extremum principles in finite deformation elastoplastic analysis
- Gao, Strang
- 1989
(Show Context)
Citation Context ...a sufficient condition for global minimizer, and min P(x) = min x∈Rn x∈R ς>0 n max L(x,ς)= max ς>0 P d (ς). (13) Furthermore, in the study of post-buckling analysis of large deformed beam theory (see =-=[2]-=-), the author discovered that if G(x,ς) � 0, then L(x,ς) is a so-called super-Lagrangian. If (¯x, ¯ς) is a critical point of L(x,ς), and ¯ς<0, then in the neighborhood of (¯x, ¯ς), we have either or P... |

17 | Canonical dual transformation method and generalized triality theory in nonsmooth global optimization
- Gao
- 2000
(Show Context)
Citation Context ...... ,p) and the input f, the dual algebraic Equation (7) has at most s = 2 p+1 − 1 real solutions: ¯ς (i) (i = 1,... ,s). For each dual solution ¯ς ∈ R, the vector ¯x defined by ¯x( ¯ς)= ( ¯ςp!) −1 f =-=(8)-=- is a critical point of the primal problem (P) and P(¯x) = P d ( ¯ς). Conversely, every critical point ¯x of the polynomial P(x) can be written in form (8) for some dual solution ¯ς ∈ R.s134 D. YANG G... |

16 |
Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications
- Gao
- 2000
(Show Context)
Citation Context ...P) can be formulated as the following (P d ) : max ς � P d (ς) =− |f|2 2ςp! − where ςp! = ςpςp−1 ···ς2ς1, and ς1 = ς, ςk = αk p� k=1 ςp! ςk! W ∗ k (ςk) � , (5) � 1 ς 2αk−1 2 � k−1 − λk , k= 2,... ,p. =-=(6)-=- W ∗ k (ςk) is a quadratic function of ςk defined by W ∗ k (ςk) = 1 2αk ς 2 k + λkςk. The dual problem is a nonlinear programming with only one variable ς ∈R, which is much easier than the primal prob... |

14 | Finite Deformation Beam Models and Triality Theory in Dynamical Post-Buckling Analysis
- Gao
(Show Context)
Citation Context ...x)2 −λ1) 2 is the well-known second-order Landau potential in phase transitions of superconductivity and shape memory alloys. In post-buckling analysis of extended beam theory developed by the author =-=[7]-=-, each potential well of W represents a possible buckled beam state. Numerical discretizations of these mechanics problems usually lead to a large-scale polynomial optimization problems of type (P). T... |

4 |
global optimization
- Gao
- 2000
(Show Context)
Citation Context ... R.s134 D. YANG GAO Proof. We first prove the vector defined by (8) solves (3). Substituting ( ¯ςp!) −1f = ¯x into the dual algebraic Equation (7), we obtain � 1 α1 2 |¯x|2 � − λ1 = α1(¯ξ1 − λ1) =¯ς. =-=(9)-=- Thus from (6) we have ¯ςk = αk(¯ξk − λk), k = 1,... ,p. (10) Substituting ¯x( ¯ς)= ( ¯ςp!) −1 f = � p� k=1 αk(¯ξk − λk) � −1 into the left hand side of the canonical Equation (3) leads to f. Thus for... |

2 |
Perfect duality theory and complete set of, solutions to a class of global optimization
- Gao
- 2003
(Show Context)
Citation Context ...nima by using the sequential canonical dual transformation. This method has been successfully applied for solving a large class of nonconvex variational analysis and global optimization problems (see =-=[3,6,10,11]-=-).sCOMPLETE SOLUTIONS AND EXTREMALITY CRITERIA 133 2. Complete Solutions By the use of the sequential canonical dual transformation developed in [5], the perfect dual problem (with zero duality gap) (... |

2 |
Complete solutions to constrained quadratic optimization problems
- Gao
- 2004
(Show Context)
Citation Context ...nima by using the sequential canonical dual transformation. This method has been successfully applied for solving a large class of nonconvex variational analysis and global optimization problems (see =-=[3,6,10,11]-=-).sCOMPLETE SOLUTIONS AND EXTREMALITY CRITERIA 133 2. Complete Solutions By the use of the sequential canonical dual transformation developed in [5], the perfect dual problem (with zero duality gap) (... |