## Global Optimization with Polynomials and the Problem of Moments (2001)

Venue: | SIAM Journal on Optimization |

Citations: | 319 - 33 self |

### BibTeX

@ARTICLE{Lasserre01globaloptimization,

author = {Jean B. Lasserre},

title = {Global Optimization with Polynomials and the Problem of Moments},

journal = {SIAM Journal on Optimization},

year = {2001},

volume = {11},

pages = {796--817}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of Karush--Kuhn--Tucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22, 90C25 PII. S1052623400366802 1.

### Citations

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Citation Context ...NTS 801 where X is a real-valued symmetric matrix and #A, B# stands for the usual Frobenius inner product trace(AB) for real-valued symmetric matrices. The reader is referred to Vandenberghe and Boyd =-=[20]-=- for a survey on semidefinite programming. We first have the following result. Proposition 3.1. Assume that Q # has a feasible solution. Then Q # is solvable and there is no duality gap, that is, inf ... |

157 |
Positive polynomials on compact semi-algebraic sets
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Citation Context ...omials. For the historical development and recent results on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], [3], 798 JEAN B. LASSERRE Jacobi [8], Putinar =-=[15]-=-, Putinar and Vasilescu [14], Simon [19], Schmudgen [16], and references therein. The paper is organized as follows. We introduce the notation and some preliminary results in section 2. The unconstrai... |

141 |
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Citation Context ...s on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], [3],s798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu [14], Simon [19], Schmüdgen =-=[16]-=-, and references therein. The paper is organized as follows. We introduce the notation and some preliminary results in section 2. The unconstrained case is treated in section 3 and the constrained cas... |

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Citation Context ...global minimum, but in general a lower bound only (if (Q 1 K ) # has a feasible solution). Solving some test problems. We have also solved the following test problems proposed in Floudas and Pardalos =-=[6]-=-. Problem 2.2 in [6]. # # # # # # # # # min x,y p(x, y) := c T x - 0.5x T Qx + d T y; 6x 1 + 3x 2 + 3x 3 + 2x 4 + x 5 # 6.5; 10x 1 + 10x 3 + y # 20; 0 # y; 0 # x i # 1, i = 1, . . . , 5 with Q := I an... |

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Squared functional systems and optimization problems. Chapter 17
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Citation Context ...oblems like quadratic, linear, and 0-1 programming as particular cases. In the one-dimensional case, that is, when n = 1, Shor [17] first showed that (1.1) reduces to a convex problem. Next, Nesterov =-=[13]-=-, invoking a well-known representation of nonnegative polynomials as a sum of squares of polynomials, provided a self-concordant barrier for the cone K 2n of nonnegative univariate polynomials so that... |

88 | The classical moment problem as a self-adjoint finite difference operator
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Citation Context ...nd recent results on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], [3], 798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu [14], Simon =-=[19]-=-, Schmudgen [16], and references therein. The paper is organized as follows. We introduce the notation and some preliminary results in section 2. The unconstrained case is treated in section 3 and the... |

45 |
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Citation Context ..., . . . , r, which includes many applications of interest and standard problems like quadratic, linear, and 0-1 programming as particular cases. In the one-dimensional case, that is, when n = 1, Shor =-=[17]-=- first showed that (1.1) reduces to a convex problem. Next, Nesterov [13], invoking a well-known representation of nonnegative polynomials as a sum of squares of polynomials, provided a self-concordan... |

44 |
Nondifferentiable Optimization and Polynomial Problems
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Citation Context ...um of squares of polynomials. Even more, as mentioned in Nesterov [13], the global unconstrained minimization of a 4-degree polynomial is an NP-hard problem. Via successive changes of variables, Shor =-=[18]-=- (see also Ferrier [5]) proposed to transform (1.2) into a quadratic, quadratically constrained optimization problem and then solve a standard convex linear matrix inequality (LMI) relaxation to obtai... |

33 |
Recursiveness, positivity, and truncated moment problems
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Citation Context ... problem on the representation of nonnegative polynomials. For the historical development and recent results on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow =-=[2]-=-, [3], 798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu [14], Simon [19], Schmudgen [16], and references therein. The paper is organized as follows. We introduce the notation and s... |

28 | A representation theorem for certain partially ordered commutative rings
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Citation Context ...egative polynomials. For the historical development and recent results on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], [3], 798 JEAN B. LASSERRE Jacobi =-=[8]-=-, Putinar [15], Putinar and Vasilescu [14], Simon [19], Schmudgen [16], and references therein. The paper is organized as follows. We introduce the notation and some preliminary results in section 2. ... |

27 |
The multidimensional moment problem and semi-groups
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Citation Context ...ials and Hilbert's 17th problem on the representation of nonnegative polynomials. For the historical development and recent results on the theory of moments, the interested reader is referred to Berg =-=[1]-=-, Curto and Fialkow [2], [3], 798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu [14], Simon [19], Schmudgen [16], and references therein. The paper is organized as follows. We intro... |

25 | Solving the moment problem by dimensional extension
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(Show Context)
Citation Context ...evelopment and recent results on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], [3], 798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu =-=[14]-=-, Simon [19], Schmudgen [16], and references therein. The paper is organized as follows. We introduce the notation and some preliminary results in section 2. The unconstrained case is treated in secti... |

21 |
Fialkow: Flat extensions of positive moment matrices: Recursively generated relations
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Citation Context ...lem on the representation of nonnegative polynomials. For the historical development and recent results on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], =-=[3]-=-, 798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu [14], Simon [19], Schmudgen [16], and references therein. The paper is organized as follows. We introduce the notation and some p... |

10 |
Optimality conditions and LMI relaxations for 0-1 programs
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Citation Context ... quadratic form with no squared terms under the integrality constraints x 2 i = 1 for all i) are encouraging. Indeed, the optimal value was obtained at the Q 2 K relaxation in all cases (see Lasserre =-=[12]-=-) for n = 5 and even n = 10. 5. Karush--Kuhn--Tucker global optimality conditions. In this section we still consider the problem PK with a compact set K defined by polynomials inequalities g i (x) # 0... |

8 | On special representations of strictly positive polynomials
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- 2001
(Show Context)
Citation Context ...stance, if there is one polynomial g i (x) such that {g i (x) # 0} is compact (take u k (x) # 0 except u i (x) # 1 in (4.2)). It is also satisfied if all the p i 's are linear (see Jacobi and Prestel =-=[9]-=-) and for 0-1 programs, that is, when K includes the inequalities x 2 i # x i and x i # x 2 i for all i. Therefore, one way to ensure that Assumption 4.1 holds is to add to the definition of K the ext... |

5 |
Hilbert’s 17th problem and best dual bounds in quadatic minimization. Cybernet Systems Anal
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- 1998
(Show Context)
Citation Context ...mials. Even more, as mentioned in Nesterov [13], the global unconstrained minimization of a 4-degree polynomial is an NP-hard problem. Via successive changes of variables, Shor [18] (see also Ferrier =-=[5]-=-) proposed to transform (1.2) into a quadratic, quadratically constrained optimization problem and then solve a standard convex linear matrix inequality (LMI) relaxation to obtain good lower bounds. B... |

5 | The Markov Moment Problem and Extremal - Krein, Nudelman - 1997 |

3 | Conditions for global optimality, in: Handbook of Global Optimization - Hiriart-Urruty - 1995 |

3 |
udgen: The K-Moment Problem for Compact Semi-Algebraic Sets, Mathematische Annalen 289
- Schm
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(Show Context)
Citation Context ...s on the theory of moments, the interested reader is referred to Berg [1], Curto and Fialkow [2], [3], 798 JEAN B. LASSERRE Jacobi [8], Putinar [15], Putinar and Vasilescu [14], Simon [19], Schmudgen =-=[16]-=-, and references therein. The paper is organized as follows. We introduce the notation and some preliminary results in section 2. The unconstrained case is treated in section 3 and the constrained cas... |

1 |
Lagrangian stability and global optimality in nonconvex quadratic minimization over Euclidian balls and spheres
- Tao, An
- 1995
(Show Context)
Citation Context ...eneral case where # 2 xx L(x # , # # ) # 0, which may also happen at a global minimizer of some nonconvex quadratic programs. For instance, the particular nonconvex quadratic problems investigated in =-=[4]-=- reduce to solving the single LMI problem Q 1 K . The di#erence between the convex and nonconvex cases is that Q 1 K provides an exact solution in the convex case, whereas one has to solve an (often f... |