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

Venue: | SIAM JOURNAL ON OPTIMIZATION |

Citations: | 556 - 47 self |

### Citations

1078 | Semidefinite programming
- Vandenberghe, Boyd
- 1996
(Show Context)
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 ... |

265 |
Positive polynomials on compact semi-algebraic sets
- Putinar
- 1993
(Show Context)
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... |

208 |
The K-moment problem for compact semi-algebraic sets
- Schmüdgen
- 1991
(Show Context)
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... |

150 |
A Collection of Test Problems for Constrained Global Optimization Algorithms
- Floudas, Pardalos
- 1990
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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... |

148 | The classical Moment Problem as a Self-Adjoint Finite Difference Operator
- Simon
- 1998
(Show Context)
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... |

128 |
Squared functional systems and optimization problems
- Nesterov
- 2000
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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... |

75 | Recursiveness, positivity, and truncated moment problems
- Curto, Fialkow
- 1991
(Show Context)
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... |

63 |
Quadratic optimization problems
- SHOR
- 1987
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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... |

58 |
Nondi¤erentiable optimization and polynomial problems., Nonconvex Optimization and Its Applications. 24
- Shor
- 1998
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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... |

42 | Solving moment problems by dimensional extension
- Putinar, Vasilescu
- 1999
(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... |

39 |
The multidimensional moment problem and semigroups. Moments in mathematics (San
- Berg
- 1987
(Show Context)
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... |

39 | A representation theorem for certain partially ordered commutative rings
- Jacobi
(Show Context)
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. ... |

34 | Flat extensions of positive moment matrices: recursively generated relations
- Curto, Fialkow
- 1998
(Show Context)
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... |

11 | The Markov Moment Problem and Extremal - Krein, Nudelman - 1977 |

10 |
Optimality conditions and LMI relaxations for 0-1 programs
- LASSERRE
- 2000
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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... |

9 | On special representations of strictly positive polynomials
- JACOBI, PRESTEL
- 2000
(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 quadratic minimization
- FERRIER
- 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 | Conditions for global optimality, in: Handbook of Global Optimization - Hiriart-Urruty - 1995 |

4 |
UDGEN: The K-moment problem for compact semi-algebraic sets
- SCHM
(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... |