Global Optimization with Polynomials and the Problem of Moments

By Jean B. Lasserre. SIAM Journal on Optimization, 2000

2,867 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a6430d5de0a3aea562052151efec5ff53d83a10f

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Citations in the corpus

Complete search in continuous global optimization and constraint satisfaction (2004, background): “Starting quite recently, a large number of papers appeared that propose the use of semidefinite relaxations, or convex conic relaxations, to solve polynomial constraint satisfaction and global optimization problems: see, e.g., Chesi and Garulli (2001), Jibetean and De Klerk (2003), Fujie and Kojima ”
Revisiting two theorems of Curto and Fialkow on moment matrices (2005, background): “Then, using Theorem 1.6, Lasserre [19] shows the finite convergence (in n steps); that is, p∗t = p∗ for t ≥ n.”
A tutorial on sum of squares techniques for systems analysis (2005, background): “Not all nonnegative polynomials can be written as SOS, apart from three special cases: (i) when n = 2, (ii) when deg(p) = 2, and (iii) when n = 3 and deg(p) = 4.”
A numerical algorithm for block-diagonal decomposition of matrix *-algebras with general irreducible components (2010, methodology): “It is an extension of the SOS–SDP relaxation method of Lasserre [21] for a polynomial optimization problem to a polynomial SDP.”
SOS approximations of nonnegative polynomials via simple high degree perturbations (2005, background): “But s.o.s. polynomials are also of primary importance for practical computation, especially in view of their numerous potential applications, notably in polynomial optimization; see e.g. [ 9 , 14, 17, 19].”
Truncated K-moment problems in several variables (2005, background): “t problem (with moments relative to monomials of the form ¯zizj). In [CuFi4] we remarked that [CuFi4, Theorem 1.6] extended to truncated moment problems in any number of real or complex variables. In [Las1], Lasserre developed applications of [CuFi4, Theorem 1.6] to optimization problems in the plan”
LMI Techniques for Optimization Over Polynomials in Control: A Survey (2010, background): “Last, Theorem 9 is obtained in [29] for the case of diagonal .”
Strong SOCP Relaxations for the Optimal Power Flow Problem (2015, background): “A way to further strengthen the SDP relaxation is to solve a hierarchy of moment relaxation problems (Lasserre 2001, Parrilo 2003).”
Hankel Matrix Rank Minimization with Applications to System Identification and Realization (2013, background): “, yi corresponds to the ith monomial; see [27, 28].”
Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization (2003, background): “3), which are shown to monotonically converge to f? in [18].”
Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals (2007, methodology): “10) by adding a redundant condition like R − x2 1 − x 2 2 ≥ 0 as introduced in [12].”