scholarly journals On the triality theory for a quartic polynomial optimization problem

2012 ◽  
Vol 8 (1) ◽  
pp. 229-242 ◽  
Author(s):  
David Yang Gao ◽  
◽  
Changzhi Wu ◽  
Keyword(s):  
Optimization Problem ◽  
Polynomial Optimization ◽  
Polynomial Optimization Problem ◽  
Triality Theory ◽  
Quartic Polynomial

Sparse Sum-of-Squares Optimization for Model Updating Through Minimization of Modal Dynamic Residuals

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
Dan Li ◽  
Yang Wang
Keyword(s):  
Optimization Problem ◽  
Model Updating ◽  
Polynomial Optimization ◽  
Element Model ◽  
Optimization Method ◽  
Sum Of Squares ◽  
Global Optimality ◽  
Polynomial Optimization Problem ◽  
Computation Efficiency ◽  
Dynamic Residuals

This research investigates the application of sum-of-squares (SOS) optimization method on finite element model updating through minimization of modal dynamic residuals. The modal dynamic residual formulation usually leads to a nonconvex polynomial optimization problem, the global optimality of which cannot be guaranteed by most off-the-shelf optimization solvers. The SOS optimization method can recast a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem. However, the size of the SDP problem can grow very large, sometimes with hundreds of thousands of variables. To improve the computation efficiency, this study exploits the sparsity in SOS optimization to significantly reduce the size of the SDP problem. A numerical example is provided to validate the proposed method.

scholarly journals A Shifted Power Method for Homogenous Polynomial Optimization over Unit Spheres

2015 ◽  
Vol 7 (2) ◽  
pp. 175 ◽  
Author(s):  
Shuquan Wang
Keyword(s):  
Global Convergence ◽  
Optimization Problem ◽  
Polynomial Optimization ◽  
Power Method ◽  
Polynomial Optimization Problem

In this paper, we propose a shifted power method for a type of polynomial optimization problem over unit spheres. The global convergence of the proposed method is established and an easily implemented scope of the shifted parameter is provided.

scholarly journals Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction

2021 ◽  
Author(s):  
María López Quijorna
Keyword(s):  
Optimization Problem ◽  
Polynomial Optimization ◽  
Gaussian Quadrature ◽  
Hankel Matrix ◽  
Optimal Solution ◽  
Quadrature Rule ◽  
Polynomial Optimization Problem ◽  
Optimal Value ◽  
Semialgebraic Subset ◽  
Gns Construction

AbstractA basic closed semialgebraic subset of $${\mathbb {R}}^{n}$$ R n is defined by simultaneous polynomial inequalities $$p_{1}\ge 0,\ldots ,p_{m}\ge 0$$ p 1 ≥ 0 , … , p m ≥ 0 . We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality we will extract the potential minimizers with a truncated version of the Gelfand–Naimark–Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will enable us to prove, with the use of the GNS truncated construction, a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.

Improved approximation results on standard quartic polynomial optimization

2016 ◽  
Vol 11 (8) ◽  
pp. 1767-1782 ◽  
Author(s):  
Chen Ling ◽  
Hongjin He ◽  
Liqun Qi
Keyword(s):  
Polynomial Optimization ◽  
Quartic Polynomial
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