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.

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 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.

scholarly journals Optimal Power Flow as a Polynomial Optimization Problem

2016 ◽  
Vol 31 (1) ◽  
pp. 539-546 ◽  
Author(s):  
Bissan Ghaddar ◽  
Jakub Marecek ◽  
Martin Mevissen
Keyword(s):  
Power Flow ◽  
Optimal Power Flow ◽  
Optimization Problem ◽  
Polynomial Optimization ◽  
Polynomial Optimization Problem ◽  
Optimal Power

scholarly journals The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

2021 ◽  
Vol 88 (1) ◽  
Author(s):  
Antoine Gautier ◽  
Matthias Hein ◽  
Francesco Tudisco
Keyword(s):  
Global Convergence ◽  
Convergence Theorem ◽  
Matrix Norm ◽  
Nonnegative Matrices ◽  
Np Hard ◽  
Power Method ◽  
Contraction Ratio ◽  
Matrix Norms ◽  
Vector Norms

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

scholarly journals A Linearized Relaxing Algorithm for the Specific Nonlinear Optimization Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Mio Horai ◽  
Hideo Kobayashi ◽  
Takashi G. Nitta
Keyword(s):  
Global Optimization ◽  
Global Convergence ◽  
Nonlinear Optimization ◽  
Branch And Bound ◽  
Nonconvex Optimization ◽  
Optimization Problem ◽  
New Method ◽  
Linear Relaxation ◽  
Global Optimization Problem ◽  
Nonconvex Global Optimization

We propose a new method for the specific nonlinear and nonconvex global optimization problem by using a linear relaxation technique. To simplify the specific nonlinear and nonconvex optimization problem, we transform the problem to the lower linear relaxation form, and we solve the linear relaxation optimization problem by the Branch and Bound Algorithm. Under some reasonable assumptions, the global convergence of the algorithm is certified for the problem. Numerical results show that this method is more efficient than the previous methods.

A MODIFIED FR CONJUGATE GRADIENT METHOD FOR COMPUTING -EIGENPAIRS OF SYMMETRIC TENSORS

2016 ◽  
Vol 94 (3) ◽  
pp. 411-420
Author(s):  
MEILAN ZENG ◽  
GUANGHUI ZHOU
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Power Method ◽  
Symmetric Tensors

This paper proposes improvements to the modified Fletcher–Reeves conjugate gradient method (FR-CGM) for computing $Z$-eigenpairs of symmetric tensors. The FR-CGM does not need to compute the exact gradient and Jacobian. The global convergence of this method is established. We also test other conjugate gradient methods such as the modified Polak–Ribière–Polyak conjugate gradient method (PRP-CGM) and shifted power method (SS-HOPM). Numerical experiments of FR-CGM, PRP-CGM and SS-HOPM show the efficiency of the proposed method for finding $Z$-eigenpairs of symmetric tensors.

scholarly journals An Allele Real-Coded Quantum Evolutionary Algorithm Based on Hybrid Updating Strategy

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yu-Xian Zhang ◽  
Xiao-Yi Qian ◽  
Hui-Deng Peng ◽  
Jian-Hui Wang
Keyword(s):  
Genetic Algorithm ◽  
Global Convergence ◽  
Convergence Rate ◽  
Local Search ◽  
Evolutionary Algorithm ◽  
Optimization Problem ◽  
Continuous Optimization ◽  
Evolutionary Process ◽  
Experimental Results ◽  
Quantum Genetic Algorithm

For improving convergence rate and preventing prematurity in quantum evolutionary algorithm, an allele real-coded quantum evolutionary algorithm based on hybrid updating strategy is presented. The real variables are coded with probability superposition of allele. A hybrid updating strategy balancing the global search and local search is presented in which the superior allele is defined. On the basis of superior allele and inferior allele, a guided evolutionary process as well as updating allele with variable scale contraction is adopted. AndHεgate is introduced to prevent prematurity. Furthermore, the global convergence of proposed algorithm is proved byMarkovchain. Finally, the proposed algorithm is compared with genetic algorithm, quantum evolutionary algorithm, and double chains quantum genetic algorithm in solving continuous optimization problem, and the experimental results verify the advantages on convergence rate and search accuracy.

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