linear equality constraints
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Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 270
Author(s):  
Chenyang Hu ◽  
Yuelin Gao ◽  
Fuping Tian ◽  
Suxia Ma

Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.


2021 ◽  
pp. 1-43
Author(s):  
Yanqin Fan ◽  
Xuetao Shi

Via generalized interval arithmetic, we propose a Generalized Interval Arithmetic Center and Range (GIA-CR) model for random intervals, where parameters in the model satisfy linear inequality constraints. We construct a constrained estimator of the parameter vector and develop asymptotically uniformly valid tests for linear equality constraints on the parameters in the model. We conduct a simulation study to examine the finite sample performance of our estimator and tests. Furthermore, we propose a coefficient of determination for the GIA-CR model. As a separate contribution, we establish the asymptotic distribution of the constrained estimator in Blanco-Fernández (2015, Multiple Set Arithmetic-Based Linear Regression Models for Interval-Valued Variables) in which the parameters satisfy an increasing number of random inequality constraints.


2021 ◽  
pp. 1-42
Author(s):  
Yijia Peng ◽  
Wanghui Bu

Abstract Workspace is an important reference for design of cable-driven parallel robots (CDPRs). Most current researches focus on calculating the workspace of redundant CDPRs. However, few literatures study the workspace of under-constrained CDPRs. In this paper, the static equilibrium reachable workspace (SERW) of spatial 3-cable under-constrained CDPRs is solved numerically since expressions describing workspace boundaries cannot be obtained in closed form. The analysis steps to solve the SERW are as follows. First, expressions which describe the SERW and its boundaries are proposed. Next, these expressions are instantiated through the novel anchor points model composed of linear equations, quadratic equations and limits of tension in cables. Then, based on the reformulated linearization technique (RLT), the constraint system is transformed into a system containing only linear equality constraints and linear inequality constraints. Finally, the framework of branch-and-prune (BP) algorithm is adopted to solve this system. The effect of the algorithm is verified by 2 examples. One is a special 3-cable CDPR in which the anchor points layouts both on the moving platform (MP) and on the base are equilateral triangles, followed by the method to extract the SERW boundary where cables do not interfere with each other. The other is a general case with randomly selected geometry arrangement. The presented method in this paper is universal for spatial 3-cable CDPRs with arbitrary geometry parameters.


Author(s):  
Geovani Nunes Grapiglia ◽  
Ya-xiang Yuan

Abstract In this paper we study the worst-case complexity of an inexact augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded we prove a complexity bound of $\mathcal{O}(|\log (\epsilon )|)$ outer iterations for the referred algorithm to generate an $\epsilon$-approximate KKT point for $\epsilon \in (0,1)$. When the penalty parameters are unbounded we prove an outer iteration complexity bound of $\mathcal{O}(\epsilon ^{-2/(\alpha -1)} )$, where $\alpha>1$ controls the rate of increase of the penalty parameters. For linearly constrained problems these bounds yield to evaluation complexity bounds of $\mathcal{O}(|\log (\epsilon )|^{2}\epsilon ^{-2})$ and $\mathcal{O}(\epsilon ^{- (\frac{2(2+\alpha )}{\alpha -1}+2 )})$, respectively, when appropriate first-order methods ($p=1$) are used to approximately solve the unconstrained subproblems at each iteration. In the case of problems having only linear equality constraints the latter bounds are improved to $\mathcal{O}(|\log (\epsilon )|^{2}\epsilon ^{-(p+1)/p})$ and $\mathcal{O}(\epsilon ^{-(\frac{4}{\alpha -1}+\frac{p+1}{p})})$, respectively, when appropriate $p$-order methods ($p\geq 2$) are used as inner solvers.


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