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 ◽  
Author(s):  
Herbert Hoijtink ◽  
Xin Gu ◽  
Joris Mulder ◽  
Yves Rosseel

The Bayes factor is increasingly used for the evaluation of hypotheses. These may betraditional hypotheses specified using equality constraints among the parameters of thestatistical model of interest or informative hypotheses specified using equality andinequality constraints. So far no attention has been given to the computation of Bayesfactors from data with missing values. A key property of such a Bayes factor should bethat it is only based on the information in the observed values. This paper will show thatsuch a Bayes factor can be obtained using multiple imputations of the missing values.


2021 ◽  
Vol 2090 (1) ◽  
pp. 012125
Author(s):  
Nikolaos P. Theodorakatos ◽  
Miltiadis Lytras ◽  
Rohit Babu

Abstract The impact of the generalized pattern search algorithm (GPSA) on power system complete observability utilizing synchrophasors is proposed in this work. This algorithmic technique is an inherent extension of phasor measurement unit (PMU) minimization in a derivative-free framework by evaluating a linear objective function under a set of equality constraints that is smaller than the decision variables in number. A comprehensive study about the utility of such a system of equality constraints under a quadratic objective has been given in our previous paper. The one issue studied in this paper is the impact of a linear cost function to detect optimality in a shorter number of iterations, whereas the cost is minimized. The GPSA evaluates a linear cost function through the iterations needed to satisfy feasibility and optimality criteria. The other issue is how to improve the performance of convergence towards optimality using a gradient-free mathematical algorithm. The GPSA detects an optimal solution in a fewer number of iterations than those spent by a recursive quadratic programming (RQP) algorithm. Numerical studies on standard benchmark power networks show significant improvement in the maximum observability over the existing measurement redundancy generated by the RQP optimization scheme already published in our former paper.


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