A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem

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.

Generalized Chain Regression-cum-Chain Ratio Estimator for Population Mean under Stratified Extreme-cum-Median Ranked Set Sampling

Estimation of population mean of study variable Y suffers loss of precision in the presence of high variation in the data set. The use of auxiliary information incorporated in construction of an estimator under ranked set sampling scheme results in efficient estimation of population mean. In this paper, we propose an efficient generalized chain regression-cum-chain ratio type estimator to estimate finite population mean of study variable under stratified extreme-cum-median ranked set sampling utilizing information on two auxiliary variables. Mean square error (MSE) of the proposed generalized estimator is derived up to first order of approximation. The applications of the proposed estimator under symmetrical and asymmetrical probability distributions are discussed using simulation study and real-life data set for comparisons of efficiency. It is concluded that the proposed generalized estimator performs efficiently as compared to some existing estimators. It is also observed that the efficiency of the proposed estimator is directly proportional to the correlations between the study variable and its auxiliary variables.

Ratio-Type Estimation Using Scrabled Auxiliary Variables in Stratification Under Simple Random Sampling and Ranked Set Sampling

This chapter introduced basic elements on stratified simple random sampling (SSRS) on ranked set sampling (RSS). The chapter extends Singh et al. results to sampling a stratified population. The mean squared error (MSE) is derived. SRS is used independently for selecting the samples from the strata. The chapter extends Singh et al. results under the RSS design. They are used for developing the estimation in a stratified population. RSS is used for drawing the samples independently from the strata. The bias and mean squared error (MSE) of the developed estimators are derived. A comparison between the biases and MSEs obtained for the sampling designs SRS and RSS is made. Under mild conditions the comparisons sustained that each RSS model is better than its SRS alternative.

Energy quadratization Runge-Kutta method for the modified phase field crystal equation

In this paper, we propose high order and unconditionally energy stable methods for a modified phase field crystal equation by applying the strategy of the energy quadratization Runge–Kutta methods. We transform the original model into an equivalent system with auxiliary variables and quadratic free energy. The modified system preserves the laws of mass conservation and energy dissipation with the associated energy functional. We present rigorous proofs of the mass conservation and energy dissipation properties of the proposed numerical methods and present numerical experiments conducted to demonstrate their accuracy and energy stability. Finally, we compare long-term simulations using an indicator function to characterize the pattern formation.

A Comparison of Model-Assisted Estimators, With and Without Data-Driven Transformations of Auxiliary Variables, With Application to Forest Inventory

Forest information is requested at many levels and for many purposes. Sampling-based national forest inventories (NFIs) can provide reliable estimates on national and regional levels. By combining expensive field plot data with different sources of remotely sensed information, from airplanes and/or satellite platforms, the precision in estimators of forest variables can be improved. This paper focuses on the design-based model-assisted approach to using NFI data together with remotely sensed data to estimate forest variables for small areas, where the variables studied are total growing stock volume, volume of Norway spruce (Picea abies), and volume of broad-leaved trees. Remote sensing variables may be highly correlated with one another and some may have poor predictive ability for target forest variables, and therefore model selection and/or coefficient shrinkage may be appropriate to improve the efficiency of model-assisted estimators of forest variables. For this purpose, one can use modern shrinkage estimators based on lasso, ridge, and elastic net regression methods. In a simulation study using real NFI data, Sentinel 2 remote-sensing data, and a national airborne laser scanning (ALS) campaign, we show that shrinkage estimators offer advantages over the (weighted) ordinary least-squares (OLS) estimator in a model-assisted setting. For example, for a sample size n of about 900 and with 72 auxiliary variables, the RMSE was up to 41% larger when based on OLS. We propose a data-driven method for finding suitable transformations of auxiliary variables, and show that it can improve estimators of forest variables. For example, when estimating volume of Norway spruce, using a smaller expert selection of auxiliary variables, transformations reduced the RMSE by up to 10%. The overall best results in terms of RMSE were obtained using shrinkage estimators and a larger set of 72 auxiliary variables. However, for this larger set of variables, the use of transformations yielded at most small improvements of RMSE, and at worst large increases of RMSE, except in combination with ridge and elastic net regression.

A MODIFIED GENERALIZED CHAIN RATIO IN REGRESSION ESTIMATOR

Generalized Chain ratio in regression type estimator is efficient for estimating the population mean. Many authors have derived a Generalized Chain ratio in regression type estimator. However, the computation of its Mean Square Error (MSE) is cumbersome based on the fact that several iterations have to be done, hence the need for a modified generalized chain ratio in regression estimator with lower MSE. This study proposed a modified generalized chain ratio in regression estimator which is less cumbersome in its computation. Two data sets were used in this study. The first data were on tobacco production by tobacco producing countries with yield of tobacco (variable of interest), area of land and production in metric tonnes as the auxiliary variables. The second data were the number of graduating pupils (variable of interest) in Ado-Odo/Ota local government, Ogun state with the number of enrolled pupils in primaries one and five as the auxiliary variables. The mean square errors in the existing and proposed estimators for various values of alpha were derived and relative efficiency was determined. The MSE for the existing estimator of tobacco production gave six values 0.0080, 0.0079, 0.0080, 0.0082, 0.0087 and 0.0093 with 0.0079 as the minimum while the proposed estimator gave 0.0054. The MSEs for the existing estimator for the graduating pupils were 20.73, 11.08, 7.49, 9.96, 18.50 and 33.10 with 7.49 as the minimum while the proposed was 6.52. The results of this study showed that the proposed estimator gave lower MSE for the two data sets, hence it is more efficient.