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This paper proposes a method for eliminating multicollinearity from linear regression models. Specifically, we select the best subset of explanatory variables subject to the upper bound on the condition number of the correlation matrix of selected variables. We first develop a cutting plane algorithm that, to approximate the condition number constraint, iteratively appends valid inequalities to the mixed integer quadratic optimization problem. We also devise a mixed integer semidefinite optimization formulation for best subset selection under the condition number constraint. Computational results demonstrate that our cutting plane algorithm frequently provides solutions of better quality than those obtained using local search algorithms for subset selection. Additionally, subset selection by means of our optimization formulation succeeds when the number of candidate explanatory variables is small.

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The purpose of this study was to clarify the relationship between the position of the center of gravity and the abdominal muscle thickness, and to provide data to guide center-of-gravity conditions. [Participants and Methods] The subjects were 52 healthy individuals (25 males and 27 females; mean age: 20.9 ± 1.3 years). Changes in muscle thicknesses of the external oblique, internal oblique, and transverse abdominal muscles were compared in the resting position and center-of-gravity front-back and left-right conditions. [Results] The main effect was on the rates of change in muscle thicknesses of the external oblique and internal oblique muscles. The rates of change in muscle thicknesses of all muscles was highest in the center-of-gravity right condition. [Conclusion] Changes in the center of gravity affect abdominal muscle thickness.