Approximate affine linear relationship between L₁ norm objective functional values and L₂ norm constraint bounds

摘要

For an optimisation problem with an norm objective function subject to an norm inequality constraint, this study shows that there is an approximately linear relationship between the norm objective functional values and the norm specifications. This relationship is verified through the use of random and real world industrial data. The obtained results can be employed for (i) estimating the norm objective functional value without solving the optimisation problem numerically; (ii) providing an insight for defining the norm specification in which a simple method is proposed in this study; and (iii) testing whether the obtained solutions are the globally optimal solutions or not. These advantages are demonstrated via the use of random data.