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

For an optimisation problem with an L₁ norm objective function subject to an L₂ norm inequality constraint, this study shows that there is an approximately linear relationship between the L₁ norm objective functional values and the L₂ 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 L₁ norm objective functional value without solving the optimisation problem numerically; (ii) providing an insight for defining the L₂ 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.