Posynomials
MonomialsDefinitionA function ![]() where ![]() where we follow the power law notation: by convention, for two vectors Examples:
Log-linearity and power lawsMonomials are closely related to linear or affine functions: indeed, if Just as linear models are important in (approximate) models between general variables, monomials play an ubiquituous role for modeling relationships between positive variables, such as prices, concentrations, energy, or geometric data such as length, area and volume, etc. Like their linear counterpart, power laws can be easily fitted to experimental data, via least-squares methods. Examples:
PosynomialsA function ![]() where The values of a posynomial can be always written as ![]() for some ![]() where we follow the above power law notation to define what Examples: Generalized PosynomialsA generalized posynomial function is any function obtained from posynomials using addition, multiplication, pointwise maximum, and raising to constant positive power. For example, the function ![]() is a generalized posynomial. Convex RepresentationMonomials and (generalized) posynomials are not convex. However, with a simple transformation of the variables, we can transform them into convex ones. Convex representation of posynomialsConsider a posynomial function
![]() where
![]() where ![]() where ![]() where Thus, we can view a posynomial as the log-sum-exp function of an affine combination of the logarithm of the original variables. Since the Remark: Why do we take the log?. Convex representation of generalized posynomialsAdding variables, and with the logarithmic change of variables seen above, we can also transform generalized posynomial inequalities into convex ones. For example, consider the posynomial ![]() where ![]() can be expressed as two posynomial constraints in Likewise, for ![]() with ![]() which in turn is equivalent to the posynomial constraint ![]() Hence, by adding as many variables as necessary, we can express a generalized posynomial constraint as a set of ordinary ones. |