Posynomials and generalized posynomials
Convex representation via the log-sum-exp function
where and . We can write in short-hand notation:
where we follow the power law notation: by convention, for two vectors , is the product .
Example. The function with domain and values is a monomial, while is not.
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.
The Stephan law of physics states that the power radiated by a body evolves as , where is the surface area of the body, and the temperature; this is a monomial in variables .
In industrial engineering, the experience curve relates the cost of producing a unit for the first time, to the cost of producing the unit for the -th time, as , where is the cumulative number of unit produced, and exponent depends on the industry. Thus is a monomial in and .
A function is a posyomial if its domain is (the set of vectors with positive components) and its values take the form of a non-negative sum of monomials:
where , , and each is a monomial.
The values of a posynomial can be always written as
for some , and . If we denote by , the -th row of , then we can write in short-hand notation
where we follow the above power law notation to define what means when are two vectors of the same size.
is a generalized posynomial.
Monomials and (generalized) posynomials are not convex. However, with a simple transformation of the variables, we can transform them into convex ones.
For a monomial with values , where and , we have
where , , and . Our transformation yields an affine function.
For a posynomial with values
where , we have
where , . The above can be written
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 function is convex, this transformation will allow us to use convex optimization to optimize posynomials.
Remark: Why do we take the log?.
For example, consider the posynomial with values
where are two posynomials. For , the constraint
can be expressed as two posynomial constraints in .
Likewise, for , consider the power constraint
with an ordinary posynomial and . Since , the above is equivalent to
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.