| # Posynomials GP > Posynomials | Standard Forms | Applications Monomials Posynomials Generalized posynomials Convex representation via the log-sum-exp function ## Monomials ### Definition A function is a monomial if its domain is (the set of vectors with positive components) and its values take the form where and . We can write in short-hand notation: where we follow the power law notation: by convention, for two vectors , is the product . Examples: The function with domain and values is a monomial, while is not. ### Log-linearity and power laws Monomials are closely related to linear or affine functions: indeed, if is a monomial in variable , then is affine in the vector . Hence monomial functions could be called ‘log-linear’’, although the term is not widely used. 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: 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 . ## Posynomials 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. Examples: Construction and operating costs for a water tank. Signal-to-noise ratios in wireless communications. ## Generalized Posynomials A generalized posynomial function is any function obtained from posynomials using addition, multiplication, pointwise maximum, and raising to constant positive power. For example, the function with values is a generalized posynomial. ## Convex Representation Monomials and (generalized) posynomials are not convex. However, with a simple transformation of the variables, we can transform them into convex ones. ### Convex representation of posynomials Consider a posynomial function . Instead of the original (positive) variables, we use the new variable , . We then take the logarithm of the function . Let us look at the effect of such transformations on monomials and posynomials. 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 where is the matrix with rows , , and is the log-sum-exp function. Recall that this function is convex. 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?. ### Convex representation of generalized posynomials Adding 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 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.