# Posynomial function

A posynomial function (also written as a posinomial function ) and the monomial function closely related to it are functions that are used in the formulation of geometric programs . They can be seen as generalizations of polynomial functions in several variables, since any real exponents are allowed.

## definition

Be for as well as for . Then the function is called ${\ displaystyle \ mathbb {R} _ {++} ^ {n}: = \ {x \ in \ mathbb {R} ^ {n} \, | \, x_ {i}> 0}$ ${\ displaystyle i = 1, \ dots, n \}}$ ${\ displaystyle c_ {k}> 0}$ ${\ displaystyle k = 1, \ dots, N}$ ${\ displaystyle f \ colon \ mathbb {R} _ {++} ^ {n} \ to \ mathbb {R}}$ ${\ displaystyle f (x_ {1}, \ dots, x_ {n}) = \ sum _ {k = 1} ^ {N} c_ {k} x_ {1} ^ {a_ {1, k}} \ cdot \ dots \ cdot x_ {n} ^ {a_ {n, k}}}$ a posynomial function. Everyone is there . If the sum consists of only one sum term, one speaks of a monomial function. ${\ displaystyle a_ {i, j} \ in \ mathbb {R}}$ ## example

The function

${\ displaystyle f (x_ {1}, x_ {2}) = {\ frac {{\ sqrt {x_ {1}}} x_ {2} + x_ {2} ^ {2 {,} 3}} {x_ {1} x_ {2}}}}$ is a posynomial function, it has the normal representation

${\ displaystyle f (x_ {1}, x_ {2}) = x_ {1} ^ {- 0 {,} 5} + x_ {1} ^ {- 1} x_ {2} ^ {1 {,} 3 }}$ The function

${\ displaystyle f (x_ {1}, x_ {2}, x_ {3}) = {\ frac {x_ {1} ^ {17} x_ {2}} {x_ {3} ^ {\ sqrt {2} }}}}$ is a monomial function, it has the normal representation

${\ displaystyle f (x_ {1}, x_ {2}, x_ {3}) = x_ {1} ^ {17} x_ {2} x_ {3} ^ {- {\ sqrt {2}}}}$ ## properties

• Posynomial functions are closed under addition, multiplication and multiplication with positive scalars.
• Monomial functions are completed under multiplication, division and positive scaling.
• The posynomial functions thus in particular form a convex cone in the vector space of all functions , the monomial functions at least a (dotted) linear sub- cone .${\ displaystyle \ mathbb {R} _ {++} ^ {n} \ to \ mathbb {R}}$ 