Signomial

A signomial is an algebraic <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of one or more independent variables. It is perhaps most easily thought of as an algebraic extension of multivariable <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>—an extension that permits exponents to be arbitrary real numbers (rather than just non-negative integers) while requiring the independent variables to be strictly positive (so that division by zero and other inappropriate algebraic operations are not encountered).
Formally, a signomial is a function with domain 
 
 
 
 
 
 R
 
 
 >
 0
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} _{>0}^{n}}
 
 which takes values

f
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        =
        
          ∑
          
            i
            =
            1
          
          
            M
          
        
        
          (
          
            
              c
              
                i
              
            
            
              ∏
              
                j
                =
                1
              
              
                n
              
            
            
              x
              
                j
              
              
                
                  a
                  
                    i
                    j
                  
                
              
            
          
          )
        
      
    
    {\displaystyle f(x_{1},x_{2},\dots ,x_{n})=\sum _{i=1}^{M}\left(c_{i}\prod _{j=1}^{n}x_{j}^{a_{ij}}\right)}

where the coefficients 
 
 
 
 
 c
 
 i
 
 
 
 
 {\displaystyle c_{i}}
 
 and the exponents 
 
 
 
 
 a
 
 i
 j
 
 
 
 
 {\displaystyle a_{ij}}
 
 are real numbers. Signomials are <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under addition, subtraction, multiplication, and scaling.
If we restrict all 
 
 
 
 
 c
 
 i
 
 
 
 
 {\displaystyle c_{i}}
 
 to be positive, then the function f is a <a href="/facts/Posynomial/lqdhJvSL">posynomial</a>. Consequently, each signomial is either a posynomial, the negative of a posynomial, or the difference of two posynomials. If, in addition, all exponents 
 
 
 
 
 a
 
 i
 j
 
 
 
 
 {\displaystyle a_{ij}}
 
 are non-negative integers, then the signomial becomes a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> whose domain is the positive <a href="/facts/Orthant/3bkJde4G">orthant</a>.
For example,

f
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        
          x
          
            3
          
        
        )
        =
        2.7
        
          x
          
            1
          
          
            2
          
        
        
          x
          
            2
          
          
            −
            1
            
              /
            
            3
          
        
        
          x
          
            3
          
          
            0.7
          
        
        −
        2
        
          x
          
            1
          
          
            −
            4
          
        
        
          x
          
            3
          
          
            2
            
              /
            
            5
          
        
      
    
    {\displaystyle f(x_{1},x_{2},x_{3})=2.7x_{1}^{2}x_{2}^{-1/3}x_{3}^{0.7}-2x_{1}^{-4}x_{3}^{2/5}}

is a signomial. 
The term "signomial" was introduced by Richard J. Duffin and Elmor L. Peterson in their seminal joint work on general algebraic optimization—published in the late 1960s and early 1970s. A recent introductory exposition involves <a href="/facts/Optimization_problem/QzwJfFEE">optimization problems</a>. <a href="/facts/Nonlinear_optimization/cSXGuX2m">Nonlinear optimization</a> problems with <a href="/facts/Constrained_optimization/tJuCmQ4b">constraints</a> and/or <a href="/facts/Objective_function/xv5ozuhl">objectives</a> defined by signomials are harder to solve than those defined by only posynomials, because (unlike posynomials) signomials cannot necessarily be made <a href="/facts/Convex_function/IbzG5SLF">convex</a> by applying a logarithmic change of variables. Nevertheless, signomial optimization problems often provide a much more accurate mathematical representation of real-world nonlinear optimization problems.

Signomial open-in-new

Signomial