Geometric programming

A geometric program (GP) is an <a href="/facts/Optimization_(mathematics)/oRn8Iv5I">optimization</a> problem of the form

minimize
                  
                
              
              
                
                  f
                  
                    0
                  
                
                (
                x
                )
              
            
            
              
                
                  
                    subject to
                  
                
              
              
                
                  f
                  
                    i
                  
                
                (
                x
                )
                ≤
                1
                ,
                
                i
                =
                1
                ,
                …
                ,
                m
              
            
            
              
              
                
                  g
                  
                    i
                  
                
                (
                x
                )
                =
                1
                ,
                
                i
                =
                1
                ,
                …
                ,
                p
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{array}{ll}{\mbox{minimize}}&f_{0}(x)\\{\mbox{subject to}}&f_{i}(x)\leq 1,\quad i=1,\ldots ,m\\&g_{i}(x)=1,\quad i=1,\ldots ,p,\end{array}}}

where 
 
 
 
 
 f
 
 0
 
 
 ,
 …
 ,
 
 f
 
 m
 
 
 
 
 {\displaystyle f_{0},\dots ,f_{m}}
 
 are <a href="/facts/Posynomials/lqdhJvSL">posynomials</a> and 
 
 
 
 
 g
 
 1
 
 
 ,
 …
 ,
 
 g
 
 p
 
 
 
 
 {\displaystyle g_{1},\dots ,g_{p}}
 
 are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from 
 
 
 
 
 
 R
 
 
 +
 +
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} _{++}^{n}}
 
 to 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 defined as

x
        ↦
        c
        
          x
          
            1
          
          
            
              a
              
                1
              
            
          
        
        
          x
          
            2
          
          
            
              a
              
                2
              
            
          
        
        ⋯
        
          x
          
            n
          
          
            
              a
              
                n
              
            
          
        
      
    
    {\displaystyle x\mapsto cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}}

where 
 
 
 
 c
 >
 0
  
 
 
 {\displaystyle c>0\ }
 
 and 
 
 
 
 
 a
 
 i
 
 
 ∈
 
 R
 
 
 
 {\displaystyle a_{i}\in \mathbb {R} }
 
. A posynomial is any sum of monomials.
Geometric programming is
closely related to <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a>: any GP can be made convex by means of a change of variables. GPs have numerous applications, including component sizing in <a href="/facts/Integrated_circuit/fjn6vg7C">IC</a> design, aircraft design, <a href="/facts/Maximum_likelihood_estimation/0Yq2dpQD">maximum likelihood estimation</a> for <a href="/facts/Logistic_regression/mGj6mc8y">logistic regression</a> in <a href="/facts/Statistics/DNPsKGYU">statistics</a>, and parameter tuning of positive <a href="/facts/Linear_dynamical_system/tYnoVnKs">linear systems</a> in <a href="/facts/Control_theory/aIuERpn6">control theory</a>.

Geometric programming open-in-new

Geometric programming