Geometric programming - Alchetron, The Free Social Encyclopedia

A geometric program (GP) is an optimization problem of the form

Minimize f 0 ( x ) subject to f i ( x ) ≤ 1 , i = 1 , … , m h i ( x ) = 1 , i = 1 , … , p where f 0 , … , f m are posynomials and h 1 , … , h p are monomials.

In the context of geometric programming (unlike all other disciplines), a monomial is defined as a function h : R + + n → R defined as

h ( x ) = c x 1 a 1 x 2 a 2 ⋯ x n a n

where c > 0 and a i ∈ R .

GPs have numerous application, such as components sizing in IC design and parameter estimation via logistic regression in statistics. The maximum likelihood estimator in logistic regression is a GP.

Convex form

Geometric programs are not (in general) convex optimization problems, but they can be transformed to convex problems by a change of variables and a transformation of the objective and constraint functions. In particular, defining y i = log ⁡ ( x i ) , the monomial f ( x ) = c x 1 a 1 ⋯ x n a n ↦ e a T y + b , where b = log ⁡ ( c ) . Similarly, if f is the posynomial

f ( x ) = ∑ k = 1 K c k x 1 a 1 k ⋯ x n a n k

then f ( x ) = ∑ k = 1 K e a k T y + b k , where a k = ( a 1 k , … , a n k ) and b k = log ⁡ ( c k ) . After the change of variables, a posynomial becomes a sum of exponentials of affine functions.

References

Geometric programming Wikipedia

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