Geometric Programming 101¶
What is a GP?¶
A Geometric Program (GP) is a type of non-linear optimization problem whose objective and constraints have a particular form.
The decision variables must be strictly positive (non-zero, non-negative) quantities. This is a good fit for engineering design equations (which are often constructed to have only positive quantities), but any model with variables of unknown sign (such as forces and velocities without a predefined direction) may be difficult to express in a GP. Such models might be better expressed as Signomials.
More precisely, GP objectives and inequalities are formed out of monomials and posynomials. In the context of GP, a monomial is defined as:
where \(c\) is a positive constant, \(x_{1..n}\) are decision variables, and \(a_{1..n}\) are real exponents. For example, taking \(x\), \(y\) and \(z\) to be positive variables, the expressions
are all monomials. Building on this, a posynomial is defined as a sum of monomials:
For example, the expressions
are all posynomials. Alternatively, monomials can be defined as the subset of posynomials having only one term. Using \(f_i\) to represent a monomial and \(g_i\) to represent a posynomial, a GP in standard form is written as:
Boyd et. al. give the following example of a GP in standard form:
Why are GPs special?¶
Geometric programs have several powerful properties:
- Unlike most non-linear optimization problems, large GPs can be solved extremely quickly.
- If there exists an optimal solution to a GP, it is guaranteed to be globally optimal.
- Modern GP solvers require no initial guesses or tuning of solver parameters.
These properties arise because GPs become convex optimization problems via a logarithmic transformation. In addition to their mathematical benefits, recent research has shown that many practical problems can be formulated as GPs or closely approximated as GPs.
What are Signomials / Signomial Programs?¶
When the coefficients in a posynomial are allowed to be negative (but the variables stay strictly positive), that is called a Signomial.
A Signomial Program has signomial constraints. While they cannot be solved as quickly or to global optima, because they build on the structure of a GP they can often be solved more quickly than a generic nonlinear program. More information can be found under Signomial Programming.
Where can I learn more?¶
To learn more about GPs, refer to the following resources:
- A tutorial on geometric programming, by S. Boyd, S.J. Kim, L. Vandenberghe, and A. Hassibi.
- Convex optimization, by S. Boyd and L. Vandenberghe.
- Geometric Programming for Aircraft Design Optimization, Hoburg, Abbeel 2014