EE 364A (Convex Optimization): Chapter 4.5 - Geometric Programming

Convex Optimization

Geometric Programming

Posynomials

Author

Chao Ma

Published

January 18, 2026

Geometric Programming

Geometric programming (GP) is a class of problems built from monomials and posynomials over positive variables. With a logarithmic change of variables, a GP becomes a convex optimization problem, so it can be solved efficiently and globally.

Monomial functions

A monomial has the form \[ f(x)=c x_1^{a_1}x_2^{a_2}\dots x_n^{a_n},\quad \mathrm{dom}\,f=\mathbb{R}_{++}^n. \]

$c>0$.
- In algebraic monomials, $c$ can be any real number.
- In GP, $c$ must be positive so we can write $c=e^{\log c}$.
$a_i\in\mathbb{R}$.

Posynomials

A posynomial is a sum of monomials: \[ f(x)=\sum_{k=1}^K c_k x_1^{a_{1k}}x_2^{a_{2k}}\dots x_n^{a_{nk}},\quad \mathrm{dom}\,f=\mathbb{R}_{++}^n. \]

Standard GP form

A geometric program is \[ \begin{aligned} \text{minimize} &\quad f_0(x) \\ \text{subject to} &\quad f_i(x)\le 1,\quad i=1,\dots,m \\ &\quad h_i(x)=1,\quad i=1,\dots,p, \end{aligned} \] where - $f_0,f_i$ are posynomials, - $h_i$ are monomials, - the domain is $\mathbb{R}_{++}^n$.

GP in convex form

GP is not a standard convex program in $x$, but it becomes convex after a log change of variables.

For a monomial \[ f(x)=c x_1^{a_1}\dots x_n^{a_n}, \] define - $y_i=\log x_i$ so $x_i=e^{y_i}$, - $b=\log c$ so $c=e^b$. Then \[ f(x)=e^{a^\top y+b}. \]

For a posynomial \[ f(x)=\sum_{k=1}^K c_k x_1^{a_{1k}}\dots x_n^{a_{nk}}, \] we get \[ f(x)=\sum_{k=1}^K e^{a_k^\top y+b_k}. \]

So the GP becomes \[ \begin{aligned} \text{minimize} &\quad \sum_{k=1}^{K_0} e^{a_{0k}^\top y+b_{0k}} \\ \text{subject to} &\quad \sum_{k=1}^{K_i} e^{a_{ik}^\top y+b_{ik}} \le 1,\quad i=1,\dots,m \\ &\quad e^{g_i^\top y+h_i}=1,\quad i=1,\dots,p. \end{aligned} \] The monomial equality constraint is equivalent to the affine constraint \[ g_i^\top y+h_i=0. \]

It is also common to take logs of the posynomial constraints and objective (monotone transformation): \[ \tilde f_i(y)=\log\Big(\sum_{k=1}^{K_i} e^{a_{ik}^\top y+b_{ik}}\Big),\quad i=0,\dots,m. \] Then the convex form is \[ \begin{aligned} \text{minimize} &\quad \tilde f_0(y) \\ \text{subject to} &\quad \tilde f_i(y)\le 0,\quad i=1,\dots,m \\ &\quad g_i^\top y+h_i=0,\quad i=1,\dots,p. \end{aligned} \] Here $\tilde f_i$ are log-sum-exp functions (convex), and the equality constraints are affine, so the problem is convex.

Examples

Cantilever beam design

We design a cantilever beam with a vertical load $F$ applied at the free end. The beam is split into $N$ equal-length segments. Segment $i$ has a rectangular cross section with width $w_i$ and height $h_i$.

The goal is to minimize total volume while respecting geometry and mechanics: \[ \min \sum_{i=1}^N w_i h_i. \]

Subject to: - Geometric bounds: \[ w_{\min} \le w_i \le w_{\max},\quad h_{\min} \le h_i \le h_{\max}. \] - Aspect ratio constraints: \[ S_{\min} \le \frac{h_i}{w_i} \le S_{\max}. \] - Stress constraints: \[ \sigma_i = \frac{6 i F}{w_i h_i^2} \le \sigma_{\max}. \] - Deflection constraint: \[ y_1 \le y_{\max}. \]

The deflections $y_i$ and slopes $v_i$ follow beam-recursion relations with boundary conditions $v_{N+1}=y_{N+1}=0$.

Why this is a GP - Each term $w_i h_i$ is a monomial, so the volume is a posynomial. - Geometric bounds and aspect ratio constraints can be written as monomial inequalities. - Stress constraints take the monomial form $c\,w_i^{-1}h_i^{-2}\le\sigma_{\max}$. - The recursive formulas for $v_i$ and $y_i$ are built from sums and products of monomials/posynomials, so $v_i$ and $y_i$ are posynomials by induction. - Therefore the deflection constraint $y_1\le y_{\max}$ is a posynomial inequality.

Since all constraints and the objective are posynomials/monomials over positive variables, the problem is a valid GP and can be solved globally after the log transform.

--- title: "EE 364A (Convex Optimization): Chapter 4.5 - Geometric Programming" author: "Chao Ma" date: "2026-01-18" categories: [Convex Optimization, Geometric Programming, Posynomials] --- # Geometric Programming Geometric programming (GP) is a class of problems built from **monomials** and **posynomials** over positive variables. With a logarithmic change of variables, a GP becomes a convex optimization problem, so it can be solved efficiently and globally. ## Monomial functions A monomial has the form $$ f(x)=c x_1^{a_1}x_2^{a_2}\dots x_n^{a_n},\quad \mathrm{dom}\,f=\mathbb{R}_{++}^n. $$ - $c>0$. - In algebraic monomials, $c$ can be any real number. - In GP, $c$ must be positive so we can write $c=e^{\log c}$. - $a_i\in\mathbb{R}$. ## Posynomials A **posynomial** is a sum of monomials: $$ f(x)=\sum_{k=1}^K c_k x_1^{a_{1k}}x_2^{a_{2k}}\dots x_n^{a_{nk}},\quad \mathrm{dom}\,f=\mathbb{R}_{++}^n. $$ ## Standard GP form A geometric program is $$ \begin{aligned} \text{minimize} &\quad f_0(x) \\ \text{subject to} &\quad f_i(x)\le 1,\quad i=1,\dots,m \\ &\quad h_i(x)=1,\quad i=1,\dots,p, \end{aligned} $$ where - $f_0,f_i$ are posynomials, - $h_i$ are monomials, - the domain is $\mathbb{R}_{++}^n$. ## GP in convex form GP is not a standard convex program in $x$, but it becomes convex after a log change of variables. For a monomial $$ f(x)=c x_1^{a_1}\dots x_n^{a_n}, $$ define - $y_i=\log x_i$ so $x_i=e^{y_i}$, - $b=\log c$ so $c=e^b$. Then $$ f(x)=e^{a^\top y+b}. $$ For a posynomial $$ f(x)=\sum_{k=1}^K c_k x_1^{a_{1k}}\dots x_n^{a_{nk}}, $$ we get $$ f(x)=\sum_{k=1}^K e^{a_k^\top y+b_k}. $$ So the GP becomes $$ \begin{aligned} \text{minimize} &\quad \sum_{k=1}^{K_0} e^{a_{0k}^\top y+b_{0k}} \\ \text{subject to} &\quad \sum_{k=1}^{K_i} e^{a_{ik}^\top y+b_{ik}} \le 1,\quad i=1,\dots,m \\ &\quad e^{g_i^\top y+h_i}=1,\quad i=1,\dots,p. \end{aligned} $$ The monomial equality constraint is equivalent to the affine constraint $$ g_i^\top y+h_i=0. $$ It is also common to take logs of the posynomial constraints and objective (monotone transformation): $$ \tilde f_i(y)=\log\Big(\sum_{k=1}^{K_i} e^{a_{ik}^\top y+b_{ik}}\Big),\quad i=0,\dots,m. $$ Then the convex form is $$ \begin{aligned} \text{minimize} &\quad \tilde f_0(y) \\ \text{subject to} &\quad \tilde f_i(y)\le 0,\quad i=1,\dots,m \\ &\quad g_i^\top y+h_i=0,\quad i=1,\dots,p. \end{aligned} $$ Here $\tilde f_i$ are log-sum-exp functions (convex), and the equality constraints are affine, so the problem is convex. # Examples ## Cantilever beam design We design a **cantilever beam** with a vertical load $F$ applied at the free end. The beam is split into $N$ equal-length segments. Segment $i$ has a rectangular cross section with width $w_i$ and height $h_i$. The goal is to **minimize total volume** while respecting geometry and mechanics: $$ \min \sum_{i=1}^N w_i h_i. $$ Subject to: - **Geometric bounds**: $$ w_{\min} \le w_i \le w_{\max},\quad h_{\min} \le h_i \le h_{\max}. $$ - **Aspect ratio constraints**: $$ S_{\min} \le \frac{h_i}{w_i} \le S_{\max}. $$ - **Stress constraints**: $$ \sigma_i = \frac{6 i F}{w_i h_i^2} \le \sigma_{\max}. $$ - **Deflection constraint**: $$ y_1 \le y_{\max}. $$ The deflections $y_i$ and slopes $v_i$ follow beam-recursion relations with boundary conditions $v_{N+1}=y_{N+1}=0$. ![Cantilever beam setup](ee364a-chapter4-5-cantilever-1.png) **Why this is a GP** - Each term $w_i h_i$ is a monomial, so the volume is a posynomial. - Geometric bounds and aspect ratio constraints can be written as monomial inequalities. - Stress constraints take the monomial form $c\,w_i^{-1}h_i^{-2}\le\sigma_{\max}$. - The recursive formulas for $v_i$ and $y_i$ are built from sums and products of monomials/posynomials, so $v_i$ and $y_i$ are posynomials by induction. - Therefore the deflection constraint $y_1\le y_{\max}$ is a posynomial inequality. Since all constraints and the objective are posynomials/monomials over positive variables, the problem is a valid GP and can be solved globally after the log transform. ![GP formulation summary](ee364a-chapter4-5-cantilever-2.png)