# 9.2 Geometric (posynomial) Optimization¶

## 9.2.1 Problem Definition¶

A *geometric optimization* problem can be stated as follows

where it is assumed that

and if \(i \neq j\), then

Hence, \(A\) is a \(T \times n\) matrix and \(c\) is a vector of length \(T\). Given \(c_k>0\) then

is called a *monomial*. A sum of monomials i.e.

is called a *posynomial* posynomial In general, problem (1) is very hard to solve. However, the posynomial case where it is required that

is relatively easy. The reason is that using a simple variable transformation a convex optimization problem can be obtained. Indeed using the variable transformation

we obtain the problem

which is a convex optimization problem that can be solved using **MOSEK**. We will call

a *term* and hence the number of terms is \(T\).

As stated, problem (2) is non-separable. However, using

we obtain the separable problem

which is a separable convex optimization problem.

A warning about this approach is that the exponential function \(e^x\) is only numerically well-defined for values of \(x\) in a small interval around \(0\) since \(e^x\) grows very rapidly as \(x\) becomes larger. Therefore numerical problems may arise when solving the problem on this form.

Applications

A large number of practical applications, particularly in electrical circuit design, can be cast as a geometric optimization problem. We will not review these applications here but rather refer the reader to [BKVH04] and the references therein.

Further Information

More information about geometric optimization problems is located in [BSS93], [BP76], [BKVH04].

Modeling tricks

A lot of tricks that can be used for modeling posynomial optimization problems are described in [BKVH04]. Therefore, in this section we cover only one important case.

### 9.2.1.1 Equalities¶

In general, equalities are not allowed in (1), i.e.

is not allowed. However, a monomial equality is not a problem. Indeed consider the example

of a monomial equality. The equality is identical to

which in turn is identical to the two inequalities

Hence, it is possible to model a monomial equality using two inequalities.

## 9.2.2 Problematic Formulations¶

Certain formulations of geometric optimization problems may cause problems for the algorithms implemented in **MOSEK**. Basically there are two kinds of problems that may occur:

- The solution vector is finite, but an optimal objective value can only be a approximated.
- The optimal objective value is finite but implies that a variable in the solution is infinite.

### 9.2.2.1 Finite Unattainable Solution¶

The following problem illustrates an unattainable solution:

Clearly, the optimal objective value is \(0\) but because of the constraint the \(x,y>0\) constraint this value can never be attained: To see why this is a problem, remember that **MOSEK** substitutes \(x=e^{t_x}\) and \(y=e^{t_y}\) and solves the problem as

The optimal solution implies that \(t_x=-\infty\) or \(t_y=-\infty\), and thus it is unattainable.

Now, the issue should be clear: If a variable \(x\) appears only with nonnegative exponents, then fixing \(x=0\) will minimize all terms in which it appears — but such a solution cannot be attained.

### 9.2.2.2 Infinite Solution¶

A similar problem will occur if a finite optimal objective value requires a variable to be infinite. This can be illustrated by the following example:

which is a valid geometric programming problem. In this case the optimal objective is \(0\), but this requires \(x=\infty\), which is unattainable.

Again, this specific case will appear if a variable \(x\) appears only with negative exponents in the problem, implying that each term in which it appears can be minimized for \(x\rightarrow \infty\).

## 9.2.3 An Example¶

Consider the example

which is not a geometric optimization problem. However, using the obvious transformations we obtain the problem

which is a geometric optimization problem.

## 9.2.4 Solving the Example¶

The problem (3) can be defined and solved in the **MOSEK** toolbox as shown in Listing 9.6.

```
function go2()
c = [1 1 3 1 1 0.1 1/3]';
a = sparse([[-1 1 0];
[2 -0.5 0];
[0 0.5 -1];
[1 -1 -2];
[-1 1 2];
[-1 0 0];
[1 0 0]]);
map = [0 1 1 2 3 4 5]';
[res] = mskgpopt(c,a,map);
fprintf('\nPrimal optimal solution to original gp:');
fprintf(' %e',exp(res.sol.itr.xx));
fprintf('\n\n');
% Compute the optimal objective value and
% the constraint activities.
v = c.*exp(a*res.sol.itr.xx);
% Add appropriate terms together.
f = sparse(map+1,1:7,ones(size(map)))*v;
% First objective value. Then constraint values.
fprintf('Objective value: %e\n',log(f(1)));
fprintf('Constraint values:');
fprintf(' %e',log(f(2:end)));
fprintf('\n\n');
% Dual multipliers (should be negative)
fprintf('Dual variables (should be negative):');
fprintf(' %e',res.sol.itr.y);
fprintf('\n\n');
```

## 9.2.5 Exporting to a File¶

It’s possible to write a geometric optimization problem to a file with the command:

```
mskgpwri(c,a,map,filename)
```

This file format is compatible with the `mskenopt`

command line tool. See the **MOSEK** Tools User’s manual for details on `mskenopt`

. This file format can be useful for sending debug information to **MOSEK** or for testing. It’s also possible to read the above format with the command:

```
[c,a,map] = mskgpread(filename)
```