Online Help

The following command allows you to use the short form of the command names in the Optimization package.

The commands in the Optimization package allow you to specify the objective function and the constraints in several different ways. This worksheet deals with the matrix form of the objective function and constraints. The other forms are algebraic form and operator form.

A linear program can be stated in the form:

$\min \mathrm{Transpose}\left(c\right)x$

subject to $\mathrm{Ax}\le b$

where $c$ is in $R^n$, $A$ is in $R^{\mathrm{mxn}}$, and $b$ is in $R^m$

For a first example we have a simplex two dimensional linear programming problem of the form described above with:

The LPSolve command returns the optimal function values, as well as the point at which the optimal value occurs.

Alternatively, we could use the first two constraints and the nonnegative option.

The first element of the solution is the minimum value that the objective function obtains while satisfying the constraints. The second element indicates a point where the minimum is reached. This point is not necessarily unique.

We can also include equality constraint as the next example shows,
so our problem becomes:

$\min \mathrm{Transpose}\left(c\right)x$

subject to $\mathrm{Ax}\le b$

$\mathrm{Ex}\=f$

where $c$ is in $R^n$, $A$ is in $R^{\mathrm{mxn}}$, and $b$ is in $R^m$, $E$ is in $R^{\mathrm{mxn}}$, and $f$ is in $R^m$

This example demonstrates the maximize option.

A quadratic program can be written in the form:

min 1/2*Transpose(x).H.x+Transpose(x).c

subject to $\mathrm{Ax}\le b$

$\mathrm{Ex}\=f$

where
H in is $R^{\mathrm{nxn}}$, c is in $R^n$, A is in $R^{\mathrm{mxn}}$, and b is in $R^m$, E is in $R^{\mathrm{mxn}}$, and f is in $R^m$

The QPSolve command returns the optimal function values, as well as the point at which the optimal value occurs.

We can also use NLPSolve in matrix form where the gradient should be provided.

We can now use the matrix form of NLPSolve