The expression f is the linear objective function to be maximized subject to the linear constraints C. The function maximize returns either a set of equations describing the optimal solution to the specified linear program, or the empty set in the case where no feasible solution to C exists, or NULL in the case where the solution is unbounded.

The equations returned by maximize can be substituted back into the objective function f to obtain the value of the objective function at the optimal solution.

A third parameter may be used to specify that all variables are constrained to be NONNEGATIVE; such constraints may also be listed explicitly. Similarly, UNRESTRICTED indicates that no sign constraint is to be placed on the variables.

A fourth and a fifth parameter may be included to specify names for returning the optimal description, and any variable transformations used to set up the problem.

The Optimization[LPSolve] command also computes solutions to linear programs.  It is generally more efficient than the simplex[minimize] command, but performs all its computations using floating-point values.

The command with(simplex,maximize) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{simplex}\right)\:$

$\mathrm{maximize}\left(x+y\,\left\{3x+4y\le 4\,4x+3y\le 5\right\}\right)$

$\left\{x=\frac{8}{7}\,y=\frac{1}{7}\right\}$

$\mathrm{maximize}\left(x-y\,\left\{3x+4y\le 4\,4x+3y\le -3\right\}\right)$

$\mathrm{maximize}\left(x-y\,\left\{3x+4y\le 4\,4x+3y\le -3\right\}\,\mathrm{NONNEGATIVE}\right)$

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