The Linear Programming Solver

Overview: LP Solver

The OPTMODEL procedure provides a framework for specifying and solving linear programs (LPs). A standard linear program has the following formulation:

$\text{[math]}$

where

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of decision variables
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the matrix of constraints
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of objective function coefficients
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of constraints right-hand sides (RHS)
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of lower bounds on variables
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of upper bounds on variables

The following LP solvers are available in the OPTMODEL procedure:

primal simplex solver
dual simplex solver
network simplex solver
interior point solver

The primal and dual simplex solvers implement the two-phase simplex method. In phase I, the solver tries to find a feasible solution. If no feasible solution is found, the LP is infeasible; otherwise, the solver enters phase II to solve the original LP. The network simplex solver extracts a network substructure, solves this using network simplex, and then constructs an advanced basis to feed to either primal or dual simplex. The interior point solver implements a primal-dual predictor-corrector interior point algorithm. If any of the decision variables are constrained to be integer-valued, then the relaxed version of the problem is solved.