CS 600 Algorithms and Data Structures

Notes on Linear Programming

Standard Form of Linear Programming Problem

Maximize $\Sigma_{j=1}^{n} c_{j} x_{j}$
Subject to $\Sigma_{j=1}^{n} a_{ij} x_{j} \leq b_{i}$ (i = 1,2,...,m)
                $x_{j} \geq 0$ (j = 1,2, ..., n).

is called the standard form of a Linear Programming (LP) problem.
The terminology is not uniform. Some people call this form canonical or symmetric form.

The function to be maximized, $\Sigma_{j=1}^{n} c_{j} x_{j}$, is called the objective function, and the inequalities are called the constraints. The constraints $x_{j} \geq 0$ (j = 1,2, ..., n) are called the nonnegativity constraints.

Values for x₁, x₂, ... x_n that satisfy all the constraints of an LP problem are said to constitute a feasible solution of that problem. A solution that is not feasible is called an infeasible solution. A feasible solution that maximizes the objective function is called an optimal solution. The corresponding value of the objective function is called the optimal value of the problem. The set of all feasible solutions is called the feasible region of the LP problem. A feasible solution is a basic feasible solution if it does not lie on any line segment connecting two other feasible solutions. Thus a basic feasible solution is on the boundary of the feasible region, in particular it is a corner point of the feasible region.

An LP problem with no feasible solutions is called infeasible; an LP problem that has feasible solutions but no optimal solutions is called unbounded.

Example: For the LP problem

Maximize   3x₁ + 5x₂
Subject to $x_{1} \leq 4$
                $2x_{2} \leq 12$
                $3x_{1} + 2x_{2} \leq 18$
                $x_{j} \geq 0$ (j = 1,2),

x₁ = 2, x₂ = 3 is a feasible solution, x₁ = 2, x₂ = 6 is an optimal solution and the optimal value is 36. x₁ = 4, x₂ = 5 is an infeasible solution.

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