A linear program may have both equality and inequality constraints. It is easy to rewrite equality constraints by inequalities. Can we rewrite inequality constraints by equalities? In fact, the standard form of linear programming only allows equality constraints and a special type of inequalities.

Now the question is how to convert a linear program into the standard form?
Clearly the third requirement is trivial. We only need to consider the first two requirement.
First, for the second requirement, consider a simple example . We can add a slack variable : together with . Note that it is equivalent to the original inequality. Suppose we have another example . Then multiply the inequality by and add a slack variable .
Next we consider the first requirement. If a variable has non-positive constraints, such as , we can easily let and substitute for in the linear program. If some variable, for example , is unconstrained in sign, we can replace it by and require and to be non-negative.

The standard form is useful in algorithm design and analysis, especially in the algorithms based on primal dual method.

We now use the form Note that the feasible set is a polyhedron since it is the intersection of some halfspaces.

When a linear program achieve its optimal value , intuitively the hyperplane passes through a vertex of the feasible set. We may use this observation to solve linear programs geometrically.

We now introduce a (ususally) efficient algorithm: the simplex method. We remark here that the simplex method is not a polynomial-time algorithm. However, it runs fast except for some artificially designed cases.

The key idea is that when we find a vertex of the feasible set, move from the current vertex to a "better" neighbor, where two vertices are neighbors if they share tight constraints.
Assume is a feasible set. Then we can start from the origin point . It is clear that there are neighbors of the origin, and each of them has zero coordiantes.
How can we know whether a neighbor is "better"? Note that our goal is to compute . If , the objective function is "better" as long as . To this end, we can choose such that and increase to until some constraint is tight. Now there are constraints tight: Then we shift coordinates so that becomes the origin point. It suffices to let , , .

As shown above, the first step is to choose a variable whose coefficient is negative, and then increase it. What should we do if we have multiple variables who have negative coefficients?

In fact, the following example shows we may encounter some tricky problems if we choose a wrong variable. Consider the linear program with the same objective function as above, but the constraints are

• Suppose in the first step, we choose to increase . Then the tight constraint should be .
• Let and . The new constraints are
• Unfortunately, we choose to increase and repeat the process again and again……

Clearly it is possible to fail in this case, which is called degeneracy. One way to break cycles is to add perturbation in , that is, let where is a Gaussian random variable.

Now the question is, what if the origin point is not feasible? If there exists a known feasible solution such that , then let and further let to guarantee that all variables are nonnegative.

However, what if there is no known feasible solutions? The following two-phase simple method gives an algorithm to find a feasible solution of a linear program.

• First, convert the linear program into the standard form: subject to and , where .
• Next, add slack variables for constraints. Then the constraints are , and .
• Now it is clear that there exists a trivial solution and .
• Finally, we can check if is possible by solving the linear program