Linear Programming

The main objective of Linear Programming is to optimize (maximize or minimize) a linear function, known as the objective function, subject to a set of linear constraints.

In a two-variable linear programming problem, the feasible region is a polygon formed by the intersection of the half-planes defined by the linear constraints.

Linear Programming deals with linear constraints. Nonlinear equality constraints are not allowed in standard linear programming problems.

The fundamental theorem of Linear Programming states that if a linear programming problem has a feasible solution, then it also has an optimal solution, either a maximum or a minimum.

The graphical method is a visual approach used to solve linear programming problems with a small number of variables (typically two). It involves plotting the feasible region and identifying the optimal solution graphically.

The dual problem of a linear programming problem is another linear programming problem that has the same constraints as the original problem but a different objective function.

A valid simplex tableau is a table that represents the current state of a linear programming problem during the simplex method. It has all non-negative entries, except for the rightmost column, which may contain negative entries.

The slack variable is a non-negative variable introduced into a linear programming problem to convert inequality constraints into equality constraints, making it easier to solve the problem using the simplex method.

Dynamic programming is a technique used to solve optimization problems with a recursive structure. It is not a method specifically designed for solving linear programming problems.

The artificial variable is a non-negative variable introduced into a linear programming problem to create an initial feasible solution when the problem does not have an obvious feasible solution.

Primal form is not a valid type of simplex tableau. The standard form, canonical form, and reduced form are all valid types of simplex tableaux.

The reduced cost is used in the simplex method to determine the direction of movement from one basic feasible solution to another. It helps identify which variable should enter or leave the basis.

The feasible region being empty is not a valid termination criterion for the simplex method. The simplex method terminates when all reduced costs are non-negative, all variables have non-negative values, or the objective function is unbounded.

The big M method is a technique used in Linear Programming to create an initial feasible solution by introducing artificial variables and assigning them a very large positive cost (M). This helps to ensure that the artificial variables are driven out of the solution as the simplex method progresses.

Nonlinear programming problems are not valid types of linear programming problems. Linear programming deals with linear objective functions and linear constraints, while nonlinear programming problems involve nonlinear functions.