Question

Why big m method is called penalty method?

Answer 1

Answer:

==>When used in the objective function, the Big M method sometimes refers to formulations of linear optimization problems in which violations of a constraint or set of constraints are associated with a large positive penalty constant, M. (hence the need for M to be "large enough.")

Step-by-step explanation:

==>The general technique is to add to the objective function a term that produces a high cost for violation of constraints. There are two types of methods: Interior penalty methods, also known as barrier methods.

==>In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists.

==>

Penalty function methods approximate a constrained problem by an unconstrained problem structured such that minimization favors satisfaction of the constraints.

The general technique is to add to the objective function a term that produces a high cost for violation of constraints. There are two types of methods:

Interior penalty methods, also known as barrier methods

Exterior penalty methods

General Form

minimize f(x)

subject to h(x)=0

g(x) <= 0

We will reformulate f(x) such that the constraints are part of the objective function. We also want to insure that the two essential conditions of Lagrange's method hold:

If an optimal solution exists that requires the constraint to be tight, then any perturbation away from constraint activity should be heavily penalized.

If an optimal solution dictates the constraint be loose at optimality, the problem should behave as though it were unconstrained by that constraint.

The reformulated objective function:

minimize:

where:

f(x) is the original objective function

s= +- 1

ti= scale factor for equality constraints

rk= monotonically increasing or decreasing inequality scale factor

penalty function involving each original inequality constraint

penalty function involving each original equality constraint

This is a generalized equation that represents both interior and exterior methods. The nature of s, r and in the general formulation depends on whether one wishes to begin with a feasible solution and iterate toward optimality, or start with an infeasible solution and proceed toward both optimality and feasibility.

The equality scale factor, ti, is the penalty for violation of an equality constraint. In the infinite barrier method, ti would be set to infinity, approximated numerically by some large number such as 1020. This method does not work well with inequality constraints.

Example

min f(x)= x2 - 10x

subject to g(x)= x-3 <=0

Penalty method:

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Answer 2

Answer:

The correct answer of this question is Big M technique is a type of linear optimization problem formulation in which breaches of a constraint

Step-by-step explanation:

Given -  Big m method is called penalty method .

To Find - Write why big m method is called penalty method.

The Big M technique is a type of linear optimization problem formulation in which breaches of a constraint or collection of constraints are accompanied with a big positive penalty constant, M.

Penalty techniques are a type of algorithm that is used to solve limited optimization issues. A penalty approach is a method that replaces a constrained optimization issue with a succession of unconstrained problems whose answers ideally converge to the original restricted problem's solution.

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