In the field of mathematical optimization, Lagrangian relaxation is a relaxation method which approximates a difficult problem of constrained optimization by a simpler problem. A solution to the relaxed problem is an approximate solution to the original problem, and provides useful information. The method penalizes violations of inequality constraints using a Lagrange multiplier, which imposes a cost on violations. These added costs are used instead of the strict inequality constraints in the optimization. In practice, this relaxed problem can often be solved more easily than the original problem. The problem of maximizing the Lagrangian function of the dual variables (the Lagrangian multipliers) is the Lagrangian dual problem. Mathematical description Suppose we are given a linear programming problem, with and , of the following form: {| border="0" cellpadding="1" cellspacing="0" |- |max | | |- |s.t. |- | | | |} If we split the constraints in such that , and we may write the system: {| border="0" cellpadding="1" cellspacing="0" |- |max | | |- |s.t. |- |(1) | | |- |(2) | | |} We may introduce the constraint (2) into the objective: {| border="0" cellpadding="1" cellspacing="0" |- |max | | |- |s.t. |- |(1) | | |} If we let be nonnegative weights, we get penalized if we violate the constraint (2), and we are also rewarded if we satisfy the constraint strictly. The above system is called the Lagrangian relaxation of our original problem. The LR solution as a bound Of particular use is the property that for any fixed set of values, the optimal result to the Lagrangian relaxation problem will be no smaller than the optimal result to the original problem. To see this, let be the optimal solution to the original problem, and let be the optimal solution to the Lagrangian relaxation. We can then see that {| border="0" cellpadding="1" cellspacing="0" |- | |} The first inequality is true because is feasible in the original problem and the second inequality is true because is the optimal solution to the Lagrangian relaxation. Iterating towards a solution of the original problem The above inequality tells us that if we minimize the maximum value we obtain from the relaxed problem, we obtain a tighter limit on the objective value of our original problem. Thus we can address the original problem by instead exploring the partially dualized problem {| border="0" cellpadding="1" cellspacing="0" |- |min | | | |s.t. | |