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The Lagrangian dual problem is obtained by forming the Lagrangian of a minimization problem by using nonnegative Lagrange multipliers to add the constraints to the objective function, and then solving for the primal variable values that minimize the original objective function. This solution gives the primal variables as functions of the ...
Figure 2: A paraboloid constrained along two intersecting lines. Figure 3: Contour map of Figure 2. The method of Lagrange multipliers can be extended to solve problems with multiple constraints using a similar argument. Consider a paraboloid subject to two line constraints that intersect at a single point. As the only feasible solution, this ...
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective, but the augmented Lagrangian method adds yet another term designed to mimic a Lagrange multiplier.
Quadratic programming is particularly simple when Q is positive definite and there are only equality constraints; specifically, the solution process is linear. By using Lagrange multipliers and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality constrained problem
A Lagrangian relaxation algorithm thus proceeds to explore the range of feasible values while seeking to minimize the result returned by the inner problem. Each value returned by P {\displaystyle P} is a candidate upper bound to the problem, the smallest of which is kept as the best upper bound.
The strong duality theorem says that if one of the two problems has an optimal solution, so does the other one and that the bounds given by the weak duality theorem are tight, i.e.: max x c T x = min y b T y. The strong duality theorem is harder to prove; the proofs usually use the weak duality theorem as a sub-routine.
Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a global maximum or minimum over the domain of the choice variables and a global minimum (maximum) over the multipliers.
Under certain conditions (called "constraint qualification"), if a problem is polynomial-time solvable, then it has strong duality (in the sense of Lagrangian duality). It is an open question whether the opposite direction also holds, that is, if strong duality implies polynomial-time solvability.