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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 ...
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1] It is named after the mathematician Joseph-Louis ...
Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem; Lagrangian, a functional whose extrema are to be determined in the calculus of variations; Lagrangian submanifold, a class of submanifolds in symplectic geometry; Lagrangian system, a pair ...
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. [3]
where the objective function is the Lagrange dual function. Provided that the functions f {\displaystyle f} and g 1 , … , g m {\displaystyle g_{1},\ldots ,g_{m}} are convex and continuously differentiable, the infimum occurs where the gradient is equal to zero.
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.
For this non-convex minimization problem, Lemaréchal applied the theory of Lagrangian duality that was described in Lasdon's Optimization Theory for Large Systems. [ 3 ] [ 4 ] Because the primal problem was non-convex, there was no guarantee that a solution to the dual problem would provide useful information about the primal.