Lagrange Multipliers Optimization Algorithm

A Lagrange Multipliers Optimization Algorithm is a constrained mathematical optimization that uses a continuous first partial derivatives of the optimization function and the constraint function.

• Context:
  • It can be implemented by a Lagrange Multipliers Optimization System (that solves a Lagrange multipliers optimization task).
• Example(s):
  • an Augmented Lagrangian Method, such as ADDM.
  • …
• Counter-Example(s):
  • Interior Point Method.
• See: Mathematical Optimization, Constrained Optimization Problem, Partial Derivative, Stationary Point.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Lagrange_multiplier Retrieved:2015-11-8.
  • In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange ) is a strategy for finding the local maxima and minima of a function subject to equality constraints. For instance (see Figure 1), consider the optimization problem :maximize f(x, y) :subject to g(x, y) 0. We need both and to have continuous first partial derivatives. We introduce a new variable () called a Lagrange multiplier and study the Lagrange function (or Lagrangian) defined by : [math]\displaystyle{ \mathcal{L}(x,y,\lambda) = f(x,y) + \lambda \cdot g(x,y), }[/math] where the term may be either added or subtracted. If f(x₀, y₀) is a maximum of f(x, y) for the original constrained problem, then there exists λ₀ such that (x₀, y₀, λ₀) is a stationary point for the Lagrange function (stationary points are those points where the partial derivatives of [math]\displaystyle{ \mathcal{L} }[/math] are zero). However, not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist.

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