The Augmented Lagrangian method is a numerical optimization technique used to solve constrained optimization problems. It is particularly useful when dealing with difficulties encountered in traditional methods, such as penalty methods or Lagrange multipliers, especially in cases of non-smooth or non-convex constraints. ### Concept: The Augmented Lagrangian method combines the ideas of Lagrange multipliers and penalty methods to tackle constrained optimization problems.
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For smooth constraints , minimizeapproximately in , then update the nonnegative Lagrange multipliers by the displayed projected rule. Equality constraints instead contribute and use the unprojected update . The construction combines Lagrangian duality with a quadratic penalty method.
For an inequality, the exact identity is . Its unshifted negative part satisfies . On a violated constraint, accurate new multipliers therefore give violation of the order of times the multiplier error. This can achieve high feasibility accuracy without a vanishingly small , reducing condition number difficulties. The negative-part qualification is essential if an iterate approaches a binding constraint from the feasible side.