Augmented Lagrangian method

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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; the difference is that the augmented Lagrangian method adds an additional term to the unconstrained objective. This additional term is designed to mimic a Lagrange multiplier. The augmented Lagrangian is not the same as the method of Lagrange multipliers.

Viewed differently, the unconstrained objective is the Lagrangian of the constrained problem, with an additional penalty term (the augmentation).

The method was originally known as the method of multipliers, and was studied much in the 1970 and 1980s as a good alternative to penalty methods. It was first discussed by Magnus Hestenes in 1969[1] and by Powell in 1969[2] The method was studied by R. Tyrrell Rockafellar in relation to Fenchel duality, particularly in relation to proximal-point methods, Moreau–Yosida regularization, and maximal monotone operators: These methods were used in structural optimization. The method was also studied and implemented by Dimitri Bertsekas, notably in his 1982 book,[3] and with respect to entropic regularization (which accelerate the rate of convergence for his "exponential method of multipliers").

Since the 1970s, sequential quadratic programming (SQP) and interior point methods (IPM) have had increasing attention, in part because they more easily use sparse matrix subroutines from numerical software libraries, and in part because IPMs have proven complexity results via the theory of self-concordant functions. The augmented Lagrangian method was rejuvenated by the optimization systems LANCELOT and AMPL, which allowed sparse matrix techniques to be used on seemingly dense but "partially separable" problems. The method is still useful for some problems.[4] As of around 2007, there has been a resurgence of Augmented Lagrangian methods (and ADMM in particular) in fields such as total-variation denoising and compressed sensing; for example, the SALSA package was proposed in 2009.

A variant of the standard Augmented Lagrangian method that uses partial updates (similar to the Gauss-Seidel method for solving linear equations) is known as the alternating direction method of multipliers or ADMM.

General method[edit]

Let us say we are solving the following constrained problem:

 \min f(\bold x)

subject to

 c_i(\bold x) = 0 ~\forall  i \in I.

This problem can be solved as a series of unconstrained minimization problems. For reference, we first list the penalty method approach:

 \min \Phi_k (\bold x) = f (\bold x) + \mu_k ~ \sum_{i\in I} ~ c_i(\bold x)^2

The penalty method solves this problem, then at the next iteration it re-solves the problem using a larger value of \mu_k (and using the old solution as the initial guess or "warm-start").

The augmented Lagrangian method uses the following unconstrained objective:

 \min \Phi_k (\bold x) = f (\bold x) + \frac{\mu_k}{2} ~ \sum_{i\in I} ~ c_i(\bold x)^2  - \sum_{i\in I} ~ \lambda_i c_i(\bold x)

and after each iteration, in addition to updating \mu_k, the variable \lambda is also updated according to the rule

\lambda_{i+1} \leftarrow \lambda_i - \mu_k c_i(\bold{x}_k)

where \bold{x}_k is the solution to the unconstrained problem at the kth step, i.e. \bold{x}_k=\text{argmin} \Phi_k(\bold x)

The variable \lambda is an estimate of the Lagrange multiplier, and the accuracy of this estimate improves at every step. The major advantage of the method is that unlike the penalty method, it is not necessary to take \mu \rightarrow \infty in order to solve the original constrained problem. Instead, because of the presence of the Lagrange multiplier term, \mu can stay much smaller.

The method can be extended to handle inequality constraints. For a discussion of practical improvements, see.[5]

Comparison with penalty methods[edit]

From,[6] it is suggested that the augmented Lagrangian method is generally preferred to the quadratic penalty method since there is little extra computational cost and the parameter \mu need not go to infinity, thus avoiding ill-conditioning.

Alternating direction method of multipliers[edit]

The alternating direction method of multipliers (ADMM) is a variant of the augmented Lagrangian scheme that uses partial updates for the dual variables. This method is often applied to solve problems such as

 \min_x f(x) + g(x).

This is equivalent to the constrained problem

 \min_{x,y} f(x) + g(y), \quad \text{subject to}\quad  x = y.

Though this change may seem trivial, the problem can now be attacked using methods of constrained optimization (in particular, the augmented Lagrangian method), and the objective function is separable in x and y. The dual update requires solving a proximity function in x and y at the same time; the ADMM technique allows this problem to be solved approximately by first solving for x with y fixed, and then solving for y with x fixed. Rather than iterate until convergence (like the Jacobi method), the algorithm proceeds directly to updating the dual variable and then repeating the process. This is not equivalent to the exact minimization, but surprisingly, it can still be shown that this method converges to the right answer (under some assumptions). Because of this approximation, the algorithm is distinct from the pure augmented Lagrangian method.

The ADMM can be viewed as an application of the Douglas-Rachford splitting algorithm, and the Douglas-Rachford algorithm is in turn an instance of the Proximal point algorithm; details can be found here.[7] There are several modern software packages that solve Basis pursuit and variants and use the ADMM; such packages include YALL1 (2009), SpaRSA (2009) and SALSA (2009).

Software[edit]

Some well-known software packages that use the augmented Lagrangian method are LANCELOT and PENNON. The software MINOS also uses an augmented Lagrangian method for some types of problems.

See also[edit]

References[edit]

  1. ^ M.R. Hestenes, "Multiplier and gradient methods", Journal of Optimization Theory and Applications, 4, 1969, pp. 303–320
  2. ^ M.J.D. Powell, "A method for nonlinear constraints in minimization problems", in Optimization ed. by R. Fletcher, Academic Press, New York, NY, 1969, pp. 283–298.
  3. ^ Dimitri P. Bertsekas, Constrained optimization and Lagrange multiplier methods, Athena Scientific, 1996 (first published 1982)
  4. ^ Nocedal & Wright (2006), chapter 17
  5. ^ Nocedal & Wright (2006), chapter 17
  6. ^ Nocedal & Wright (2006), chapter 17
  7. ^ Eckstein, J.; Bertsekas, D. P. (1992). "On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators". Mathematical Programming 55: 293. doi:10.1007/BF01581204. 

Bibliography[edit]