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 ; the difference is that the augmented Lagrangian method adds yet another term, 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 and s as a good alternative to penalty methods. It was first discussed by Magnus Hestenes in  and by Powell in 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 by Dimitri Bertsekas , notably in his book,  together with extensions involving nonquadratic regularization functions, such as entropic regularization , which gives rise to the "exponential method of multipliers," a method that handles inequality constraints with a twice differentiable augmented Lagrangian function. Since the s, 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 method is still useful for some problems.
Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization
In particular, a variant of the standard augmented Lagrangian method that uses partial updates similar to the Gauss-Seidel method for solving linear equations known as the alternating direction method of multipliers or ADMM gained some attention. This problem can be solved as a series of unconstrained minimization problems.
For reference, we first list the k th step of the penalty method approach:. The method can be extended to handle inequality constraints. For a discussion of practical improvements, see. 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. 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. Stochastic optimization considers the problem of minimizing a loss function with access to noisy samples of the gradient of the function.
Practical Augmented Lagrangian Methods for Constrained Optimization
The goal is to have an estimate of the optimal parameter minimizer per new sample. ADMM is originally a batch method. However, with some modifications it can also be used for stochastic optimization. Since in stochastic setting we only have access to noisy samples of gradient, we use an inexact approximation of the Lagrangian as.
The alternating direction method of multipliers ADMM is a popular method for online and distributed optimization on a large scale,  and is employed in many applications, e. Regularized optimization problems are especially relevant in the high dimensional regime since regularization is a natural mechanism to overcome ill-posedness and to encourage parsimony in the optimal solution, e.
Due to the efficiency of ADMM in solving regularized problems, it has a good potential for stochastic optimization in high dimensions. However, conventional stochastic ADMM methods suffer from curse of dimensionality. Their convergence rate is proportional to square of the dimension and in practice they scale poorly. Recently, a general framework has been proposed for stochastic optimization in high-dimensions that solves this bottleneck by adding simple and cheap modifications to ADMM. The modifications are in terms of added projection which goes a long way and results in logarithmic dimension dependency.
The specific cases of sparse optimization framework and noisy decomposition framework are discussed further. Most approaches dene a sequence of such penalty functions, in which the penalty terms for the constraint violations are multiplied by a positive coefcient. By making this coefcient larger, we penalize constraint violations more severely, thereby forcing the minimizer of the penalty function closer to the feasible region for the constrained problem.
The simplest penalty function of this type is the quadratic penalty function, in which the penalty terms are the squares of the constraint violations.
Augmented Lagrangian method
Because the penalty terms in In searching for xk , we can use the minimizers xk1 , xk2 , etc. In Figure There is also a local maximizer near x 0. The minimizer in this gure is much closer to the solution 1, 1 T of the problem The situation is not always so benign as in Example For a given value of the penalty parameter , the penalty function may be unbounded below even if the original constrained problem has a unique solution. For such values of , the iterates generated by an unconstrained minimization method would usually diverge. This deciency is, unfortunately, common to all the penalty functions discussed in this chapter.
In this case, Q may be less smooth than the objective and constraint functions. Framework A practical implementation must include safeguards that increase the penalty parameter and possibly restore the initial point when the constraint violation is not decreasing rapidly enough, or when the iterates appear to be diverging. When only equality constraints are present, Q x; k is smooth, so the algorithms for unconstrained minimization described in the rst chapters of the book can be used to identify the approximate solution xk. However, the minimization of Q x; k becomes more difcult to perform as k becomes large, unless we use special techniques to calculate the search 2 directions.
For one thing, the Hessian x x Q x; k becomes arbitrarily ill conditioned near the minimizer. This property alone is enough to make many unconstrained minimization algorithms such as quasi-Newton and conjugate gradient perform poorly. Newton's method, on the other hand, is not sensitive to ill conditioning of the Hessian, but it, too, may encounter 2 difculties for large k for two other reasons. First, ill conditioning of x x Q x; k might be expected to cause numerical problems when we solve the linear equations to calculate the Newton step. Second, even when x is close to the minimizer of Q ; k , the quadratic Taylor series approximation to Q x; k about x is a reasonable approximation of the true function only in a small neighborhood of x.
This property can be seen in Figure Since Newton's method is based on the quadratic model, the steps that it generates may not make rapid progress toward the minimizer of Q x; k. We restrict our attention to the equality-constrained problem For the rst result we assume that the penalty function Q x; k has a nite minimizer for each value of k. Theorem Suppose that each xk is the exact global minimizer of Q x; k dened by Let x be a global solution of Since this result requires us to nd the global minimizer for each subproblem, this desirable property of convergence to the global solution of In contrast to Theorem This observation is important for the analysis of augmented Lagrangian methods in Section Suppose that the tolerances and penalty parameters in Framework On the other hand, if a limit point x is feasible and the constraint gradients ci x are linearly independent, then x is a KKT point for the problem By differentiating Q x; k in If, on the other hand, the constraint gradients ci x are linearly independent at a limit point x , we have from Hence, the second KKT condition We need to check the rst KKT condition By multiplying Hence, x is a KKT point for It is reassuring that, if a limit point x is not feasible, it is at least a stationary point for the function c x 2.
Newton-type algorithms can always be attracted to infeasible points of this type. We see the same effect in Chapter 11, in our discussion of methods for nonlinear equations that use the sum-of-squares merit function r x 2. Such methods cannot be guaranteed to nd a root, and can be attracted to a stationary point or minimizer of the merit function. In the case in which the nonlinear program An understanding of the properties of this matrix, and the similar Hessians that arise in other penalty and barrier methods, is essential in choosing effective algorithms for the minimization problem and for the linear algebra calculations at each iteration.
When x is close to the minimizer of Q ; k and the conditions of Theorem We see from this expression that x x Q x; k is approximately equal to the sum of - a matrix whose elements are independent of k the Lagrangian term , and - a matrix of rank E whose nonzero eigenvalues are of order k the second term on the right-hand side of The number of constraints E is usually smaller than n. In this case, the last term in One consequence of the ill conditioning is possible inaccuracy in the calculation of the Newton step for Q x; k , which is obtained by solving the following system: 2 x x Q x; k p x Q x; k.
For the same reason, iterative methods can be expected to perform poorly unless accompanied by a preconditioning strategy that removes the systematic ill conditioning. There is an alternative formulation of the equations We note, however, that neither system may yield a good search direction p because the coefcients k ci x in the summation term of the upper left block of This fact may cause the quadratic model on which p is based to be an inadequate model of Q ; k , so the Newton step may be intrinsically an unsuitable search direction.
We discussed possible remedies for this difculty above, in our comments following Framework To compute the step via A similar system must be solved to calculate the sequential quadratic programming SQP step In fact, when k is large, On the other hand, when k is small, This situation is undesirable because the steps may not make signicant progress toward the feasible region, resulting in inefcient global behavior. This property is desirable because it makes the performance of penalty methods less dependent on the strategy for updating the penalty parameter.
Cites per any
The quadratic penalty function of Section In this section we discuss nonsmooth exact penalty functions, which have proved to be useful in a number of practical contexts. A popular nonsmooth penalty function for the general nonlinear programming problem Its name derives from the fact that the penalty term is times the 1 norm of the constraint violation.
The following result establishes the exactness of the 1 penalty function. For a proof see [, Theorem 4. Suppose that x is a strict local solution of the nonlinear programming problem Denition Penalty function for problem If a point is infeasible for For the function in Example The following result complements Theorem Then, if x is feasible for the nonlinear program Is x is not feasible for Suppose rst that x is feasible. We have from A. We leave verication of Consider any direction p in the linearized feasible direction set F x of Denition We can now apply Farkas' Lemma Lemma As we noted earlier see Theorem We leave the second part of the proof concerning infeasible x as an exercise.
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