# Augmented Lagrangian method

Manopt.augmented_Lagrangian_methodFunction
augmented_Lagrangian_method(M, f, grad_f, p=rand(M); kwargs...)
augmented_Lagrangian_method(M, cmo::ConstrainedManifoldObjective, p=rand(M); kwargs...)

perform the augmented Lagrangian method (ALM) [LB19].

The aim of the ALM is to find the solution of the constrained optimisation task

\begin{aligned} \min_{p ∈\mathcal{M}} &f(p)\\ \text{subject to } &g_i(p)\leq 0 \quad \text{ for } i= 1, …, m,\\ \quad &h_j(p)=0 \quad \text{ for } j=1,…,n, \end{aligned}

where M is a Riemannian manifold, and $f$, $\{g_i\}_{i=1}^m$ and $\{h_j\}_{j=1}^p$ are twice continuously differentiable functions from M to ℝ. In every step $k$ of the algorithm, the AugmentedLagrangianCost $\mathcal{L}_{ρ^{(k-1)}}(p, μ^{(k-1)}, λ^{(k-1)})$ is minimized on $\mathcal{M}$, where $μ^{(k-1)} ∈ \mathbb R^n$ and $λ^{(k-1)} ∈ ℝ^m$ are the current iterates of the Lagrange multipliers and $ρ^{(k-1)}$ is the current penalty parameter.

The Lagrange multipliers are then updated by

$$$λ_j^{(k)} =\operatorname{clip}_{[λ_{\min},λ_{\max}]} (λ_j^{(k-1)} + ρ^{(k-1)} h_j(p^{(k)})) \text{for all} j=1,…,p,$$$

and

$$$μ_i^{(k)} =\operatorname{clip}_{[0,μ_{\max}]} (μ_i^{(k-1)} + ρ^{(k-1)} g_i(p^{(k)})) \text{ for all } i=1,…,m,$$$

where $λ_{\min} \leq λ_{\max}$ and $μ_{\max}$ are the multiplier boundaries.

Next, the accuracy tolerance $ϵ$ is updated as

$$$ϵ^{(k)}=\max\{ϵ_{\min}, θ_ϵ ϵ^{(k-1)}\},$$$

where $ϵ_{\min}$ is the lowest value $ϵ$ is allowed to become and $θ_ϵ ∈ (0,1)$ is constant scaling factor.

Last, the penalty parameter $ρ$ is updated as follows: with

$$$σ^{(k)}=\max_{j=1,…,p, i=1,…,m} \{\|h_j(p^{(k)})\|, \|\max_{i=1,…,m}\{g_i(p^{(k)}), -\frac{μ_i^{(k-1)}}{ρ^{(k-1)}} \}\| \}.$$$

ρ is updated as

$$$ρ^{(k)} = \begin{cases} ρ^{(k-1)}/θ_ρ, & \text{if } σ^{(k)}\leq θ_ρ σ^{(k-1)} ,\\ ρ^{(k-1)}, & \text{else,} \end{cases}$$$

where $θ_ρ ∈ (0,1)$ is a constant scaling factor.

Input

• M a manifold $\mathcal M$
• f a cost function $F:\mathcal M→ℝ$ to minimize
• grad_f the gradient of the cost function

Optional (if not called with the ConstrainedManifoldObjective cmo)

• g: (nothing) the inequality constraints
• h: (nothing) the equality constraints
• grad_g: (nothing) the gradient of the inequality constraints
• grad_h: (nothing) the gradient of the equality constraints

Note that one of the pairs (g, grad_g) or (h, grad_h) has to be provided. Otherwise the problem is not constrained and a better solver would be for example quasi_Newton.

Optional

• ϵ: (1e-3) the accuracy tolerance
• ϵ_min: (1e-6) the lower bound for the accuracy tolerance
• ϵ_exponent: (1/100) exponent of the ϵ update factor; also 1/number of iterations until maximal accuracy is needed to end algorithm naturally
• θ_ϵ: ((ϵ_min / ϵ)^(ϵ_exponent)) the scaling factor of the exactness
• μ: (ones(size(h(M,x),1))) the Lagrange multiplier with respect to the inequality constraints
• μ_max: (20.0) an upper bound for the Lagrange multiplier belonging to the inequality constraints
• λ: (ones(size(h(M,x),1))) the Lagrange multiplier with respect to the equality constraints
• λ_max: (20.0) an upper bound for the Lagrange multiplier belonging to the equality constraints
• λ_min: (- λ_max) a lower bound for the Lagrange multiplier belonging to the equality constraints
• τ: (0.8) factor for the improvement of the evaluation of the penalty parameter
• ρ: (1.0) the penalty parameter
• θ_ρ: (0.3) the scaling factor of the penalty parameter
• equality_constraints: (nothing) the number $n$ of equality constraints.
• gradient_range (nothing, equivalent to NestedPowerRepresentation specify how gradients are represented
• gradient_equality_range: (gradient_range) specify how the gradients of the equality constraints are represented
• gradient_inequality_range: (gradient_range) specify how the gradients of the inequality constraints are represented
• inequality_constraints: (nothing) the number $m$ of inequality constraints.
• sub_grad: (AugmentedLagrangianGrad(problem, ρ, μ, λ)) use augmented Lagrangian gradient, especially with the same numbers ρ,μ as in the options for the sub problem
• sub_kwargs: ((;)) keyword arguments to decorate the sub options, for example the debug= keyword.
• sub_stopping_criterion: (StopAfterIteration(200) |StopWhenGradientNormLess(ϵ) |StopWhenStepsizeLess(1e-8)) specify a stopping criterion for the subsolver.
• sub_problem: (DefaultManoptProblem(M,ConstrainedManifoldObjective(subcost, subgrad; evaluation=evaluation))) problem for the subsolver
• sub_state: (QuasiNewtonState) using QuasiNewtonLimitedMemoryDirectionUpdate with InverseBFGS and sub_stopping_criterion as a stopping criterion. See also sub_kwargs.
• stopping_criterion: (StopAfterIteration(300) | (StopWhenSmallerOrEqual(ϵ, ϵ_min) & StopWhenChangeLess(1e-10))) a functor inheriting from StoppingCriterion indicating when to stop.

For the ranges of the constraints' gradient, other power manifold tangent space representations, mainly the ArrayPowerRepresentation can be used if the gradients can be computed more efficiently in that representation.

With equality_constraints and inequality_constraints you have to provide the dimension of the ranges of h and g, respectively. If not provided, together with M and the start point p0, a call to either of these is performed to try to infer these.

Output

the obtained (approximate) minimizer $p^*$, see get_solver_return for details

source

## State

Manopt.AugmentedLagrangianMethodStateType
AugmentedLagrangianMethodState{P,T} <: AbstractManoptSolverState

Describes the augmented Lagrangian method, with

Fields

a default value is given in brackets if a parameter can be left out in initialization.

• p: a point on a manifold as starting point and current iterate
• sub_problem: an AbstractManoptProblem problem for the subsolver
• sub_state: an AbstractManoptSolverState for the subsolver
• ϵ: (1e–3) the accuracy tolerance
• ϵ_min: (1e-6) the lower bound for the accuracy tolerance
• λ: (ones(n)) the Lagrange multiplier with respect to the equality constraints
• λ_max: (20.0) an upper bound for the Lagrange multiplier belonging to the equality constraints
• λ_min: (- λ_max) a lower bound for the Lagrange multiplier belonging to the equality constraints
• μ: (ones(m)) the Lagrange multiplier with respect to the inequality constraints
• μ_max: (20.0) an upper bound for the Lagrange multiplier belonging to the inequality constraints
• ρ: (1.0) the penalty parameter
• τ: (0.8) factor for the improvement of the evaluation of the penalty parameter
• θ_ρ: (0.3) the scaling factor of the penalty parameter
• θ_ϵ: (((ϵ_min/ϵ)^(ϵ_exponent)) the scaling factor of the accuracy tolerance
• penalty: evaluation of the current penalty term, initialized to Inf.
• stopping_criterion: ((StopAfterIteration(300) | (StopWhenSmallerOrEqual(ϵ, ϵ_min) &StopWhenChangeLess(1e-10))) a functor inheriting from StoppingCriterion indicating when to stop.

Constructor

AugmentedLagrangianMethodState(M::AbstractManifold, co::ConstrainedManifoldObjective, p; kwargs...)

construct an augmented Lagrangian method options with the fields and defaults as stated before, where the manifold M and the ConstrainedManifoldObjective co can be helpful for manifold- or objective specific defaults.

augmented_Lagrangian_method

source

## Helping functions

Manopt.AugmentedLagrangianCostType
AugmentedLagrangianCost{CO,R,T}

Stores the parameters $ρ ∈ ℝ$, $μ ∈ ℝ^m$, $λ ∈ ℝ^n$ of the augmented Lagrangian associated to the ConstrainedManifoldObjective co.

This struct is also a functor (M,p) -> v that can be used as a cost function within a solver, based on the internal ConstrainedManifoldObjective it computes

$$$\mathcal L_\rho(p, μ, λ) = f(x) + \frac{ρ}{2} \biggl( \sum_{j=1}^n \Bigl( h_j(p) + \frac{λ_j}{ρ} \Bigr)^2 + \sum_{i=1}^m \max\Bigl\{ 0, \frac{μ_i}{ρ} + g_i(p) \Bigr\}^2 \Bigr)$$$

Fields

• co::CO, ρ::R, μ::T, λ::T as mentioned in the formula, where $R$ should be the

number type used and $T$ the vector type.

Constructor

AugmentedLagrangianCost(co, ρ, μ, λ)
source
Manopt.AugmentedLagrangianGradType
AugmentedLagrangianGrad{CO,R,T}

Stores the parameters $ρ ∈ ℝ$, $μ ∈ ℝ^m$, $λ ∈ ℝ^n$ of the augmented Lagrangian associated to the ConstrainedManifoldObjective co.

This struct is also a functor in both formats

• (M, p) -> X to compute the gradient in allocating fashion.
• (M, X, p) to compute the gradient in in-place fashion.

additionally this gradient does accept a positional last argument to specify the range for the internal gradient call of the constrained objective.

based on the internal ConstrainedManifoldObjective and computes the gradient $\operatorname{grad} \mathcal L_{ρ}(p, μ, λ)$, see also AugmentedLagrangianCost.

Fields

• co::CO, ρ::R, μ::T, λ::T as mentioned in the formula, where $R$ should be the

number type used and $T$ the vector type.

Constructor

AugmentedLagrangianGrad(co, ρ, μ, λ)
source

## Technical details

The augmented_Lagrangian_method solver requires the following functions of a manifold to be available

## Literature

[LB19]
C. Liu and N. Boumal. Simple algorithms for optimization on Riemannian manifolds with constraints. Applied Mathematics & Optimization (2019), arXiv:1091.10000.