Options

ALADIN-$\alpha$ comes with several ALADIN variants. These variants can be activated by setting options. All options with possible values can be found in loadDefOpts.m. Here, we give a little more detailled description of these options.

Basic options

ALADIN option	default values	alternative
rho0	1e2	double > 0
rhoUpdate	1.1	double > 0
rhoMax	1e8	double > 0
Sig	‘const’	‘dyn’
lamInit	‘false’	‘true’
term_eps	0	double > 0
maxiter	30	integer > 0
mu0	1e3	double > 0
muUpdate	2	double > 0
muMax	2*1e6	double > 0
solveQP	‘MA57’	‘ipopt’, ‘pinv’, ‘linsolve’, ‘sparseBs’, ‘MOSEK’, ‘quadprog’
loc Sol	‘MA57’	‘ipopt’,’sqpmethod’
reg	‘true’	‘false’
regParam	1e-4	double > 0
actMargin	-1e-6	double < 0
plot	‘true’	‘false’

Extensions

ALADIN option	default values	alternative
slack	‘standard’	‘redSpace’
hessian	‘standard’
Hess	‘standard’	‘DBFGS’, $\text{ }$ ‘BFGS’
BFGSinit	‘ident’	‘exact’
parfor	‘false’	‘true’
DelUp	‘false’	‘true’
reuse	‘false’	‘true’
commCount	‘false’	‘true’

Bi-Level options

ALADIN option	default values	alternative
innnerAlg	‘none’	‘D-CG’, ‘D-ADMM’
rhoADM	2e-2
warmStart	‘true’	‘false’
innerIter	200	integer > 0

Parameter choices for initialization

During the initialization process, the options for the parameters $\Sigma, \rho 0, \mu 0$ and $\varepsilon$ are of relevance. The default settings are given by

ALADIN option	default values	alternatives
Sig	‘const’	‘dny’
rho0	1e2	$\geq 0$
mu0	1e3	$\geq 0$
term_eps	0	$\geq 0$

The parameter $\texttt{rho0}$ represents the penalization of the distance (augmented step size) $\left|x_i-z_i^k\right|_{\Sigma_i}^2$ during the first parallel step. The option $\texttt{Sig}$ determines the augemented norm. In the first step, $\texttt{Sig}$ is the identity matrix, thus the norm $|x_i - z_i|$ is evaluated. Changing sigma leads to penalization of the distance $|\Sigma_i(x_i - z_i)| = |x_i - z_i|_{\Sigma_i}$.

First, we recall, that the algorithm consists of several stepts, which are see here. Keeping the algorithm in mind, we can now focus on the options that can be selected.

Basic Options

Parameter $\rho$

During the Parallelizable Step $k$, the optimization problems

$$\begin{aligned} \min_{x_i\in [\underline {x_i}, \overline x_i]} &f_i(x_i) + (\lambda^k)^\top A_i x_i + \frac{\rho^k}{2}\left\|x_i-z_i^k\right\|_{\Sigma_i}^2 \;\; \\ \text{s.t.}\quad & g_i(x_i) = 0, \; \;h_i(x_i)\leq 0,\; \;\; \underline{x_i} \leq x_i \leq \overline{x}_i. \end{aligned}$$

need to be solved for fixed $z_i$. The parameter $\rho^k$ represents the penalization of the distance $\left|x_i-z_i^k\right|_{\Sigma_i}^2$. As long as $\texttt{rho}$ is smaller than $\texttt{rhoMax}$, it is increased by factor $\texttt{rhoUpdate}$.

ALADIN option	default values	alternative
rho0	1e2	double > 0
rhoUpdate	1.1	double > 0
rhoMax	1e8	double > 0

Dynamic $\Sigma$

The second parameter relevant for the Parallelizable Step is the scaling matrix $\Sigma$. During the first iteration, $\Sigma$ equals the identity matrix. When the alternative option ‘dyn’ was selected, in each step the it is checked whether the step sizes for all variables decrease. In case that a step size is not decreasing in a sufficient manner, $\Sigma$ is changed dynamically to increase the negative impact of the large step size on the objective function in the parallel step.

ALADIN option	default value	alternative
Sig	‘const’	‘dyn’

Parameter $\lambda$

The parameter $\lambda$ takes over the function of the lagrange multipliers. We can decide whether we want to hand over a specific one or not.

ALADIN option	default value	alternative
lamInit	‘false’	‘true’

Termination criterion

Executing the ALADIN algorithm, in each step a termination criterion is checked. The termination criterion is split into two parts. Part one checks, whether the maximum number $\texttt{maxiter}$ of iterations is reached. Additionatlly, a termination bound $\varepsilon > 0$ can be handed over. Then, in each step, it is terminated, if $\left\|\sum_{i\in \mathcal{R}}A_ix^k_i -b \right\|\leq \epsilon \text{ and } \left\| x^k - z^k \right \|\leq \epsilon\;,$ holds true.

ALADIN option	default value	alternative
term_eps	0	double > 0
maxiter	30	integer > 0

Parameter $\mu$

During the consensus step, the coordination QP

$$\begin{aligned} &\underset{\Delta x,s}{\min}\;\;\sum_{i\in \mathcal{R}}\left\{\frac{1}{2}\Delta x_i^\top B^k_i\Delta x_i + {g_i^k}^\top \Delta x_i\right\} + (\lambda^k)^\top s + \frac{\mu^k}{2}\|s\|^2_2 \\ & \begin{aligned} \text{subject to}\; \sum_{i\in \mathcal{R}}A_i(x^k_i+\Delta x_i) &= s \qquad |\; \lambda^{\mathrm{QP} k},\\ C^k_i \Delta x_i &= 0 \qquad \forall i\in \mathcal{R},\\ \end{aligned} \end{aligned}$$

has to be executed. Setting $s:= \sum A_ix_i - b \overset{!}{=} 0$, the parameter $\mu$ is similarly to the parameter $\rho$ from above a penalty parameter. It can be set in the same manner as $\rho$.

ALADIN option	default values	alternative
mu0	1e3	double > 0
muUpdate	2	double > 0
muMax	2*1e6	double > 0

Solvers

To solve the two optimization problems in each iteration step, the desired solvers can be indicated via setting the option opts.solveQP and opts.locSol:

1. QP Solver

ALADIN option	default value	alternatives
solveQP	‘MA57’	‘ipopt’, ‘pinv’, ‘linsolve’, ‘sparseBs’, ‘MOSEK’, ‘quadprog’

2. Local Solver

ALADIN option	default value	alternatives
locSol	‘ipopt’	‘ipopt’, ‘sqpmethod’

Regularization Parameters

Sometimes the Hessian matrices of the given problems do not have full rank, such that some important matrix operations are not readily available. Remedy can be obtained by regularization approaches that increase the rank of the matrix. The option can be set as follows:

ALADIN option	default value	alternative
reg	‘true’	‘false’
regParam	1e-4

Active Margin Detection

ALADIN, beeing a solver for constraint optimization, needs an active margin detection. The tolerance is handed over by the option $\texttt{actMargin}$

ALADIN option	default value
actMargin	-1e-6

Plots

Plotting results is nice, because one can immediately see the results. However, push up windows spreading plots over your screen can be annoying and slow down the algorithm, so we included an option to deactivate plots :)

ALADIN option	default value	alternative
plot	‘true’	‘false’

Extensions

parfor Option

The parallelizable step from the ALADIN Algorithm can be executed in the parallel threads in matlab using its implemented parfor -loop. This can be activeted by:

ALADIN option	default value	alternative
parfor	‘false’	‘true’

Bilevel Options

Parameter combinations

Note that ss ALADIN-$\alpha$ is still in a prototypical phase of development, it is not guaranteed that all combinations of options work. We tried to make ALADIN-$\alpha$ as stable as possible running tests with a high code coverage, but we are at the moment not able to guarantee that all combinations of options work