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ALADIN-\alpha

ALADIN-\alpha is a rapid-prototyping toolbox for distributed and decentralized non-convex optimization. ALADIN-\alpha provides an implementation of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm and the Alternating Direction of Multipliers Method (ADMM) with a unified interface. Moreover, a bi-level ALADIN variant is included in ALADIN-\alpha allowing for decentralized non-convex optimization. Application examples from various fields highlight the broad applicability of ALADIN-\alpha. The toolbox is described in more detail here.

ALADIN-\alpha solves problem of the form

\begin{aligned} &\min_{x_1,\dots,x_{n_s}} && \sum_{i\in \mathcal{S}} f_i(x_i,p_i) \\ &\;\;\text{subject to}&&g_{i}(x_i,p_i) = 0 \quad &&\mid \kappa_i, &\forall i \in \mathcal{R}, \\ &&&h_{i}(x_i,p_i) \leq 0 \quad \;\,&& \mid \gamma_i, &\forall i \in \mathcal{R}, \\ &&&\underline{x}_i \leq x_i \leq \overline{x}_i\;\,&& \mid\eta_i, &\forall i \in \mathcal{R}, \\ &&&\sum_{i\in \mathcal{S}}A_i x_i=0\;&&\mid\lambda. \end{aligned}

in a distributed fashion.

Eearly-stage version of ALADIN-\alpha

Note that ALADIN-\alpha is still in a prototypical phase of development.

An example

Here’s an example how to use ALADIN-\alpha. Let us consider an inequality-constrained non-convex problem

\begin{aligned} & \min_{x \in \mathbb{R},y \in \mathbb{R}^2} 2 \,(x - 1)^2 + (y(2) - 2)^2\\ \;\;\text{subject to} \;\; & - 1 - y(1)\,y(2) \leq 0, \quad -1.5 + y(1) y(2) \leq 0, \\ & 1\,x \;\;+\; \;(\,-1 \;\; 0 \,)\,y = 0, \end{aligned}

which is in the above form. In MATLAB code, this looks as follows:

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% define local objective functions
f1 = @(x) 2 * ( x - 1)^2;
f2 = @(y) (y(2) - 2)^2;

% local nonlinear inequality constraints
h1 = @(x) [];
h2 = @(y) [-1 -  y(1) * y(2) ;-1.5 + y(1) * y(2)];

% coupling matrices
A1  =    1;
A2  =   [-1   0];

% collect variables in sProb struct
sProb.locFuns.ffi  = {f1, f2};
sProb.locFuns.hhi  = {h1, h2};
sProb.AA           = {A1, A2};

That’s all! Now we are ready to solve our problem with the run_ALADIN function.

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sol_ALADIN = run_ALADINnew( sProb );

If the option plot is true, ALADIN-M shows progress by the following plot while iterating. A sample plot is shown below.

The resulting solver console output is shown next.

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   ========================================================      
   ==               This is ALADIN-alpha v0.1            ==      
   ========================================================      
   QP solver:        MA57
   Local solver:     ipopt
   Inner algorithm:  none

   No termination criterion was specified.
   Consensus violation: 1.1368e-10

   Maximum number of iterations reached.

   -----------------   ALADIN-M timing   ------------------
                       t[s]        %tot         %iter
   Tot time:......:    2.5                                 
   Prob setup:....:    0.1         2.3                     
   Iter time:.....:    2.4         97.6                    
   ------
   NLP time:......:    0.9                      37.3       
   QP time:.......:    0.0                      1.5        
   Reg time:......:    0.0                      0.4        
   Plot time:.....:    1.4                      56.2       

   ========================================================

For further examples checkout the /examples and the /test folder!

How to install

Clone https://github.com/alexe15/ALADIN.m and add /ALADIN.m to your MATLAB path.

Requirements

  • MATLAB
  • CasADi
  • MATLAB symbolic toolbox (only for examples)

The current version is tested with MATLAB R2019b and CasADi 3.5.1.