# MLDO Problem One

## Original Problem: All At Once

Can we have a less generic name of this problem? Is it reported in the literature? Do we know the real solution of this analytical problem? You can call it problem 1 if you'd like.

It does not matter how I would like to call it. I would like this page to be a seed for a technical report or a paper. Therefore it makes sense (at least to me) to use proper names from the beginning. It would also help with naming Wiki pages in a meaningful way

sd: (The problem is a reduced version of a problem used earlier in literature. It is presented in Tosserams, S., Etman, L. F. P., and Rooda, J. E., 2006, “An Augmented Lagrangian Relaxation for Analytical Target Cascading Using the Alternating Directions Method of Multipliers,” Struct. Multidiscip. Optim., 313, pp. 176–189.)

$\underset{Z_1,Z_2,...,Z_7}{min} f = {z_1}^2 + {z_2}^2$

$g_1 = \dfrac{{z_3}^{-2} + {z_4}^{2}}{{z_5}^{2}} - 1 {\le} 0$

$g_2 = \dfrac{{z_5}^{2} + {z_6}^{-2}}{{z_7}^{2}} - 1 {\le} 0$

$h_1 = {z_1}^{2} - {z_3}^{2} - {z_4}^{-2} - {z_5}^{2} = 0$

$h_2 = {z_2}^{2} - {z_5}^{2} - {z_6}^{2} - {z_7}^{2} = 0$

$z_1,z_2,......z_7 {\ge} 0$

$x^{*} = \quad \left \lbrack 2.15,2.06,1.32,0.76,1.07,1.00,1.47 \right \rbrack$

• Tomasz needs a formulation suitable for DOT

The DOT formulation would be one with only <= design constraints. For that, each of the two equality constraints must be written twice, once as <=0 and another time as >=0. Then the >=0 constraint must be multiplied times -1 so it can also be written at <=0 constraint. That way, the problem can be solved using DOT.

• Does '*' means the value of the design variables at the optimum? If yes, is it the "well known solution reported in the literature", or is it the solution obtained by Saber, if different?

I believe this is a convex programming problem, which means it only has one global optimum at the x* location.

This is not what Tomasz asked. Are we sure that the Saber's code reproduces the "one global optimum"? Does DOT reproduces it as well?

## Decomposition 1: AL ATC Schema 1

Tomasz's guess is that it is Augmented Lagrangian method at two levels.

### level 1

$obj = {z_1}^2 + \lambda * (z_5 - {z_5}^L) + [W * (z_5 - {z_5}^L)]^2$

$g_1 = \dfrac{{z_3}^{-2}+{z_4}^2}{{z_5}^2} - 1 \le 0$

$h_1 = {z_1}^2 - {z_3}^2 - {z_4}^{-2} - {z_5}^2 = 0$

#### Matlab implementation

function f = obj1(x,Landa,W,r)
z1 = x(1); z3 = x(2); z4 = x(3); t = x(4);
f = z1^2 + Landa * ( t - r ) + W^2 * ( t - r )^2;

function [c,ceq] = const1(x,Landa,W,r)
z1 = x(1); z3 = x(2); z4 = x(3); t = x(4);

% Inequality constraints
c(1) = (z3^-2 + z4^2)* t^-2 - 1 ;

% Equality Constraints
ceq(1) = z1^2 - z3^2 - z4^-2 - t^2;


### level 2

$obj = {z_2}^2 + \lambda * (z_5 - {z_5}^L) + [W * (z_5 - {z_5}^U)]^2$

$g_1 = \dfrac{{z_5}^2+{z_6}^{-2}}{{z_7}^2} - 1 \le 0$

$h_1 = {z_2}^2 - {z_5}^2 - {z_6}^2 - {z_7}^2 = 0$

#### Matlab implementation

function f = obj2(x,Landa,W,t)
z2 = x(1); z6 = x(3); z7 = x(4); r = x(2);
f = z2^2 + Landa * ( t - r ) + W^2 * ( t - r )^2;

function [c,ceq] = const2(x,Landa,W,t)
z2 = x(1); z6 = x(3); z7 = x(4); r = x(2);

% Inequality constraints
c(1) = (r^2 + z6^-2)* z7^-2 - 1 ;

% Equality Constraints
ceq(1) = z2^2 - r2^2 - z6^2 - z7^2;


## Decomposition 2: AL ATC Schema 2

Tomasz's guess is that it is Augmented Lagrangian method at three levels. This case in not shown in Figure 7 of "DQA Approximation..." paper by Li, Lu, and Michalek. Or is it AL-BCD?

### Level 1 Element 1

$\underset{Z_1,Z_3,Z_4,Z_5}{min} f = {z_1}^2 + {\pi}_{AL} = {z_1}^2 + {\lambda}.{z_5} + {\left ( W.{\left ( {z_5} - {z_5}^L \right )} \right )}^2$

$g_1 = \dfrac{{z_3}^{-2}+{z_4}^2}{{z_5}^2} - 1 \le 0$

$h_1 = {z_1}^2 - {z_3}^2 - {z_4}^{-2} - {z_5}^2 = 0$

### Level 2 Element 2

$\underset{Z_2,Z_6,Z_5,Z_7}{min} f = {z_2}^2 + {\pi}_{AL} = {z_2}^2 + {\lambda}.{z_5} + {\left ( W.{\left ( {z_5} - {z_5}^U \right )} \right )}^2$

$g_2 = \dfrac{{z_5}^2+{z_6}^{-2}}{{z_7}^2} - 1 \le 0$

$h_2 = {z_2}^2 - {z_5}^2 - {z_6}^{-2} - {z_7}^2 = 0$

• Tomasz needs a formulation suitable for DOT.
• Tomasz needs Matlab implementation of this schema