`%plot --format svg`

# Optimization

Many problems involve finding a minimum or maximum based on given constraints. Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest.

The Lennard-Jones potential is commonly used to model interatomic bonding.

$E_{LJ}(x)=4\epsilon \left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^{6}\right)$

Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no force applied to the chain and even for tension, F. This will allow us to calculate the nonlinear spring constant of a 1-D gold chain.

Computational Tools to Study and Predict the Long-Term Stability of Nanowires. By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva

Single atom gold chain mechanics

### First, let's find the minimum energy $\min(E_{LJ}(x))$

## Brute force

```
setdefaults
epsilon = 0.039; % kcal/mol
sigma = 2.934; % Angstrom
x=linspace(2.8,6,200); % bond length in Angstrom
Ex = lennard_jones(x,sigma,epsilon);
[Emin,imin]=min(Ex);
plot(x,Ex,x(imin),Emin,'o')
```

```
x(imin-1)
x(imin)
x(imin+1)
```

```
ans = 3.2824
ans = 3.2985
ans = 3.3146
```

## Golden Search Algorithm

We can't just look for a sign change for the problem (unless we can take a derivative) so we need a new approach to determine whether we have a maximum between the two bounds.

Rather than using the midpoint of initial bounds, here the problem is more difficult. We need to compare the values of 4 function evaluations. The golden search uses the golden ratio to determine two interior points.

Start with bounds of 2.5 and 6 Angstrom.

```
% define Au atomic potential
epsilon = 0.039; % kcal/mol
sigma = 2.934; % Angstrom
Au_x= @(x) lennard_jones(x,sigma,epsilon);
% calculate golden ratio
phi = 1/2+sqrt(5)/2;
% set initial limits
x_l=2.8;
x_u=6;
% Iteration #1
d=(phi-1)*(x_u-x_l);
x1=x_l+d; % define point 1
x2=x_u-d; % define point 2
% evaluate Au_x(x1) and Au_x(x2)
f1=Au_x(x1);
f2=Au_x(x2);
plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')
hold on;
if f2<f1
plot([x_u,x1],[Au_x(x_u),f1],'k-')
x_u=x1;
else
plot([x_l,x2],[Au_x(x_l),f2],'k-')
x_l=x1;
end
hold off
%old_min = current_min;
current_min=min([f1,f2,Au_x(x_l),Au_x(x_u)])
%error_app=abs((current_min-old_min)/current_min)
```

```
current_min = -0.019959
```

```
% Iteration #2
d=(phi-1)*(x_u-x_l);
x1=x_l+d; % define point 1
x2=x_u-d; % define point 2
% evaluate Au_x(x1) and Au_x(x2)
f1=Au_x(x1);
f2=Au_x(x2);
plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')
hold on;
if f2<f1
plot([x_u,x1],[Au_x(x_u),f1],'k-')
x_u=x1;
else
plot([x_l,x2],[Au_x(x_l),f2],'k-')
x_l=x1;
end
hold off
old_min = current_min;
current_min=min([f1,f2,Au_x(x_l),Au_x(x_u)])
error_app=abs((current_min-old_min)/current_min)
```

```
current_min = -0.033707
error_app = 0.40787
```

```
% Iteration #3
d=(phi-1)*(x_u-x_l);
x1=x_l+d; % define point 1
x2=x_u-d; % define point 2
% evaluate Au_x(x1) and Au_x(x2)
f1=Au_x(x1);
f2=Au_x(x2);
plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')
hold on;
if f2<f1
plot([x_u,x1],[Au_x(x_u),f1],'k-')
x_u=x1;
else
plot([x_l,x2],[Au_x(x_l),f2],'k-')
x_l=x1;
end
hold off
old_min = current_min;
current_min=min([f1,f2,Au_x(x_l),Au_x(x_u)])
error_app=abs((current_min-old_min)/current_min)
```

```
current_min = -0.038904
error_app = 0.13359
```

```
% Iteration #3
d=(phi-1)*(x_u-x_l);
x1=x_l+d; % define point 1
x2=x_u-d; % define point 2
% evaluate Au_x(x1) and Au_x(x2)
f1=Au_x(x1);
f2=Au_x(x2);
plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')
hold on;
if f2<f1
plot([x_u,x1],[Au_x(x_u),f1],'k-')
x_u=x1;
else
plot([x_l,x2],[Au_x(x_l),f2],'k-')
x_l=x1;
end
hold off
old_min = current_min;
current_min=min([f1,f2,Au_x(x_l),Au_x(x_u)])
error_app=abs((current_min-old_min)/current_min)
```

```
current_min = -0.038904
error_app = 0
```

## Parabolic Interpolation

Near a minimum/maximum, the function resembles a parabola. With three data points, it is possible to fit a parabola to the function. So, given a lower and upper bound, we can choose the midpoint and fit a parabola to the 3 x,f(x) coordinates.

$ x_{4} =x_{2} - \frac{1}{2} \frac{ \left(x_{2} -x_{1} \right)^{2} \left[f \left(x_{2} \right)-f \left(x_{3} \right) \right]- \left(x_{2} -x_{3} \right)^{2} \left[f \left(x_{2} \right)-f \left(x_{1} \right) \right]}{ \left(x_{2} -x_{1} \right) \left[f \left(x_{2} \right)-f \left(x_{3} \right) \right]- \left(x_{2} -x_{3} \right) \left[f \left(x_{2} \right)-f \left(x_{1} \right) \right]}$

Where $x_{4}$ location of the maximum or minimum of the parabola.

```
% define Au atomic potential
epsilon = 0.039; % kcal/mol
sigma = 2.934; % Angstrom
Au_x= @(x) lennard_jones(x,sigma,epsilon);
% set initial limits
x_l=2.8;
x_u=3.5;
% Iteration #1
x1=x_l;
x2=mean([x_l,x_u]);
x3=x_u;
% evaluate Au_x(x1), Au_x(x2) and Au_x(x3)
f1=Au_x(x1);
f2=Au_x(x2);
f3=Au_x(x3);
p = polyfit([x1,x2,x3],[f1,f2,f3],2);
x_fit = linspace(x1,x3,20);
y_fit = polyval(p,x_fit);
plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')
hold on
if f2<f1 && f2<f3
x4=x2-0.5*((x2-x1)^2*(f2-f3)-(x2-x3)^2*(f2-f1))/((x2-x1)...
*(f2-f3)-(x2-x3)*(f2-f1));
f4=Au_x(x4);
if x4>x2
plot(x4,f4,'*',[x1,x2],[f1,f2])
x1=x2;
f1=f2;
else
plot(x4,f4,'*',[x3,x2],[f3,f2])
x3=x2;
f3=f2;
end
x2=x4; f2=f4;
else
error('no minimum in bracket')
end
hold off
```

```
p = polyfit([x1,x2,x3],[f1,f2,f3],2);
x_fit = linspace(x1,x3,20);
y_fit = polyval(p,x_fit);
plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')
hold on
if f2<f1 && f2<f3
x4=x2-0.5*((x2-x1)^2*(f2-f3)-(x2-x3)^2*(f2-f1))/((x2-x1)...
*(f2-f3)-(x2-x3)*(f2-f1));
f4=Au_x(x4);
if x4>x2
plot(x4,f4,'*',[x1,x2],[f1,f2])
x1=x2;
f1=f2;
else
plot(x4,f4,'*',[x3,x2],[f3,f2])
x3=x2;
f3=f2;
end
x2=x4; f2=f4;
else
error('no minimum in bracket')
end
hold off
```

```
p = polyfit([x1,x2,x3],[f1,f2,f3],2);
x_fit = linspace(x1,x3,20);
y_fit = polyval(p,x_fit);
plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')
hold on
if f2<f1 && f2<f3
x4=x2-0.5*((x2-x1)^2*(f2-f3)-(x2-x3)^2*(f2-f1))/((x2-x1)...
*(f2-f3)-(x2-x3)*(f2-f1));
f4=Au_x(x4);
if x4>x2
plot(x4,f4,'*',[x1,x2],[f1,f2])
x1=x2;
f1=f2;
else
plot(x4,f4,'*',[x3,x2],[f3,f2])
x3=x2;
f3=f2;
end
x2=x4; f2=f4;
else
error('no minimum in bracket')
end
hold off
```

Parabolic interpolation does not converge in many scenarios even though it it a bracketing method. Instead, functions like `fminbnd`

in Matlab and Octave use a combination of the two (Golden Ratio and Parabolic)

## Using the solutions to minimization for the nonlinear spring constant

Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain

$E_{total}(\Delta x) = E_{LJ}(x_{min}+\Delta x) - F \cdot \Delta x$

```
epsilon = 0.039; % kcal/mol
epsilon = epsilon*6.9477e-21; % J/atom
epsilon = epsilon*1e18; % aJ/J
% final units for epsilon are aJ
sigma = 2.934; % Angstrom
sigma = sigma*0.10; % nm/Angstrom
x=linspace(2.8,6,200)*0.10; % bond length in um
Ex = lennard_jones(x,sigma,epsilon);
%[Emin,imin]=min(Ex);
[xmin,Emin] = fminbnd(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6)
plot(x,Ex,xmin,Emin,'o')
ylabel('Lennard Jones Potential (aJ/atom)')
xlabel('bond length (nm)')
Etotal = @(dx,F) lennard_jones(xmin+dx,sigma,epsilon)-F.*dx;
```

```
xmin = 0.32933
Emin = -2.7096e-04
```

```
N=50;
dx = zeros(1,N); % [in nm]
F_applied=linspace(0,0.0022,N); % [in nN]
for i=1:N
optmin=goldmin(@(dx) Etotal(dx,F_applied(i)),-0.001,0.035);
dx(i)=optmin;
end
plot(dx,F_applied)
xlabel('dx (nm)')
ylabel('Force (nN)')
```

```
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```

For this function, it is possible to take a derivative and compare the analytical result:

```
dx_full=linspace(0,0.06,50);
F= @(dx) 4*epsilon*6*(sigma^6./(xmin+dx).^7-2*sigma^12./(xmin+dx).^13)
plot(dx_full,F(dx_full),dx,F_applied)
```

```
F =
@(dx) 4 * epsilon * 6 * (sigma ^ 6 ./ (xmin + dx) .^ 7 - 2 * sigma ^ 12 ./ (xmin + dx) .^ 13)
```

## Curve-fitting

Another example is minimizing error in your approximation of a function. If you have data (now we have Force-displacement data) we can fit this to a function, such as:

$F(x) = K_{1}\Delta x + \frac{1}{2} K_{2}(\Delta x)^{2}$

```
function SSE = sse_of_parabola(K,xdata,ydata)
% calculate the sum of squares error for a parabola given a function, func, and xdata and ydata
% output is SSE=sum of squares error
K1=K(1);
K2=K(2);
y_function = K1*xdata+1/2*K2*xdata.^2;
SSE = sum((ydata-y_function).^2);
end
```

```
[K,SSE_min]=fminsearch(@(K) sse_of_parabola(K,dx,F_applied),[1,1]);
fprintf('\nThe nonlinear spring constants are K1=%1.2f nN/nm and K2=%1.2f nN/nm^2\n',K)
fprintf('The mininum sum of squares error = %1.2e',SSE_min)
```

```
The nonlinear spring constants are K1=0.16 nN/nm and K2=-5.98 nN/nm^2
The mininum sum of squares error = 7.35e-08
```

`plot(dx,F_applied,'o',dx,K(1)*dx+1/2*K(2)*dx.^2)`