We want to solve following optimization problem

$$ \text{minimize} 3x^3y+y^2z^2-2xy-4xz^3 \quad \text{s.t.} \quad x^2+y^2+z^2=1 $$

The strategy to find the *global* optimum is to use the method of Lagrange multipliers to find *all* critical points of the objective function such that the equality constraint is satisfied. We start with defining our Lagrangian.

```
using HomotopyContinuation, DynamicPolynomials
@polyvar x y z
J = 3x^3*y+y^2*z^2-2x*y-x*4z^3
g = x^2+y^2+z^2-1
# Introduce auxillary variable for Lagrangian
@polyvar λ
# define Lagrangian
L = J - λ * g
```

```
3x³y - 4xz³ + y²z² - x²λ - y²λ - z²λ - 2xy + λ
```

In order to compute all critical points we have to solve the square system of equations

$$ \nabla_{(x,y.z,\lambda)}L = 0 $$

For this we first compute the gradient of $L$ and then use the `solve`

routine to find all critical points.

```
# compute the gradient
∇L = differentiate(L, [x, y, z, λ])
# Now we solve the polynomial system
result = solve(∇L)
```

```
Result with 26 solutions
==================================
• 26 non-singular solutions (22 real)
• 0 singular solutions (0 real)
• 54 paths tracked
• random seed: 556761
```

We see that from the theoretical 54 possible (complex) critical points there are only 26. Also we check the number of *real* critical points by

```
nreal(result)
```

```
22
```

and see that there are 22.

In order to find the global minimum we now have to evaluate all *real* solutions and find the value where the minimum is attained.

```
reals = real_solutions(result);
# Now we simply evaluate the objective J and find the minimum
minval, minindex = findmin(map(s -> J(s[1:3]), reals))
```

```
(-1.32842129069426, 21)
```

```
minarg = reals[minindex][1:3]
```

```
3-element Array{Float64,1}:
-0.4968755205596932
-0.09460829635048905
-0.8626494000056988
```

We found that the minimum of $J$ over the unit sphere is attained at $(0.496876, 0.0946083, 0.862649)$ with objective value $-1.32842$.

**Cite this example:**

```
@Misc{ optimization2020 ,
author = { Sascha Timme },
title = { Minimizing a function over the sphere },
howpublished = { \url{ https://www.JuliaHomotopyContinuation.org/examples/optimization/ } },
note = { Accessed: July 1, 2020 }
}
```