# MMA.jl

This module implements the MMA Algorithm in Julia as described by Krister Svanberg in [1].

The code in this module was made for a course in Structural Optimization and should be seen as educational. For real use it is likely better to use a more mature code base, for example NLopt.jl which contain a more modern MMA algorithm than the one implemented here.

## Usage

Ẁe will solve the nonlinear constrained minimization problem given here

```
using MMA
# Define objective function
function f(x::Vector, grad::Vector)
if length(grad) != 0
grad[1] = 0.0
grad[2] = 0.5/sqrt(x[2])
end
sqrt(x[2])
end
# Define a constraint function
function g(x::Vector, grad::Vector, a, b)
if length(grad) != 0
grad[1] = 3a * (a*x[1] + b)^2
grad[2] = -1
end
(a*x[1] + b)^3 - x[2]
end
# Create the MMAModel with a relative tolerance on x
ndim = 2
m = MMAModel(ndim, f, xtol = 1e-6, store_trace=true)
# Add box constraints to the variables
box!(m, 1, 0.0, 100.0)
box!(m, 2, 0.0, 100.0)
# Add two nonlinear inequalities
ineq_constraint!(m, (x,grad) -> g(x,grad,2,0))
ineq_constraint!(m, (x,grad) -> g(x,grad,-1,1))
# Solve the problem
x0 = [1.234, 2.345]
results = optimize(m, x0)
# Print the results
print(results)
#Results of Optimization Algorithm
# * Algorithm: MMA._MMA
# * Starting Point: [1.234,2.345]
# * Minimizer: [0.3333332274782007,0.29629691337859027]
# * Minimum: 5.443316e-01
# * Iterations: 7
# * Convergence: true
# * |x - x'| < 1.0e-06: true
# |x - x'| = 8.33e-08
# * |f(x) - f(x')| / |f(x)| < 1.5e-08: false
# |f(x) - f(x')| / |f(x)| = 7.65e-08
# * |g(x)| < 1.5e-08: false
# |g(x)| = 9.19e-01
# * stopped by an increasing objective: true
# * Reached Maximum Number of Iterations: false
# * Objective Calls: 8
# * Gradient Calls: 8
# Print the trace
#println(results.trace)
#Iter Function value Gradient norm
#------ -------------- --------------
# 1 3.948114e-01 1.266427e+00
# 2 1.799667e-01 2.778292e+00
# 3 4.353204e-01 1.148579e+00
# 4 5.338213e-01 9.366429e-01
# 5 5.442500e-01 9.186955e-01
# 6 5.443315e-01 9.185578e-01
# 7 5.443316e-01 9.185577e-01
```

## References

[1] The method of moving asymptotes - a new method for structural optimization

### Author

Kristoffer Carlsson - kristoffer.carlsson@chalmers.se