Constrained Optimization

It is possible to transform a constrained optimization problem (1) to an unconstrained one with some conditions on the solutions. A popular method to do so is known as barrier function where the objective function is modified to penalize the search method if any of the constraints is violated.

Barrier Function Method

Let \(\phi\) be a function that takes over relatively small values (compare to values of \(f\)) over positive reals and big values for negative ones. Then the function \(f(x)+\sum_{i=1}^m\phi(g_i(x))\) is fairly large values if \(x\) is outside of the feasibility region. Therefore, it is likely that a search method tends to focus on the values inside the feasibility region. Similarly, let \(\psi\) be a function that returns large positive values apart from \(0\). Then \(f(x)+\sum_{j=1}^k\psi(h_j(x))\) is large outside of the feasibility region and rather small on the feasibility region.

Since the values of continuous functions changes gradually, it seems implausible to be able to choose a function that successfully accomplish the above task. So, we employ an increasing sequence of positive numbers \((\sigma_n)\) that approaches to \(+\infty\) and find the optimum value for the function

\[\Lambda_n(x)=f(x)+\frac{1}{\sigma_n}\sum_{i=1}^m\phi(g_i(x))+\sigma_n\sum_{j=1}^k\psi(h_j(x)),\]

then the optimum values of \(\Lambda_n\) approaches the optimum value of \(f\) inside the feasibility region. Note that if the optimum value is located on the boundary of the feasibility region, this method would only produce an approximate interior substitute. This is why the method is referred to as interior point method.

The following options for the barrier function are implemented:

• For the inequality conditions:

  • Carrol: is the standard barrier function defined by \(-\frac{1}{g_i(x)}\)
  • Logarithmic: is the standard barrier function defined by \(-\log(g_i(x))\)
  • Expn: is the standard barrier function defined by \(e^{-g_i(x)+\epsilon}\)

• For the equality condition:

  • Courant: function which is simply defined by \(h_j^2(x)\).

Note

The value of \(\epsilon\) in the code is taken to be equal to \(\epsilon=\frac{1}{\sigma_n}\).