Write-up coming soon.
We aim to solve the following problem
The optimal solution always lies on the boundary \(\lvert x\rvert^p+\lvert y\rvert^p = C^p\) whenever the unconstrained minimizer \((x_0, y_0)\) lies outside the feasible set. The shape of the constraint set changes significantly with \(p\): as \(p\to 1\) the ball becomes a diamond and solutions tend to concentrate on corners, while as \(p\to\infty\) it becomes a square aligned with the axes. The curvatures \(a\) and \(b\) control the eccentricity of the objective’s level ellipses, which in turn determines where on the boundary the optimum falls.