Minima and Maxima in Higher Dimensions

In two dimensions, but the same reasoning applies to any dimension, the problem of maximizing has to be extended to increasingly complex functions.

The most immediate solution is to analyze the point along each spatial direction independently: in this way, the problem reduces entirely to the one-dimensional case.

If instead a broader neighborhood is to be exploited, the next simplest model to use is the paraboloid, a quadric expressed in the form

$$m_0 x^2 + m_1 x + m_2 y^2 + m_3 y + m_4 = z$$ (1.94)

where the points $(x,y)$ are always understood as deviations from the point to be modeled and $z$ is the value that the function takes at that particular point. Compared to the solution with completely separated axes, with this equation, points not on the axes also actively contribute to the solution of the problem. Clearly, if only the 5 points along the axes are included in the system, the solution will be exactly the same as that seen in the previous section.

Each element of the problem therefore provides a constraint in the form

$$\mathbf{m} \cdot \mathbf{x}_i=z_i$$ (1.95)

and all the constraints together generate a potentially overdetermined linear system. In this case, there are no remarkable results for obtaining a closed form solution, but the simplest approach is to precalculate a factorization of the system formed by the elements $\mathbf{x}_i$, representing a particular neighborhood of $(0,0)$, in order to speed up the subsequent resolution when the values $z_i$ are known.

The equation (1.94) has a zero gradient at the point

$$\left( - \frac{m_1}{2 m_0}, - \frac{m_3}{2 m_2} \right)$$ (1.96)

just as in the one-dimensional case, since the two components, that along $x$ and that along $y$, remain separate during evaluation. This result extends to n-dimensional cases.