Find critical points, please share any tricks you know of

[omg!]

1. Find the minimum point of
$$I(p, q, r, s)=\int_{\mathbb{R}^2}\,dxdy\,\left[exp(-(x-d)^2-y^2)+exp(-(x+d)^2-y^2)-exp(-(x-p)^2-(y-q)^2)-exp(-(x-r)^2+(y-s)^2)\right]^2$$
$$+\int_{\mathbb{R}^2}\,dxdy\,\left[exp(-(x-\delta_x)^2-(y-\delta_y)^2)-exp(-(x-p)^2+(y-q)^2)\right]^2$$
for given $$d, \delta_x, \delta_y$$.

The Attempt at a Solution

i don't have any ideas how to tackle this problem in a intelligent manner. what else can one do beyond calculating the gradient with respect to p, q, r, s?

thanks for any assistance

 

[fzero]

You will have to compute the gradients to find the critical points. But you should definitely do all the integrations first. You need to expand all terms and then complete the squares in the exponents.