I'm confronted with a problem in which I must minimize a travel path going from some transmitter $T$ to a receiver $R$, going through an interface -- i.e. a change in velocity causing refraction -- and having to pass through a point P of an axis aligned box.
In certain situation, the point $P$ must be reached after a reflection occurred on an interface located behind the axis aligned box. In diagrams, the kind of situation I'm confronted with can be represented as follow:
Note that, after having reached point P, P starts to act as a point source for the ray going from P to R. That is, the path from P-->I_R-->R is independent from the path from T-->I_T-->C-->P. Note further that to each segment in the path correspond a particular velocity.
The problem can thus be formulated as:
- Given $T$ and $R$, find $P$ in the axis aligned box such that $f^T_R(P)$ is minized where:
$$f^T_R(P) = d(T, I_T(P))\frac{1}{v_1} + d(I_T(P), C)\frac{1}{v_2} + d(C(P),P)\frac{1}{v_3} + d(P,I_R(P))\frac{1}{v_4} + d(I_R(P),R)\frac{1}{v_1}$$
where $d(x,y)$ is the euclidian distance in 2D and where $I_T(P), I_R(P)$ give the interface point for a given $P$. These last two function are defined implicitly by the quartic equation derive in this question.
I'm wondering what method I could use to guarantee that I converge (or compute) the global minimum of this problem. I already know that, under some situation, the minimum $P$ need not be unisuqe. In particular, we can posit $v_2 = v_3 = v_4$ and $T = R$. Then, placing the axis aligned box directly to the right of $T$, any point perpendicular to the interface going to $C$ would minimize $f^T_T$.
tl;dr: What method should I use to guarantee that I converge to the global minimum value of this function? It's unclear to me that the problem is convex and while the global minimum must exist I'm afraid I may either diverge or converge to the wrong point.
N.B. My initial hope had been to show that the function was convex by showing that various terms of the sum were, but the path from $T$ to $C$ is not. To show this, pose $T = (-1,0)$, $C = (k,y(\theta))$ and $\theta$ the angle of incidence of the ray going from $T$ to the interface. Then $$f(\theta) = \frac{1}{\cos{\theta}} + \frac{k}{\sqrt{1-(\frac{\sin{\theta}}{n})^2}}$$
where $n = \frac{v_1}{v_2}$ is travel time from $T$ to $C$. This function is not convex for $\theta \in (-\frac{\pi}{2}, \frac{\pi}{2})$ given large enough values of $k$.
