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I’m struggling to convert the following problem into an optimization problem. Consider a river. You’re traveling on that river. The river has multiple bends. Essentially, given a starting point on the river and an endpoint on the river, I need to determine what the shortest path is to get their.

enter image description hereHere is a picture of what my problem is. In light black, you see the outline of the river. You can also see the three river bends. The two black circles represent the starting point and ending point. The link in dark black shows a non-optimal path I can take. In red you see the optimal solution. I need to find this optimal solution.

My thoughts: At first I was thinking about minimizing the distance to each river bend. However, this won’t give an optimal solution. Another approach I was thinking about was to find the path that minimizes the number of turns you have to make since a straight line has the minimum distance. However, I’m not sure how to turn this into an optimization problem like an LP, QP, etc.

user12888
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    !) The optimal path goes always with segments between vertices. 2) If the path goes along a margin, it goes along all the respective segment. – Cesareo Oct 09 '18 at 09:12
  • @Cesareo Thanks! But, how can I encode all of this into an optimization problem? – user12888 Oct 09 '18 at 16:02
  • Suppose that the trajectory in thick black is a string. Tensioning the cord between the two ends will show the optimal trajectory. I think that this problem with a look ahead procedure can be solved. – Cesareo Oct 09 '18 at 16:44
  • Some python functions can suffice. – Cesareo Oct 09 '18 at 16:48

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