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We want to build a bridge over a river of width 2`, with a pillar in the middle of the river. The bridge is symmetric and drops (linearly) to a minimum height of h meters below the initial level of the road, as depicted here:

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How can I formulate this as a conic program and use the dual to find the optimal cost savings of the project compared to when the bridge is flat, $h=0$?

Any comment on how to find this optimal savings using the dual is much appreciated.

I think we may be able to do this by imposing a barrier function like $-log(x)$, but I'm not sure how it would work. Would I then need to find KKT points? If so, how do I know this is optimal?

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