# Finding optimal savings with dual formulation (word problem)

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:

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?