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While the underlying optimization problems are non-convex, we develop a technique to evaluate bottleneck problems in closed form by equivalently expressing them in terms of lower convex or upper concave envelope of certain functions. By applying this technique to a binary case, we derive closed form expressions for several bottleneck problems.In optical communications, four-dimensional (4D) modulation formats encode information onto the quadrature components of two arbitrary orthogonal states of polarisation of the optical field. Many a