Lambert Function

Lambert function (W(x)) satisfies the equation: x = W(x) exp(W(x)) for x ≥ −1/e. It is positive for x > 0 and is strictly concave.
How do I depict it as an objective in cvx?

Concave? Yes.

Expressible in CVX? I very much doubt it. But if you find a way, please let us know.

See page 126 in

I am not sure CVX can handle the exponential cone needed.

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This is good! We do have exponential cone support, of course subject to our standard disclaimers about the successive approximation method.