Minimizing convex function exp(-x)*(sqrt(x^2 + a^2)-x)


I would like to minimize the function f(x) = exp(-x)*(sqrt(x^2 + a^2)-x) where a is fixed. Calculations and plots show f convex for any value of a. Do you have any idea if the minimization problem can be expressed with a combination of exponential cones ?


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How about dividing it into two monotone pieces of x, then minimize/maximize x?

Thank you for your suggestion @jackfsuia. The problem is that next to this function f(x) i have another one taking the same argument x (the sum represents a dissipation of energy i want to minimize). Therefore I can not use an alternate minimization/maximization scheme just on the above one f(x)=exp(-x)*(sqrt(x^2 + a^2)-x).

The scheme should still be applicable to your convex objective function (f(x)+something else), if I understand your words correctly.