I have the following minimisation problem:
\(\displaystyle T = \min_{t \in [0, \infty)} t + k(t) \quad \text{ subject to } e^{\lambda k}(1+\mu_{t}) - 1 \geq \bar{\mu} \quad \text{ where } \mu_{t} = \frac{\psi(1-e^{\lambda t})}{1-\psi} \)
Note that \(\displaystyle \lambda > 0, \psi \in (\frac{1}{2}, 1), \bar{\mu}\in(0,1)\)
I am also generally looking for an expression for the function \(\displaystyle T = t + k(t) \).
This is my naive approach to solving this, with the constraint binding at the minimum:
I can see a key issue in line (3). The logarithm restricts the expression to be positive - in particular, \(\displaystyle e^{\lambda t}\psi < 1 \). But it should be that \(\displaystyle t \in [0, \infty) \) or at the very least, have a larger domain that in (3).
Does anyone have any suggestions on how to characterise \(\displaystyle T = t + k(t) \) and/or the optimisation problem?
Thanks
Update - I have solved this problem, but I don't know how to delete the post
\(\displaystyle T = \min_{t \in [0, \infty)} t + k(t) \quad \text{ subject to } e^{\lambda k}(1+\mu_{t}) - 1 \geq \bar{\mu} \quad \text{ where } \mu_{t} = \frac{\psi(1-e^{\lambda t})}{1-\psi} \)
Note that \(\displaystyle \lambda > 0, \psi \in (\frac{1}{2}, 1), \bar{\mu}\in(0,1)\)
I am also generally looking for an expression for the function \(\displaystyle T = t + k(t) \).
This is my naive approach to solving this, with the constraint binding at the minimum:
- \(\displaystyle e^{\lambda k} (\frac{e^{\lambda t}\psi - 1}{\psi - 1})-1 = \bar{\mu} \)
- \(\displaystyle e^{\lambda k} = \frac{(\bar{\mu}+1)(1-\psi)}{1-e^{\lambda t}\psi}\)
- \(\displaystyle k = \frac{1}{\lambda} \ln(\frac{(\bar{\mu}+1)(1-\psi)}{1-e^{\lambda t}\psi})\)
- \(\displaystyle T = t + \frac{1}{\lambda} \ln(\frac{(\bar{\mu}+1)(1-\psi)}{1-e^{\lambda t}\psi})\)
- \(\displaystyle \frac{\partial T}{\partial t} = \frac{1}{1 - e^{\lambda t} \psi} \)
- But (5) can never = 0. Further, my previous work shows that the minimum must be finite. I am not sure how to interpret a hyperbolic first derivative. Also the size of \(\displaystyle \lambda >0 \) should not (I think) substantially affect the solution.
I can see a key issue in line (3). The logarithm restricts the expression to be positive - in particular, \(\displaystyle e^{\lambda t}\psi < 1 \). But it should be that \(\displaystyle t \in [0, \infty) \) or at the very least, have a larger domain that in (3).
Does anyone have any suggestions on how to characterise \(\displaystyle T = t + k(t) \) and/or the optimisation problem?
Thanks
Update - I have solved this problem, but I don't know how to delete the post
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