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I would like to find an optimal probability density function (PDF) $f$. Given $b$,

$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\int_{0}^{x}f(t)dt \leq C, \text{ for any } x \in [0,\infty) \\\\ & \frac{1}{b} \left(\displaystyle\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt \right) \leq C, \text{ for any }x\in(b,\infty) \\\\& \displaystyle\int_{0}^{\infty}f(t)dt=1\end{array} $$

I do not know how to solve it. Are there any systematic ways or general ideas of dealing with such a functional optimization problem?

My idea

Here I can introduce another optimization variable $B\geq b$ such that the probability density is restricted within the range of $[0,B]$, i.e. $\int_{0}^{B}f(t)dt=1$. Then the original optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C, \text{ for any }x\in(b,B]$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

We take a look at the second constraint $\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C$. Taking the derivative of the left hand side with respect to x, we find that its derivative is $f(x)\geq 0$, which implies that the left hand side is non-decreasing with respect to $x\in (b,B]$. Hence the second constraint can be reduced to $1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$ (by taking $x=B$). Hence the optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad 1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

The reason why I have this optimization problem

The reason why I have this optimization problem is that I am trying to solve a cost optimization problem for a distributed storage system, constant $b$ here is the cost of transferring the data object among different servers. $f(t)$ here is the pdf of the storage duration of the data copy in a server (and storage cost is proportional to the storage duration, here we assume that the storage cost rate is 1 per unit of time), and after time $t$ the data will be dropped in the server so that there will be no storage cost anymore for our server. So our algorithm is something like a randomized online algorithm, whose target is to find the optimal pdf $f(t)$ to minimized the overall cost of storing and transferring data in the system. Through some simplifications of my modelling, I finally derive this functional optimization problem, but do not know what to do next.

This problem has infinitely many constraints (i.e., for each continuous value $x\in [0,\infty)$, we have constraints like the first and the second constraint), and this is the mainly obstacle of solving this optimization problem.

I would like to find an optimal probability density function (PDF) $f$. Given $b$,

$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\int_{0}^{x}f(t)dt \leq C, \text{ for any } x \in [0,\infty) \\\\ & \frac{1}{b} \left(\displaystyle\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt \right) \leq C, \text{ for any }x\in(b,\infty) \\\\& \displaystyle\int_{0}^{\infty}f(t)dt=1, \\ & f(t)\geq 0, \ \text{for any $t\geq 0$.}\end{array} $$

I do not know how to solve it. Are there any systematic ways or general ideas of dealing with such a functional optimization problem?

My idea

Here I can introduce another optimization variable $B\geq b$ such that the probability density is restricted within the range of $[0,B]$, i.e. $\int_{0}^{B}f(t)dt=1$. Then the original optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C, \text{ for any }x\in(b,B]$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

$\quad f(t)\geq 0, \ \text{for any $t\geq 0$.}$

We take a look at the second constraint $\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C$. Taking the derivative of the left hand side with respect to x, we find that its derivative is $f(x)\geq 0$, which implies that the left hand side is non-decreasing with respect to $x\in (b,B]$. Hence the second constraint can be reduced to $1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$ (by taking $x=B$). Hence the optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad 1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

$\quad f(t)\geq 0, \ \text{for any $t\geq 0$.}$

The reason why I have this optimization problem

The reason why I have this optimization problem is that I am trying to solve a cost optimization problem for a distributed storage system, constant $b$ here is the cost of transferring the data object among different servers. $f(t)$ here is the pdf of the storage duration of the data copy in a server (and storage cost is proportional to the storage duration, here we assume that the storage cost rate is 1 per unit of time), and after time $t$ the data will be dropped in the server so that there will be no storage cost anymore for our server. So our algorithm is something like a randomized online algorithm, whose target is to find the optimal pdf $f(t)$ to minimized the overall cost of storing and transferring data in the system. Through some simplifications of my modelling, I finally derive this functional optimization problem, but do not know what to do next.

This problem has infinitely many constraints (i.e., for each continuous value $x\in [0,\infty)$, we have constraints like the first and the second constraint), and this is the mainly obstacle of solving this optimization problem.

My opimization problem is related to finding an optimal probability density function (PDF) $f$ to minimize the following objective.

$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b\cdot\int_{0}^{x}f(t)dt}{x} \leq C, \text{ for any } x \in [0,\infty) \\ & \quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt}{b}\leq C, \text{ for any }x\in(b,\infty) \\ & \int_{0}^{\infty}f(t)dt=1\end{array} $$

where $b$ is a known constant.

I want to find the optimal PDF to minimize $C$, but do not know how to solve it. Are there any systematic ways or general ideas of dealing with such a functional optimization problem?

$\textbf{My idea:}$
Here I can introduce another optimization variable $B\geq b$ such that the probability density is restricted within the range of $[0,B]$, i.e. $\int_{0}^{B}f(t)dt=1$. Then the original optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C, \text{ for any }x\in(b,B]$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

We take a look at the second constraint $\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C$. Taking the derivative of the left hand side with respect to x, we find that its derivative is $f(x)\geq 0$, which implies that the left hand side is non-decreasing with respect to $x\in (b,B]$. Hence the second constraint can be reduced to $1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$ (by taking $x=B$). Hence the optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad 1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

$\textbf{The reason why I have this optimization problem:}$ The reason why I have this optimization problem is that I am trying to solve a cost optimization problem for a distributed storage system, constant $b$ here is the cost of transferring the data object among different servers. $f(t)$ here is the pdf of the storage duration of the data copy in a server (and storage cost is proportional to the storage duration, here we assume that the storage cost rate is 1 per unit of time), and after time $t$ the data will be dropped in the server so that there will be no storage cost anymore for our server. So our algorithm is something like a randomized online algorithm, whose target is to find the optimal pdf $f(t)$ to minimized the overall cost of storing and transferring data in the system. Through some simplifications of my modelling, I finally derive this functional optimization problem, but do not know what to do next.

This problem has infinitely many constraints (i.e., for each continuous value $x\in [0,\infty)$, we have constraints like the first and the second constraint), and this is the mainly obstacle of solving this optimization problem.

I would like to find an optimal probability density function (PDF) $f$. Given $b$,

$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\int_{0}^{x}f(t)dt \leq C, \text{ for any } x \in [0,\infty) \\\\ & \frac{1}{b} \left(\displaystyle\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt \right) \leq C, \text{ for any }x\in(b,\infty) \\\\& \displaystyle\int_{0}^{\infty}f(t)dt=1\end{array} $$

I do not know how to solve it. Are there any systematic ways or general ideas of dealing with such a functional optimization problem?

My idea

Here I can introduce another optimization variable $B\geq b$ such that the probability density is restricted within the range of $[0,B]$, i.e. $\int_{0}^{B}f(t)dt=1$. Then the original optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C, \text{ for any }x\in(b,B]$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

We take a look at the second constraint $\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C$. Taking the derivative of the left hand side with respect to x, we find that its derivative is $f(x)\geq 0$, which implies that the left hand side is non-decreasing with respect to $x\in (b,B]$. Hence the second constraint can be reduced to $1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$ (by taking $x=B$). Hence the optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad 1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

The reason why I have this optimization problem

The reason why I have this optimization problem is that I am trying to solve a cost optimization problem for a distributed storage system, constant $b$ here is the cost of transferring the data object among different servers. $f(t)$ here is the pdf of the storage duration of the data copy in a server (and storage cost is proportional to the storage duration, here we assume that the storage cost rate is 1 per unit of time), and after time $t$ the data will be dropped in the server so that there will be no storage cost anymore for our server. So our algorithm is something like a randomized online algorithm, whose target is to find the optimal pdf $f(t)$ to minimized the overall cost of storing and transferring data in the system. Through some simplifications of my modelling, I finally derive this functional optimization problem, but do not know what to do next.

This problem has infinitely many constraints (i.e., for each continuous value $x\in [0,\infty)$, we have constraints like the first and the second constraint), and this is the mainly obstacle of solving this optimization problem.

My opimization problem is related to finding an optimal probability density function (pdf) $f(t)$ to minimize the following objective.

$\min\limits_{f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,\infty)$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt}{b}\leq C, \text{ for any }x\in(b,\infty)$

$\quad\int_{0}^{\infty}f(t)dt=1$

Here $b$ is a known constant.

I want to find the optimal pdf to minimize $C$, but do not know how to solve it. Are there any systematic ways or general ideas of dealing with such a functional optimization problem?

$\textbf{My idea:}$
Here I can introduce another optimization variable $B\geq b$ such that the probability density is restricted within the range of $[0,B]$, i.e. $\int_{0}^{B}f(t)dt=1$. Then the original optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C, \text{ for any }x\in(b,B]$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

We take a look at the second constraint $\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C$. Taking the derivative of the left hand side with respect to x, we find that its derivative is $f(x)\geq 0$, which implies that the left hand side is non-decreasing with respect to $x\in (b,B]$. Hence the second constraint can be reduced to $1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$ (by taking $x=B$). Hence the optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad 1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

$\textbf{The reason why I have this optimization problem:}$ The reason why I have this optimization problem is that I am trying to solve a cost optimization problem for a distributed storage system, constant $b$ here is the cost of transferring the data object among different servers. $f(t)$ here is the pdf of the storage duration of the data copy in a server (and storage cost is proportional to the storage duration, here we assume that the storage cost rate is 1 per unit of time), and after time $t$ the data will be dropped in the server so that there will be no storage cost anymore for our server. So our algorithm is something like a randomized online algorithm, whose target is to find the optimal pdf $f(t)$ to minimized the overall cost of storing and transferring data in the system. Through some simplifications of my modelling, I finally derive this functional optimization problem, but do not know what to do next.
This problem has infinitely many constraints (i.e., for each continuous value $x\in [0,\infty)$, we have constraints like the first and the second constraint), and this is the mainly obstacle of solving this optimization problem.

My opimization problem is related to finding an optimal probability density function (PDF) $f$ to minimize the following objective.

$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b\cdot\int_{0}^{x}f(t)dt}{x} \leq C, \text{ for any } x \in [0,\infty) \\ & \quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt}{b}\leq C, \text{ for any }x\in(b,\infty) \\ & \int_{0}^{\infty}f(t)dt=1\end{array} $$

where $b$ is a known constant.

I want to find the optimal PDF to minimize $C$, but do not know how to solve it. Are there any systematic ways or general ideas of dealing with such a functional optimization problem?

$\textbf{My idea:}$
Here I can introduce another optimization variable $B\geq b$ such that the probability density is restricted within the range of $[0,B]$, i.e. $\int_{0}^{B}f(t)dt=1$. Then the original optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C, \text{ for any }x\in(b,B]$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

We take a look at the second constraint $\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{B}f(t)dt}{b}\leq C$. Taking the derivative of the left hand side with respect to x, we find that its derivative is $f(x)\geq 0$, which implies that the left hand side is non-decreasing with respect to $x\in (b,B]$. Hence the second constraint can be reduced to $1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$ (by taking $x=B$). Hence the optimization problem can be rewritten as

$\min\limits_{B,f(t)} C$

s.t. $1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in[0,B]$

$\quad 1+\frac{\int_{0}^{B}t\cdot f(t)dt}{b}\leq C$

$\quad\int_{0}^{B}f(t)dt=1$,

$\quad B\geq b$

$\textbf{The reason why I have this optimization problem:}$ The reason why I have this optimization problem is that I am trying to solve a cost optimization problem for a distributed storage system, constant $b$ here is the cost of transferring the data object among different servers. $f(t)$ here is the pdf of the storage duration of the data copy in a server (and storage cost is proportional to the storage duration, here we assume that the storage cost rate is 1 per unit of time), and after time $t$ the data will be dropped in the server so that there will be no storage cost anymore for our server. So our algorithm is something like a randomized online algorithm, whose target is to find the optimal pdf $f(t)$ to minimized the overall cost of storing and transferring data in the system. Through some simplifications of my modelling, I finally derive this functional optimization problem, but do not know what to do next.

This problem has infinitely many constraints (i.e., for each continuous value $x\in [0,\infty)$, we have constraints like the first and the second constraint), and this is the mainly obstacle of solving this optimization problem.