Continuous optimization - MathOverflow

Continuous optimization

2010-08-19T12:57:10Z

I'm interested in the solution to the following problem:

I have initial capital $C$ which I can invest into $M$ classes of resources, each unit of a class $m_i$ matures at time $t_i$, has a return of $r_i$ and a cost $c_i$. After the asset matures it the proceeds can be re-invested. What is the optimal strategy to invest $C$ in terms of profit at time $t$ and for $t\rightarrow\infty$?

I am interested in both cases where $m_i$ is in the non-negative reals and the case when $m_i$ is a member of the non-negative integers.

What is the field that studies this type of continual optimization problem?

Answer by Gilead for Continuous optimization

2010-08-19T15:26:24Z

This is how I would develop the formulation (conceptually).

Investment costs, $Q$: $$Q = \sum_{i} m_{i} c_{i}$$
Investment returns for $i$ at time $t$: we know that the return for $i$ is $m_{i} r_{i}$ if $t \geq t_{i}$, otherwise it is 0. To model this if logical condition: $$ R = \sum_{i} \delta_{i} m_{i} r_{i} $$ $$ \delta_{i} = 1, \text{ if } t \geq t_{i} $$ $$ \delta_{i} = 0, \text{ if } t < t_{i} $$ where $R$ = overall returns. In this case, $t$ and $t_{i}$ are parameters, therefore $\delta_{i}$ are parameters too, not binary variables. They can be pre-calculated for a specified $t$.

In summary, your problem can be represented as follows:

$$ \max_{m_{i}} (R - Q)$$ s.t. $$ Q = \sum_{i} m_{i} c_{i}$$ $$ R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$ Q \leq C $$ $$ m_{i} \geq 0\quad \forall i$$

where $C, c_{i}, r_{i}, \delta_{i}, t, t_{i}$ are parameters.

So, if $m_{i} \in \mathbb{R}$, this would be a linear program. If $m_{i} \in \mathbb{N}$, this becomes a mixed-integer linear program. (Note: for tractability in the integer case, you may need to specify a reasonably small upper-bound for $m_{i}$ or use partial integer variables)

For $t \rightarrow \infty$ (i.e. all the investments have reached maturity), simply set all $\delta_{i} = 1$.

Answer by KalEl for Continuous optimization

2010-08-19T16:16:42Z

Assuming $c_i$ is the cost for investment per unit till maturity. Then for a particular class $m_i$, the gain per unit investment per time is $$\frac{(1+r_i)-c_i}{t_i}$$

Consequently, minimize the quantity over all $i$ and invest all your money there.