Continuous optimization - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:21:52Z http://mathoverflow.net/feeds/question/36078 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36078/continuous-optimization Continuous optimization bugnotme 2010-08-19T12:57:10Z 2010-08-20T23:16:08Z <p>I'm interested in the solution to the following problem:</p> <p>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 $t\rightarrow\infty$?</p> <p>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.</p> <p>What is the field that studies this type of continual optimization problem?</p> http://mathoverflow.net/questions/36078/continuous-optimization/36089#36089 Answer by Gilead for Continuous optimization Gilead 2010-08-19T15:26:24Z 2010-08-19T15:43:51Z <p>This is how I would develop the formulation (conceptually). </p> <ul> <li><p>Investment costs, $Q$: $$Q = \sum_{i} m_{i} c_{i}$$</p></li> <li><p>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 <code>if</code> logical condition: $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$\delta_{i} = 1, \text{ if } t \geq t_{i}$$ $$\delta_{i} = 0, \text{ if } t &lt; t_{i}$$ 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 $t$.</p></li> </ul> <hr> <p>In summary, your problem can be represented as follows:</p> <p>$$\max_{m_{i}} (R - Q)$$ s.t. $$Q = \sum_{i} m_{i} c_{i}$$ $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$Q \leq C$$ $$m_{i} \geq 0\quad \forall i$$</p> <p>where $C, c_{i}, r_{i}, \delta_{i}, t, t_{i}$ are parameters.</p> <p>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)</p> <p>For $t \rightarrow \infty$ (i.e. all the investments have reached maturity), simply set all $\delta_{i} = 1$.</p> http://mathoverflow.net/questions/36078/continuous-optimization/36092#36092 Answer by KalEl for Continuous optimization KalEl 2010-08-19T16:16:42Z 2010-08-19T16:16:42Z <p>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 $$\frac{(1+r_i)-c_i}{t_i}$$</p> <p>Consequently, minimize the quantity over all $i$ and invest all your money there.</p>