Monthly Archives: June 2011

A simple optimization problem I don’t know how to solve (from DCP)

Inspired by the recent 8F workshop, I’m trying to write up theory challenges arising from global health. And I’m trying to do it with less background research, because avoiding foolishness is a recipe for silence.

This is the what I called the “simplest open problem in DCP optimization” in a recent post about DCP (Disease Control Priorities), but with more reflection, I should temper that claim. I’m not sure it is the simplest. I’m not sure it is an open problem. And I’m pretty sure that if we solve it, the DCP optimizers will come back with something more complicated.

But it is a nice, clean problem to start with. I’m calling it “Fully Stochastic Knapsack”. It looks just like the plain, old knapsack problem:
\max \bigg\{ \sum_{i=1}^n v_ix_i \qquad s.t. \quad \sum_{i=1}^n w_ix_i \leq W, \quad x_i \in \{0,1\} \bigg\}
The fully stochastic part is that everything that usually would be input data is now a probability distribution, and the parameters of the distribution are the input data.

This makes even deciding what to maximize a challenge. I was visiting the UW Industrial Engineering Dept yesterday, and Zelda Zabinsky pointed me to this nice INFORMS tutorial by Terry Rockafeller on “coherent approaches” to this.

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Filed under combinatorial optimization, global health