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I have an optimization problem with an uncommon utility: to find a $\beta$ that maximizes

$$ r^{T}\cdot H(X\cdot\beta) $$

where $H()$ is a Heaviside step function as in wiki

$r$ is a vector of size 1000

$X$ is a 1000x50 "tall" matrix

$\beta$ is a vector of size 50

I am familiar with gradient descent, which is how I usually solve an optimization problem. But Heaviside function does not work with gradient descent. So I am wondering if anyone here could shed some light on how to solve such optimization problem.

Thanks

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You can solve the problem via integer linear programming as follows, assuming $r_i \ge 0$ for all $i$. Let $M_i$ be a (small) upper bound on $-(X \cdot \beta)_i$. Let binary decision variable $y_i$ indicate whether $(X \cdot \beta)_i \ge 0$. The problem is to maximize $$\sum_{i=1}^{1000} r_i y_i$$ subject to $$-(X \cdot \beta)_i \le M_i(1 - y_i)$$ for all $i$. This "big-M" constraint enforces $y_i=1 \implies (X \cdot \beta)_i \ge 0$.

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  • $\begingroup$ HI Rob, thanks for answering this. but if y_{i} = 0, then X*beta>= - 1 * M_{i}; which at least overlaps with the case of y_{i} = 1 , right? $\endgroup$ – user152503 Jul 25 at 9:47
  • $\begingroup$ Yes, $y_i=0$ forces only a redundant constraint. As long as the “reward” $r_i\ge 0$, this big-M constraint is sufficient. $\endgroup$ – RobPratt Jul 25 at 13:28

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