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Consider the following problem $$\min_{x,y}f(x)g(y)\\ \text{s.t. } x\in\mathcal{X},y\in\mathcal{Y}$$

where $\mathcal{X},\mathcal{Y}$ are two convex sets, $f(x),g(y)>0$. If I want to solve this problem, can I solve the following problem, equivalently? $$\min_{x,y}\log f(x)+\log g(y)\\ \text{s.t. } x\in\mathcal{X},y\in\mathcal{Y}$$

Michael Hardy
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Dave
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1 Answers1

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Yes, the problem are equivalent.

Minimizing $h(x) > 0$ is equivalent to minimizing $\log h(x)$.

To recover the objective function of the original problem, we just have to evaluate the exponential function on the reduced problem.

Siong Thye Goh
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  • Thank you. By the way, does this mean that I can respectively solve the following two sub-problems? $$\min_{x}\log f(x)\ s.t.x\in\mathcal{X}$$ and $$\min_{y}\log g(y)\ s.t.y\in\mathcal{Y}$$ – Dave Nov 24 '17 at 02:48
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    Yup, this is a separable problem. Optimization of $x$ and $y$ can be performed separately, and in fact, in parallel. – Siong Thye Goh Nov 24 '17 at 02:53
  • Dear @Siong Thye Goh, Thanks a lot. If the objective function has a more general form, like as $f(x)+g(x)h(y)$, then can the following problem still be separable? $$\min_{x,y}f(x)+h(x)g(y)\ \text{s.t. } x\in\mathcal{X},y\in\mathcal{Y}$$ – Dave Nov 25 '17 at 08:13
  • I don't see how to separate $x$ from $y$ in the objective function. – Siong Thye Goh Nov 25 '17 at 08:34
  • For example, I can minimize $f(x)+\log h(x)+\log g(y)$ over the constraint set. Does that work? – Dave Nov 25 '17 at 09:35
  • Nope, it doesn't. We can't apply $\log$ to part of the objective function and still ensure that it is equivalent. – Siong Thye Goh Nov 25 '17 at 09:48