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I have two optimization problems.

1-) Mean-variance optimization

$J_{MV} = J_M - \gamma J_V$,

where $J_M$ is mean, $J_V$ is the variance term and $\gamma$ is the weight on variance term.

2-) The worst-case optimization:

$J_{WC} = max_u min_\theta J(u,\theta_i)$

where $\theta_i$ is the $i^{th}$ randomly chosen member in the uncertainty ensemble (scenario or sample).

I now obtained two optimal solution, $u^*_{WC}$ and $u^*_{MV}$. Now I apply these inputs to the system and calculate each objective function from each $\theta_i$. The objective function looks to have a normal distribution.

Question: I observed the results for both mean-variance (approach 1) and WC (approach 2) to give me the very similar results. For both cases, the worst-case is increased but there is a big decrease in the best case. Can anyone explain, why the result for both the approaches are same and is it because objective function has a normal distribution?

I hope my question is clear. Thanks a lot in advance.

Mohsin
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  • Isn't it strange if it is really a normal distribution (with possibly arbitrary large (and small) objective)? – borisd Mar 04 '15 at 16:34
  • It depends upon the uncertainty ensemble that you have consider. The normal distribution of objective is achieved when I only optimized the mean value, i.e., $J_M$. Later I use these two strategies, i.e., mean-variance and wrost case and obtain $u^_{wc}, u^_{MV}$, they result me in the same output. – Mohsin Mar 05 '15 at 09:38

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