How to solve this optimization problem with abs object function?

 0 Helo, every one. May I ask for help about how to solve this problem? \begin{align} max_{x_i} \quad |\sum_{i=1}^{4} a_i x_i | \\ s.t. \quad \sum_{i=1}^4 x_i^2=1\end{align} The goal is to find the optimal $$x_i$$, the $$a_i$$ is known. Thank you very much. asked 26 Sep '14, 06:30 Lee 13●3 accept rate: 0%

 2 Intuitively, the optimal $$x$$ will be proportional to $$a$$. A hint to get you started, in case this is homework: first ignore the absolute value and use the method of Lagrange multipliers to show that $$x_i = a_i / \sqrt(\sum_j a_j^2)$$. By the way, there is nothing special about 4 here; you can replace it with arbitrary $$n$$. answered 27 Sep '14, 12:36 Rob Pratt 1.2k●2●6 accept rate: 28% That formula for x guarantees objective value 1, which is not likely to be optimal. (30 Sep '14, 10:43) Paul Rubin ♦♦ The corresponding optimal objective is $$\sqrt(\sum_j a_j^2)$$, not 1. (30 Sep '14, 13:59) Rob Pratt Sorry, you're right about the sum. Oops. (02 Oct '14, 11:22) Paul Rubin ♦♦
 1 This looks like a homework problem, so I'll just give a hint: there is an "obvious" optimal solution that does not require the use of Lagrange multipliers. It relies on the inequality $|\sum_{i} z_{i}| \le \sum_{i} |z_{i}|.$ answered 30 Sep '14, 10:50 Paul Rubin ♦♦ 14.6k●4●12 accept rate: 19%
 1 Here is yet another way to look at it. Assume we have an optimal solution. If sum ai xi is negative, then we can multiply x by -1 and get another optimal solution. It shows that we can get rid of the absolute value in the objective. In that case the objective is the inner product of vectors a and x, given a is fixed and the length of x is fixed. That inner product is maximal when the two vectors are collinear. answered 07 Oct '14, 07:16 jfpuget 2.5k●3●10 accept rate: 8%
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