# How to linearize this optimization problem

 0 Maximize $$\prod_{t=1}^{T}{(1+p_th_t)}^{c_tw_t}$$ subject to: C1: $$\sum_{t=1}^{T}c_tp_t\le P_{\rm{Max}}$$ Here $$w_t>0$$ and $$h_t>0$$ are given constants and they are positive. $$T$$ is an integer multiple of $$N$$. asked 21 Feb '15, 00:34 dip 31●2●12 accept rate: 0% $$z_t = c_t w_t$$ is a linear constraint, and just does what you need. (21 Feb '15, 01:15) fbahr ♦ 1 There must be additional constraints to your problem. Indeed, if ht > 0, then your objective is an increasing function of pt. What prevents pt to be as large as we want? If ht < 0, then pt < -1/ht or wt is non negative integer. If you don't, then your objective is ill defined (23 Feb '15, 08:04) jfpuget @dip You did not address the case where ht < 0 (23 Feb '15, 10:41) jfpuget

 1 You can get rid of the ct variables altogether, thanks to the trick mcg gave in the cvx blog @Mark_Stone pointed to. Indeed, (1 + pt.ht)^(ct.wt) = (1 + pt.ct.ht) ^ wt Check for ct = 0 and for ct = 1 to see why. Then all yor constraints involve the products ct.pt Given pt is continuous, pt can range over all the values ct.pt can take. answered 23 Feb '15, 11:02 jfpuget 2.5k●3●10 accept rate: 8% Please answer my questions. 1) Are the ht positive? 2) Are there additional constraints besides the one we see here? I am asking because if ht >0 and there are no other constraint than the one with Pmax then your problem has an analytical solution IMHO. This solution is a derivative of geometric mean as @Mark L Stone suggested. (24 Feb '15, 04:57) jfpuget @jfpuget, Thanks for your concern. I have updated my problem. And to answer your quesions, 1) Yes, h_t and w_t are strictly positive. 2) There is one more constraint, please note C3. (24 Feb '15, 07:29) dip
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