Is this function convex?
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From: W^3 on 28 May 2010 14:28 In article <ba752028-85d1-46ca-8044-1c4bb7d7d5d3(a)z17g2000vbd.googlegroups.com>, ben <bwcrain(a)gmail.com> wrote: > Let x1, x2,...xm be positive variables (actually, >= 1). The function > is: > > 1/(x1*x2*...*xm). > > Intuitively it's gotta be convex, but I can't prove it. Tried every > test for convexity I can find, including proof by induction. But they > become intractable (at least for me). Any suggestions? Assuming the obvious domains, log(1/x) is convex, so each log(1/x_k) is convex. A sum of convex functions is convex, as is a convex increasing function of a convex function. Therefore exp(sum log(1/x_k)) = 1/(x1*x2*...*xm) is convex.
From: ben on 28 May 2010 18:14 On May 28, 2:28 pm, W^3 <aderamey.a...(a)comcast.net> wrote: > In article > <ba752028-85d1-46ca-8044-1c4bb7d7d...(a)z17g2000vbd.googlegroups.com>, > > ben <bwcr...(a)gmail.com> wrote: > > Let x1, x2,...xm be positive variables (actually, >= 1). The function > > is: > > > 1/(x1*x2*...*xm). > > > Intuitively it's gotta be convex, but I can't prove it. Tried every > > test for convexity I can find, including proof by induction. But they > > become intractable (at least for me). Any suggestions? > > Assuming the obvious domains, log(1/x) is convex, so each log(1/x_k) > is convex. A sum of convex functions is convex, as is a convex > increasing function of a convex function. Therefore exp(sum > log(1/x_k)) = 1/(x1*x2*...*xm) is convex. Thanx to all my respondents. Makes me feel a bit foolish, now that I see how easy it is, taking logs. But, a lesson learned, not to be forgotten....
From: Niels Diepeveen on 28 May 2010 21:15 ben wrote: > Let x1, x2,...xm be positive variables (actually, >= 1). The function > is: > > 1/(x1*x2*...*xm). > > Intuitively it's gotta be convex, but I can't prove it. Tried every > test for convexity I can find, including proof by induction. But they > become intractable (at least for me). Any suggestions? There is a heuristic that says: If there's a product you can't deal with, try taking the logarithm. In this case 1/(x_1 x_2 x_3 ... x_m) = exp(-log x_1 - log x_2 ... - log x_m) The negative log is convex, so -log x_1 - log x_2 ... is a sum of convex functions, therefore convex. Finally, the composition of a non-decreasing convex function, such as exp, and a convex function is convex. -- Niels Diepeveen
From: Niels Diepeveen on 28 May 2010 21:23 Niels Diepeveen wrote: > ben wrote: > >> Let x1, x2,...xm be positive variables (actually, >= 1). The function >> is: >> >> 1/(x1*x2*...*xm). >> >> Intuitively it's gotta be convex, but I can't prove it. Tried every >> test for convexity I can find, including proof by induction. But they >> become intractable (at least for me). Any suggestions? > > There is a heuristic that says: If there's a product you can't deal > with, try taking the logarithm. > > In this case > 1/(x_1 x_2 x_3 ... x_m) = exp(-log x_1 - log x_2 ... - log x_m) > > The negative log is convex, so > -log x_1 - log x_2 ... > is a sum of convex functions, therefore convex. > > Finally, the composition of a non-decreasing convex function, such as > exp, and a convex function is convex. > Oops. I just saw that someone had already posted the same solution while I wasn't looking. Sorry about that. -- Niels Diepeveen
From: David C. Ullrich on 29 May 2010 06:26
On Fri, 28 May 2010 15:14:53 -0700 (PDT), ben <bwcrain(a)gmail.com> wrote: >On May 28, 2:28�pm, W^3 <aderamey.a...(a)comcast.net> wrote: >> In article >> <ba752028-85d1-46ca-8044-1c4bb7d7d...(a)z17g2000vbd.googlegroups.com>, >> >> �ben <bwcr...(a)gmail.com> wrote: >> > Let x1, x2,...xm be positive variables (actually, >= 1). �The function >> > is: >> >> > 1/(x1*x2*...*xm). >> >> > Intuitively it's gotta be convex, but I can't prove it. Tried every >> > test for convexity I can find, including proof by induction. But they >> > become intractable (at least for me). �Any suggestions? >> >> Assuming the obvious domains, log(1/x) is convex, so each log(1/x_k) >> is convex. A sum of convex functions is convex, as is a convex >> increasing function of a convex function. Therefore exp(sum >> log(1/x_k)) = 1/(x1*x2*...*xm) is convex. > >Thanx to all my respondents. Makes me feel a bit foolish, now that I >see how easy it is, taking logs. Heh - imagine how _I_ feel. Bad luck that the first thing I thought of worked... >But, a lesson learned, not to be >forgotten.... |