MaplePrimes - comments on Post, convexity

MaplePrimes - comments on Post, convexity http://www.mapleprimes.com/posts/36587-Convexity en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 17:39:47 GMT Wed, 10 Jun 2026 17:39:47 GMT The latest comments added to the Post, convexity http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, convexity http://www.mapleprimes.com/posts/36587-Convexity why? http://www.mapleprimes.com/posts/36587-Convexity?ref=Feed:MaplePrimes:convexity:Comments#comment62796 . The latest comments added to the Post, convexity 62796 Tue, 13 Oct 2009 23:40:39 Z Axel Vogt Axel Vogt why http://www.mapleprimes.com/posts/36587-Convexity?ref=Feed:MaplePrimes:convexity:Comments#comment62797 E.g. for the <a href="http://en.wikipedia.org/wiki/Legendre_transform">Legendre transformation</a>. The latest comments added to the Post, convexity 62797 Wed, 14 Oct 2009 00:18:48 Z jakubi jakubi convexity http://www.mapleprimes.com/posts/36587-Convexity?ref=Feed:MaplePrimes:convexity:Comments#comment62798 There's a lot of scope for improving this code below. (A probabilistic fallback might also be worthwhile. I suspect that a fallback that using testeq & signum might not work; testeq is rather limited.) Note that `is` can depend upon Digits. <pre> > isconvex := proc(expr,A::realcons:=-infinity,B::realcons:=infinity) > local a,b,t,x,use_expr,res; > x := indets(expr,name) minus {constants}; > if nops(x) = 1 then > x := op(x); > else error; > end if; > # For more than one variable, one could test > # positive-definiteness of the Hessian. > res := is( diff(expr,x,x) >=0 ) > assuming x>=A, x<=B; > if res<>FAIL then > return res; > else > userinfo(1,'isconvex',`falling back to the definition`); > # Is it better to expand, or simplify, or put > # Normalizer calls around the two-arg evals or > # their difference, oder...? > use_expr:=expand(expr); > is( eval(use_expr,x=(1-t)*a+t*b) <= > (1-t)*eval(use_expr,x=a)+t*eval(use_expr,x=b) ) > assuming a<=b, t>=0, t<=1, b<=B, a>=A; > #else > ##Is global optimization another fallback? > end if; > end proc: > # note: this shows that none of the examples > # below had to fall back to the defn after failing > # a 2nd deriv test. But many such will exist. > infolevel[isconvex]:=1: > isconvex( y^2 ); true > > isconvex( y^3 ); false > isconvex( y^3, 0 ); true > > isconvex( (y-3)^2 ); true > > isconvex( (y-3)^4 ); true > > isconvex( exp(y) ); true > isconvex( -exp(y) ); false > > isconvex( -sin(y), 0, Pi ); true </pre> acer The latest comments added to the Post, convexity 62798 Thu, 15 Oct 2009 01:10:22 Z acer acer parameters http://www.mapleprimes.com/posts/36587-Convexity?ref=Feed:MaplePrimes:convexity:Comments#comment62799 Another issue would be parameters, on which assumptions may hold as here: <pre> isconvex( (x-a)^2 ) assuming a::real; Error, (in assuming) when calling 'isconvex'. Received: 'testing against an invalid type' </pre> Probably, it would be better to state explicitly the variable as another argument. The latest comments added to the Post, convexity 62799 Thu, 15 Oct 2009 01:35:46 Z jakubi jakubi abs http://www.mapleprimes.com/posts/36587-Convexity?ref=Feed:MaplePrimes:convexity:Comments#comment62800 Fails for abs though, <pre> isconvex(abs(x)); isconvex: falling back to the definition FAIL isconvex(abs(x^3)); isconvex: falling back to the definition FAIL </pre> And while replacing abs(x) with sqrt(x^2) gives the correct answer, <pre> isconvex(sqrt(x^2)); true </pre> the following is wrong, <pre> isconvex(-sqrt(x^2)); true </pre> That was caused by the following Maple feature <pre> diff(sqrt(x^2),x,x); 2 x 1 - ------- + ------- 2 3/2 2 1/2 (x ) (x ) normal(%); 0 is(%%<=0); true is(%%%=0); true </pre> Alec The latest comments added to the Post, convexity 62800 Thu, 15 Oct 2009 02:38:39 Z alec alec also wrong http://www.mapleprimes.com/posts/36587-Convexity?ref=Feed:MaplePrimes:convexity:Comments#comment62801 I think inequality should be strict, otherwise: <pre> isconvex(x); true </pre> The latest comments added to the Post, convexity 62801 Thu, 15 Oct 2009 02:43:48 Z jakubi jakubi