Re: strict inequalities

*To*: mathgroup at smc.vnet.net*Subject*: [mg19679] Re: strict inequalities*From*: Adam Strzebonski <adams at wolfram.com>*Date*: Thu, 9 Sep 1999 02:19:49 -0400*Organization*: Wolfram Research, Inc.*References*: <7qkq26$h30@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Joerg Rudolf Mueller wrote: > > Hello Mathematica-Users > > Is there a possibility to solve a set of (linear) > equations and to find a solution that satisfies certain > strict inequalities (e.g. x<y) AND non-strict > inequalities (e.g. x<=z)? In version 4 it can be done using InequalitySolve (or Experimental`CylindricalAlgebraicDecomposition). In[1]:= <<Algebra`InequalitySolve` In[2]:= InequalitySolve[x^2+y^2<4 && x+y<=1 && x-y^3==1, {x, y}] 2 4 6 Out[2]= Root[-63 - 2 #1 + 49 #1 - 12 #1 + #1 & , 1] < x <= 1 && 3 > y == Root[1 - x + #1 & , 1] > Is there a possibility to solve an optimization-problem > with strict AND non-strict inequalities? In Mathematica V4 you can use Experimental`Minimize or Experimental`Infimum. Experimental`Minimize[f, ineqs, vars] gives the infimum of f on the solution set of ineqs and, if possible, a point at which the infimum is attained. Experimental`Infimum[f, ineqs, vars] gives the infimum of f on the solution set of ineqs. f should be an algebraic function in vars, and ineqs should be a logical combination of algebraic equations and inequalities in vars. Here are a few examples. In[1]:= <<Experimental` In[2]:= Minimize[-(x^2+y^2), 1 < x^4+y^4 <= 2, {x, y}] Out[2]= {-2, {y -> -1, x -> -1}} In[3]:= Minimize[x^2+y^2, 1 < x^4+y^4 <= 2, {x, y}] Out[3]= {1, {x -> Indeterminate, y -> Indeterminate}} Since the first inequality is strict, the infimum is not attained in the set of points satisfying the constraints. In[5]:= Infimum[x^2+y^2, 1 < x^4+y^4 <= 2, {x, y}] Out[5]= 1 The computation of Infimum may be significantly faster than the computation of Minimize, especially if all constraints are strict inequalities. Best Regards, Adam Strzebonski Wolfram Research > > If you know about Farkas' "alternativ theorems" > - in German we call it "Alternativsitze" - you'll > know that it's necessary to attend the strictness. > > "ConstrainedMin/Max" unfortunately doesn't work to my contentedness. > (I need s.th. like ConstrainedMin/Max that doesn't ignore the strictness > > of inequalities). > > In using "SemialgebraicComponents" (in packet > Algebra`AlgebraicInequalities`) > I can only give strict inequalities - constraints of > the form (x<=z) are not possible here. > > With "InequalitySolve" (in packet Algebra`InequalitySolve`) > I can't solve optimization-problems. > > best regards, Joerg Mueller.