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MathGroup Archive 1999

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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.


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