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Re: How to remove unneeded constraints

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87940] Re: How to remove unneeded constraints
  • From: Szabolcs Horvát <szhorvat at gmail.com>
  • Date: Mon, 21 Apr 2008 03:22:51 -0400 (EDT)
  • Organization: University of Bergen
  • References: <fueeo7$b82$1@smc.vnet.net>

Kristian Schmidt wrote:
> Hello
> 
> The problem I am working on is a pretty large problem, which I am solving by dividing it into a lot of subproblems.
> 
> As such, I have a large list of constraints that apply to the entire problem, but are not relevant for each individual subproblem.
> 
> This becomes a problem when I use Refine, as it takes a very long time when you have a lot of conditions, even though the conditions don't pertain to the problem. Example:
> In[25]:= Table[Timing@Refine[p>q,Map[Subscript[x,#]>0&,Range[i]]],{i,1000,5000,1000}]
> Out[25]= {{0.547,p>q},{1.797,p>q},{3.875,p>q},{8.594,p>q},{10.468,p>q}}
> 
> And it only gets worse.
> 
> However, for each individual call to Refine I make, I only need a small subset of the total constraints.
> 
> So what I want to do is something like this:
> expr = some expression of n different variables;
> cond = DeleteCases[totalConstraints, all cases which do not contain a variable from expr];
> result = Refine[expr,cond];
> 
> I have no idea how to construct a pattern powerful enough to do what is required for the DeleteCases call, though.
> 
> All of the variables are of the form Subscript[s,_,_,_] or Subscript[b,_,_] and the conditions can also contain expressions of several variables.
> 
> I'd be most grateful for any help. Thank you.
> 

The following should work if the constraints are just a linear 
concatenation of equations/inequalities:

In[1]:= expr = Subscript[a, 1] + Subscript[a, 2] Subscript[a, 3]
Out[1]= Subscript[a, 1] + Subscript[a, 2] Subscript[a, 3]

In[2]:= constraints = And @@ (Subscript[a, #] > 0 & /@ Range[5000]);

In[3]:=
  Timing[
   filteredConstraints = With[
     {relevantVariables = Union@Cases[expr, Subscript[__], Infinity]},
     Select[constraints, ! FreeQ[#, Alternatives @@ relevantVariables] &]
    ]
  ]

Out[3]= {0.047,
  Subscript[a, 1] > 0 && Subscript[a, 2] > 0 && Subscript[a, 3] > 0}

In[4]:= Refine[expr > 0, filteredConstraints] // Timing
Out[4]= {0., True}

In[5]:= Refine[expr > 0, constraints] // Timing
Out[5]= {10.797, True}


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