Difference between revisions of "ApCoCoA-1:GLPK.L01PSolve"

From ApCoCoAWiki
Line 7: Line 7:
 
<description>
 
<description>
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 +
 +
<par/>
 +
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It operates in two stages. Firstly, it models the problem of finding one solution of given polynomial system into a mixed integer linear programming problem. For this the system is first converted into an equivalent CNF form and then the CNF form is converted into a system of equalities and inequalities. Secondly, the mixed integer linear programming model is solved using glpk.
 +
  
 
<itemize>
 
<itemize>
 
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item>  
 
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item>  
 
<item>@param <em>CuttingNumber</em>: Maximal support-length of the linear polynomials for conversion to CNF. The possible value could be from 3 to 6. </item>
 
<item>@param <em>CuttingNumber</em>: Maximal support-length of the linear polynomials for conversion to CNF. The possible value could be from 3 to 6. </item>
−
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; and 2 - Double Linear Partner;</item>
+
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item>
 
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item>
 
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item>
 
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item>
 
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item>
Line 17: Line 21:
  
 
<example>
 
<example>
−
-- We want to maximize the Function y = - 1/2x,  
+
Use Z/(2)[x[1..4]];
−
-- with the two conditions y ≤ 6 - 3/4x and y ≥ 1 - x and the bounds 0 ≤ x ≤ 6 and 1/3 ≤ y ≤ 4.
+
F:=[
 +
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1,
 +
    x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1,  
 +
    x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1,
 +
    x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1
 +
    ];
  
−
-- We prename the input of GLPK.MIPSolve-function.
+
CuttingNumber:=6;
−
Use S::=QQ[x,y];
+
QStrategy:=0;
−
OF := 1/2x + y;
+
CStrategy:=0;
−
LE := [3/4x + y - 6];
+
MinMax:=<quotes>Max</quotes>;
−
GE := [x + y - 1];
 
−
Bounds:=[[0,6], [1/3,4]];
 
−
IntNum:=[x,y];
 
  
 
-- Then we compute the solution with
 
-- Then we compute the solution with
−
GLPK.MIPSolve(OF, [], LE, GE, Bounds, IntNum, [], <quotes>Max</quotes>);
+
 
 +
GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax)
 +
 
 +
-- The result will be the following:
  
  
−
-- And we achieve:
 
−
Solution Status: INTEGER OPTIMAL
 
−
Value of objective function: 5
 
−
[x - 2, y - 4]
 
 
</example>
 
</example>
  

Revision as of 14:57, 7 December 2010

GLPK.L01PSolve

Solve a system of polynomial equations over F_2 for one solution in F_2^n.

Syntax

GLPK.L01PSolve(F:LIST, CuttingNumber:INT, QStrategy:INT, CStrategy:INT, MinMax:STRING)

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.


This function finds one solution in F_2^n of a system of polynomial equations over the field F_2. It operates in two stages. Firstly, it models the problem of finding one solution of given polynomial system into a mixed integer linear programming problem. For this the system is first converted into an equivalent CNF form and then the CNF form is converted into a system of equalities and inequalities. Secondly, the mixed integer linear programming model is solved using glpk.


  • @param F: A List containing the polynomials of the given system.

  • @param CuttingNumber: Maximal support-length of the linear polynomials for conversion to CNF. The possible value could be from 3 to 6.

  • @param QStrategy: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;

  • @param CStrategy: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;

  • @param MinMax: Optimization direction i.e. minimization ("Min") or maximization ("Max").

Example

Use Z/(2)[x[1..4]];
F:=[
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, 
    x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1
    ];

CuttingNumber:=6;
QStrategy:=0;
CStrategy:=0;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax)

-- The result will be the following:





Retrieved from "http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.L01PSolve&oldid=11448"