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

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(New page: <command> <title>GLPK.MIPSolve</title> <short_description>Solving linear programmes.</short_description> <syntax> GLPK.MIPSolve(Objective_f:POLY, EQ_Poly:LIST, LE_Poly:LIST, GE_Poly:LIST, ...)
 
m (replaced <quotes> tag by real quotes)
 
(9 intermediate revisions by 4 users not shown)
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 +
{{Version|1}}
 
<command>
 
<command>
−
<title>GLPK.MIPSolve</title>
+
<title>GLPK.L01PSolve</title>
−
<short_description>Solving linear programmes.</short_description>
+
<short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description>
 
<syntax>
 
<syntax>
−
GLPK.MIPSolve(Objective_f:POLY, EQ_Poly:LIST, LE_Poly:LIST, GE_Poly:LIST, Bounds:LIST, IntNum:LIST, Binaries:LIST, MinMax:STRING)
+
GLPK.L01PSolve(F:LIST, CuttingNumber:INT, QStrategy:INT, CStrategy:INT, MinMax:STRING):LIST
 
</syntax>
 
</syntax>
 
<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>Objective_f</em>: A linear polynomial which is equivalent to the linear objective function.</item>
+
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item>  
−
<item>@param <em>EQ_Poly</em>: List of linear polynomials, which are equivalent to the equality-part in the list of conditions.</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>LE_Poly</em>: List of linear polynomials, which are equivalent to the lower or equal-part in the list of conditions.</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>GE_Poly</em>: List of linear polynomials, which are equivalent to the greater or equal-part in the list of conditions.</item>
+
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item>
−
<item>@param <em>Bounds</em>: List of lists with two elements. Each List contains the lower and upper bounds for each variable. You can choose between INT or RAT for the type of each bound, if you type in a (empty) string, then it means minus infinity (first place) or plus infinity (second place).</item>
+
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization ("Min") or maximization ("Max").</item>
−
<item>@param <em>IntNum</em>: List of variables, which should be integer. <em>Note</em>: For each variable in this list, the borders get rounded (lower bound: up and upper bound: down). In the case that the lower rounded bound becomes greater then the upper rounded bound, glpk returns: Solution Status: INTEGER UNDEFINED - Value of objective function: 0.</item>
+
<item>@return A list containing a zero of the system F.</item>
−
<item>@param <em>Binaries</em>: List of variables, which should be binaries (0 or 1).</item>
 
−
<item>@param <em>MinMax</em>:  Minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>), that's the question.</item>
 
−
<item>@return List of linear polynomials, the zeros of the polynomials are the points where the optimal value of the objective function is achieved</item>
 
 
</itemize>
 
</itemize>
  
 
<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
 +
    ];
 +
 
 +
CuttingNumber:=6;
 +
QStrategy:=0;
 +
CStrategy:=0;
 +
MinMax:="Max";
 +
 
 +
-- Then we compute the solution with
 +
 
 +
GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);
 +
 
 +
-- The result will be the following:
 +
 
 +
Converting to CNF with CuttingLength: 6, QStrategy: Standard, CStrategy: Standard.
 +
Converting CNF to system of equalities and inequalities...
 +
Model is ready to solve with GLPK...
 +
Solution Status: INTEGER OPTIMAL
 +
Value of objective function: 2
 +
[0, 1, 0, 1]
 +
-------------------------------
 +
</example>
 +
 
 +
 
 +
<example>
 +
Use S::=Z/(2)[x[1..5]];
 +
F:=[
 +
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],
 +
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,
 +
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],
 +
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,
 +
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]
 +
];
 +
 
 +
CuttingNumber:=6;
 +
QStrategy:=1;
 +
CStrategy:=0;
 +
MinMax:="Max";
 +
 
 +
-- Then we compute the solution with
 +
 
 +
GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);
 +
 
 +
-- The result will be the following:
 +
 
 +
Converting to CNF with CuttingLength: 6, QStrategy: LinearPartner, CStrategy: Standard.
 +
Converting CNF to system of equalities and inequalities...
 +
Model is ready to solve with GLPK...
 +
Solution Status: INTEGER OPTIMAL
 +
Value of objective function: 4
 +
[1, 1, 1, 1, 0]
 +
-------------------------------
 +
</example>
 +
 
 +
<example>
 +
Use ZZ/(2)[x[1..3]];
 +
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,
 +
      x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],
 +
      x[1]x[2] + x[2]x[3] + x[2]
 +
    ];
  
−
-- We prename the input of GLPK.MIPSolve-function.
+
CuttingNumber:=5;
−
Use S::=QQ[x,y];
+
QStrategy:=0;
−
OF := 1/2x + y;
+
CStrategy:=1;
−
LE := [3/4x + y - 6];
+
MinMax:="Max";
−
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);
  
−
-- And we achieve:
+
-- The result will be the following:
 +
 
 +
Converting to CNF with CuttingLength: 5, QStrategy: Standard, CStrategy: CubicParnterDegree2.
 +
Converting CNF to system of equalities and inequalities...
 +
Model is ready to solve with GLPK...
 
Solution Status: INTEGER OPTIMAL
 
Solution Status: INTEGER OPTIMAL
−
Value of objective function: 5
+
Value of objective function: 1
−
[x - 2, y - 4]
+
[0, 0, 1]
 +
-------------------------------
 
</example>
 
</example>
 +
  
 
</description>
 
</description>
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   <type>apcocoaserver</type>
 
   <type>apcocoaserver</type>
 
  <type>linear_programs</type>
 
  <type>linear_programs</type>
 +
<type>poly_system</type>
 
</types>
 
</types>
−
<key>mipsolve</key>
+
<key>l01psolve</key>
 
<key>solve linear programm</key>
 
<key>solve linear programm</key>
 
<key>solve lp</key>
 
<key>solve lp</key>
−
<key>GLPK.MIPSolve</key>
+
<key>GLPK.l01pSolve</key>
−
<wiki-category>Package_glpk</wiki-category>
+
<wiki-category>ApCoCoA-1:Package_glpk</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:31, 29 October 2020

This article is about a function from ApCoCoA-1.

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):LIST

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").

  • @return A list containing a zero of the system F.

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:="Max";

-- Then we compute the solution with

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

-- The result will be the following:

Converting to CNF with CuttingLength: 6, QStrategy: Standard, CStrategy: Standard.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 2
[0, 1, 0, 1]
-------------------------------


Example

Use S::=Z/(2)[x[1..5]];
F:=[
 x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],
 x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,
 x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],
 x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,
 x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]
];

CuttingNumber:=6;
QStrategy:=1;
CStrategy:=0;
MinMax:="Max";

-- Then we compute the solution with

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

-- The result will be the following:

Converting to CNF with CuttingLength: 6, QStrategy: LinearPartner, CStrategy: Standard.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 4
[1, 1, 1, 1, 0]
-------------------------------

Example

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

CuttingNumber:=5;
QStrategy:=0;
CStrategy:=1;
MinMax:="Max";

-- Then we compute the solution with

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

-- The result will be the following:

Converting to CNF with CuttingLength: 5, QStrategy: Standard, CStrategy: CubicParnterDegree2.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 1
[0, 0, 1]
-------------------------------





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