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**SCIP Optimization Suite**

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**SCIP Optimization Suite**

Three main software zimpl: Compiler of ZIMPL modeling language soplex: LP solver (implementation of Simplex) scip: An advanced implementation of B&B to solve ILP All these are available in single packages SCIP optimization suite Source code zimpl, soplex, scip are standalone applications Binary (Linux & Windows) Single executable scip application that is linked by zimpl & soplex 2

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**How Does It Work? As a programming library As a standalone solver**

It has API, you can call the functions in C, C++ As a standalone solver Develop a model in zimpl language Compile/Translate your model: zimpl model.zpl Solve it LP problems: soplex model.lp ILP, MIP problems: scip -f model.lp scip by itself calls zimpl if the input file is not .lp scip -f model.zpl 3

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**What we need Parameters Variables Sets Objective function Constraints**

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Sets Set of numbers Set of strings Set of tuples 6

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Set operations 7

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**{<1, “hi”>, <2, “hi”>, <3, “hi”>, <1, “ha”>, …}**

{1, 6, 7, 8, 9} 8

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**Indexed Sets The arrays of ZIMPL Each element has its own index**

A set indexes another set Refer to i-th element by S[i] Example set I := {1, 2, 4}; set A[I] := <1> {10}, <2> {20}, <4> {30,40,50}; set B[ <i> in I] := {10 * i}; 9

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**Parameters set A := {1,2,3}; param B := 10;**

param C[A] := <1> 10, <2> 20, <3> 30; param D[A] := <1> 100 default 0; param E := min A; param F := max <i> in A : C[i]; 10

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**These operations are used in zimpl models to generate the numerical model.**

Most operations are applicable only on parameters, cannot be used for variables, because they are not linear 11

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**Variables “real”, “binary”, or “integer” var x1; var x2 integer;**

Default is “real” var x1; var x2 integer; var x3 binary; set A := {1,2,3}; var x4[A] real; var x5[A * A] integer >=0 <= 10; 12

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**Objective “maximize” or “minimize” var x1; var x2; var x3;**

maximize obj1: x1 + x2 + x3; minimize obj2: 2*x1 + 3*x2; 13

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Objective set A := {1,2,3}; param B[A] := <1> 10, <2> 20, <3> 30; var X[A]; maximize obj1: sum <i> in A: X[i]; minimize obj2: sum <i> in A: B[i] * X[i]; 14

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**Constraint subto name: constraint subto c1: x1 <= 10;**

“<=“ , “=>”, “==“ There is not “>” and “<“ subto c1: x1 <= 10; subto c2: x1 + x2 <= 20; subto c3: x1 + x2 + x3 == 100; 15

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Constraint set A := {1,2,3}; param B[A] := <1> 10, <2> 20, <3> 30; var X[A]; subto c1: forall <i> in A: X[i] <= B[i]; 16

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**Expressions forall expression sum expression if expression**

forall <i> in S: x[i] <= b[i]; sum expression sum <i> in S: x[i]; if expression forall <i> in S: x[i] <= if (i mod 3 == 0) then A[i] else B[i] end; 17

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Example 18

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**param c[I] := <1> 1, <2> 20, <3> 300; **

set I := {1,2,3}; set J := {1,2}; param c[I] := <1> 1, <2> 20, <3> 300; param A[J * I] := <1,1> 1, <1,2> 2, <1,3> 3, <2,1> 30, <2,2> 20, <2,3> 10; param b[J] := <1> 20, <2> 200; var X[I]; maximize obj: sum <i> in I: c[i] * X[i]; subto const: forall <j> in J: sum <i> in I: A[j,i] * X[i] <= b[j]; 19

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**Realistic Problems A model for the problem Multiple instances**

Each instance has its own data Separation between model and data Create a general model Read data from file 20

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**Reading Set and Parameters from file**

“read filename as template” set A := {read "a.txt" as "<1n>"}; a.txt 1 2 3 4 5 21

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**set A := {read "a.txt" as "<1n>"}; **

param B[A] := read "b.txt" as "<1n> 2s"; a.txt b.txt aa bb cc 22

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**set I := {read "I.txt" as "<1n>"}; **

set J := {read "J.txt" as "<1n>"}; param c[I] := read "c.txt" as "<1n> 2n"; param A[J * I] := read "A.txt" as "<1n,2n> 3n"; param b[J] := read "b.txt" as "<1n> 2n"; var X[I]; maximize obj: sum <i> in I: c[i] * X[i]; subto const: forall <j> in J: sum <i> in I: A[j,i] * X[i] <= b[j]; 24

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**Integrality Complexity: An Example**

Consider the LP problem I = 1000 J = 100 If variables are “real” Solution time = 0.21 sec. Objective = If variables are “integer” Solution time = sec. Objective = 1724 26

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