intlinprog

Mixed-integer linear programming (MILP)

Mixed-integer linear programming solver.

Finds the minimum of a problem specified by

$\min_{x} f^{T} x subject to {\begin{array}{l} x (intcon) are integers \\ A \cdot x \leq b \\ A e q \cdot x = b e q \\ l b \leq x \leq u b . \end{array}$

f, x, intcon, b, beq, lb, and ub are vectors, and A and Aeq are matrices.

You can specify f, intcon, lb, and ub as vectors or arrays. See Matrix Arguments.

Syntax

x = intlinprog(f,intcon,A,b)
example
x = intlinprog(f,intcon,A,b,Aeq,beq)
x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub)
example
x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub,options)
example
x = intlinprog(problem)
example
[x,fval,exitflag,output] = intlinprog(___)
example

Description

example

x = intlinprog(f,intcon,A,b) solves min f'*x such that the components of x in intcon are integers, and A*x ≤ b.

x = intlinprog(f,intcon,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. Set A = [] and b = [] if no inequalities exist.

example

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is always in the range lb ≤ x ≤ ub. Set Aeq = [] and beq = [] if no equalities exist.

example

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub,options) minimizes using the optimization options specified in options. Use optimoptions to set these options. Set lb = [] and ub = [] if no bounds exist.

example

x = intlinprog(problem) uses a problem structure to encapsulate all solver inputs. You can import a problem structure from an MPS file using mpsread.

example

[x,fval,exitflag,output] = intlinprog(___), for any input arguments described above, returns fval = f'*x, a value exitflag describing the exit condition, and a structure output containing information about the optimization process.

Examples

collapse all

Solve an MILP with Linear Inequalities

Solve the problem

$\min_{x} 8 x_{1} + x_{2} subject to {\begin{array}{l} x_{2} is an integer \\ x_{1} + 2 x_{2} \geq - 14 \\ - 4 x_{1} - x_{2} \leq - 33 \\ 2 x_{1} + x_{2} \leq 20. \end{array}$

Write the objective function vector and vector of integer variables.

f = [8;1];
intcon = 2;

Convert all inequalities into the form A*x <= b by multiplying "greater than" inequalities by -1.

A = [-1,-2;
    -4,-1;
    2,1];
b = [14;-33;20];

Call intlinprog.

x = intlinprog(f,intcon,A,b)

LP:                Optimal objective value is 59.000000.                                            


Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value;
options.TolGapAbs = 0 (the default value). The intcon variables are integer within tolerance,
options.TolInteger = 1e-05 (the default value).


x =

    6.5000
    7.0000

Solve an MILP with All Types of Constraints

Solve the problem

$\min_{x} (- 3 x_{1} - 2 x_{2} - x_{3}) subject to {\begin{array}{l} x_{3} binary \\ x_{1}, x_{2} \geq 0 \\ x_{1} + x_{2} + x_{3} \leq 7 \\ 4 x_{1} + 2 x_{2} + x_{3} = 12 \end{array}$

Write the objective function vector and vector of integer variables.

f = [-3;-2;-1];
intcon = 3;

Write the linear inequality constraints.

A = [1,1,1];
b = 7;

Write the linear equality constraints.

Aeq = [4,2,1];
beq = 12;

Write the bound constraints.

lb = zeros(3,1);
ub = [Inf;Inf;1]; % Enforces x(3) is binary

Call intlinprog.

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub)

LP:                Optimal objective value is -12.000000.                                           


Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value;
options.TolGapAbs = 0 (the default value). The intcon variables are integer within tolerance,
options.TolInteger = 1e-05 (the default value).


x =

         0
    5.5000
    1.0000

Solve an MILP with Nondefault Options

Solve the problem

$\min_{x} (- 3 x_{1} - 2 x_{2} - x_{3}) subject to {\begin{array}{l} x_{3} binary \\ x_{1}, x_{2} \geq 0 \\ x_{1} + x_{2} + x_{3} \leq 7 \\ 4 x_{1} + 2 x_{2} + x_{3} = 12 \end{array}$

without showing iterative display.

Specify the solver inputs.

f = [-3;-2;-1];
intcon = 3;
A = [1,1,1];
b = 7;
Aeq = [4,2,1];
beq = 12;
lb = zeros(3,1);
ub = [Inf;Inf;1]; % enforces x(3) is binary

Specify no display.

options = optimoptions('intlinprog','Display','off');

Run the solver.

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub,options)

Use a Problem Structure

Solve the problem

$\min_{x} (- 3 x_{1} - 2 x_{2} - x_{3}) subject to {\begin{array}{l} x_{3} binary \\ x_{1}, x_{2} \geq 0 \\ x_{1} + x_{2} + x_{3} \leq 7 \\ 4 x_{1} + 2 x_{2} + x_{3} = 12 \end{array}$

using iterative display. Use a problem structure as the intlinprog input.

Specify the solver inputs.

f = [-3;-2;-1];
intcon = 3;
A = [1,1,1];
b = 7;
Aeq = [4,2,1];
beq = 12;
lb = zeros(3,1);
ub = [Inf;Inf;1]; % enforces x(3) is binary
options = optimoptions('intlinprog','Display','off');

Insert the inputs into a problem structure. Include the solver name.

problem = struct('f',f,'intcon',intcon,...
    'Aineq',A,'bineq',b,'Aeq',Aeq,'beq',beq,...
    'lb',lb,'ub',ub,'options',options,...
    'solver','intlinprog');

Run the solver.

x = intlinprog(problem)

Examine the MILP Solution and Process

Call intlinprog with more outputs to see solution details and process.

The goal is to solve the problem

$\min_{x} (- 3 x_{1} - 2 x_{2} - x_{3}) subject to {\begin{array}{l} x_{3} binary \\ x_{1}, x_{2} \geq 0 \\ x_{1} + x_{2} + x_{3} \leq 7 \\ 4 x_{1} + 2 x_{2} + x_{3} = 12 \end{array}$

Specify the solver inputs.

f = [-3;-2;-1];
intcon = 3;
A = [1,1,1];
b = 7;
Aeq = [4,2,1];
beq = 12;
lb = zeros(3,1);
ub = [Inf;Inf;1]; % enforces x(3) is binary

Call intlinprog with all outputs.

[x,fval,exitflag,output] = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub)

LP:                Optimal objective value is -12.000000.                                           


Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value;
options.TolGapAbs = 0 (the default value). The intcon variables are integer within tolerance,
options.TolInteger = 1e-05 (the default value).


x =

         0
    5.5000
    1.0000


fval =

  -12.0000


exitflag =

     1


output = 

        relativegap: 0
        absolutegap: 0
      numfeaspoints: 1
           numnodes: 0
    constrviolation: 1.7764e-15
            message: 'Optimal solution found.

Intlinprog stopped at the root node because the objective v...'

The output structure shows numnodes is 0. This means intlinprog solved the problem before branching. This is one indication that the result is reliable. Also, the absolutegap and relativegap fields are 0. This is another indication that the result is reliable.

Related Examples

Input Arguments

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`f` — Coefficient vectorreal vector

Coefficient vector, specified as a vector of doubles representing the objective function, f'*x. The notation assumes that f is a column vector, but you are free to use a row vector.

f can also be an array. Internally, intlinprog converts an array f to the vector f(:).

If you specify f = [], intlinprog tries to find a feasible point without trying to minimize an objective function.

Example: f = [4;2;-1.7];

Data Types: double

`intcon` — Vector of integer constraintsvector of integers

Vector of integer constraints, specified as a vector of positive integers. The values in intcon indicate the components of the decision variable x that are integer-valued. intcon has values from 1 through numel(f).

intcon can also be an array. Internally, intlinprog converts an array intcon to the vector intcon(:).

Example: intcon = [1,2,7] means x(1), x(2), and x(7) take only integer values.

Data Types: double

`A` — Linear inequality constraint matrixreal matrix

Linear inequality constraint matrix, specified as a matrix of doubles. A represents the linear coefficients in the constraints A*x ≤ b. A has size M-by-N, where M is the number of constraints and N = numel(f). To save memory, A can be sparse.

Example: A = [4,3;2,0;4,-1]; means three linear inequalities (three rows) for two decision variables (two columns).

Data Types: double

`b` — Linear inequality constraint vectorreal vector

Linear inequality constraint vector, specified as a vector of doubles. b represents the constant vector in the constraints A*x ≤ b. b has length M, where A is M-by-N.

Example: [4,0]

Data Types: double

`Aeq` — Linear equality constraint matrix`[]` (default) | real matrix

Linear equality constraint matrix, specified as a matrix of doubles. Aeq represents the linear coefficients in the constraints Aeq*x = beq. Aeq has size Meq-by-N, where Meq is the number of constraints and N = numel(f). To save memory, Aeq can be sparse.

Example: A = [4,3;2,0;4,-1]; means three linear inequalities (three rows) for two decision variables (two columns).

Data Types: double

`beq` — Linear equality constraint vector`[]` (default) | real vector

Linear equality constraint vector, specified as a vector of doubles. beq represents the constant vector in the constraints Aeq*x = beq. beq has length Meq, where Aeq is Meq-by-N.

Example: [4,0]

Data Types: double

`lb` — Lower bounds`[]` (default) | real vector or array

Lower bounds, specified as a vector or array of doubles. lb represents the lower bounds elementwise in lb ≤ x ≤ ub.

Internally, intlinprog converts an array lb to the vector lb(:).

Example: lb = [0;-Inf;4] means x(1) ≥ 0, x(3) ≥ 4.

Data Types: double

`ub` — Upper bounds`[]` (default) | real vector or array

Upper bounds, specified as a vector or array of doubles. ub represents the upper bounds elementwise in lb ≤ x ≤ ub.

Internally, intlinprog converts an array ub to the vector ub(:).

Example: ub = [Inf;4;10] means x(2) ≤ 4, x(3) ≤ 10.

Data Types: double

`options` — Options for `intlinprog`options created using `optimoptions`

Options for intlinprog, specified as the output of optimoptions.

Option	Description	Default
`BranchingRule`	Rule for choosing the component for branching: `'maxpscost'` — The fractional component with maximum pseudocost. See Branch and Bound. `'mostfractional'` — The component whose fractional part is closest to `1/2`. `'maxfun'` — The fractional component with maximal corresponding component in the absolute value of objective vector `f`.	`'maxpscost'`
`CutGeneration`	Level of cut generation (see Cut Generation): `'none'` — No cuts. Makes `CutGenMaxIter` irrelevant. `'basic'` — Normal cut generation. `'intermediate'` — Use more cut types. `'advanced'` — Use most cut types.	`'basic'`
`CutGenMaxIter`	Number of passes through all cut generation methods before entering the branch-and-bound phase, an integer from `1` through `50`. Disable cut generation by setting the `CutGeneration` option to `'none'`.	`10`
`Display`	Level of display (see Iterative Display): `'off'` or `'none'` — No iterative display `'final'` — Show final values only `'iter'` — Show iterative display	`'iter'`
`Heuristics`	Algorithm for searching for feasible points (see Heuristics for Finding Feasible Solutions): `'none'` `'rss'` `'round'` `'rins'`	`'rss'`
`HeuristicsMaxNodes`	Strictly positive integer that bounds the number of nodes `intlinprog` can explore in its branch-and-bound search for feasible points. See Heuristics for Finding Feasible Solutions.	`50`
`IPPreprocess`	Types of integer preprocessing (see Mixed-Integer Program Preprocessing): `'none'` — Use very few integer preprocessing steps. `'basic'` — Use a moderate number of integer preprocessing steps. `'advanced'` — Use all available integer preprocessing steps.	`'basic'`
`LPMaxIter`	Strictly positive integer, the maximum number of simplex algorithm iterations per node during the branch-and-bound process.	`3e4`
`LPPreprocess`	Type of preprocessing for the solution to the relaxed linear program (see Linear Program Preprocessing): `'none'` — No preprocessing. `'basic'` — Use preprocessing.	`'basic'`
`MaxNodes`	Strictly positive integer that is the maximum number of nodes `intlinprog` explores in its branch-and-bound process.	`1e7`
`MaxNumFeasPoints`	Strictly positive integer. `intlinprog` stops if it finds `MaxNumFeasPoints` integer feasible points.	`Inf`
`MaxTime`	Positive real that is the maximum time in seconds that `intlinprog` runs.	`7200`
`NodeSelection`	Choose the node to explore next. `'simplebestproj'` — Best projection. See Branch and Bound. `'minobj'` — Explore the node with the minimum objective function. `'mininfeas'` — Explore the node with the minimal sum of integer infeasibilities. See Branch and Bound.	`'simplebestproj'`
`ObjectiveCutOff`	Real greater than `-Inf`. During the branch-and-bound calculation, `intlinprog` discards any node where the linear programming solution has an objective value exceeding `ObjectiveCutOff`.	`Inf`
`OutputFcn`	Specify one or more functions that an optimization function calls at events, either as a function handle or as a cell array of function handles. `@savemilpsolutions` collects the integer-feasible points in the `xIntSol` matrix in your workspace, where each column is one integer feasible point. For information on writing a custom output function, see `intlinprog` Output Functions and Plot Functions.	`[]`
`PlotFcns`	Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Pass a function handle or a cell array of function handles. `@optimplotmilp` plots the internally-calculated upper and lower bounds on the objective value of the solution. For information on writing a custom plot function, see `intlinprog` Output Functions and Plot Functions.	`[]`
`RelObjThreshold`	Nonnegative real. `intlinprog` changes the current feasible solution only when it locates another with an objective function value that is at least `RelObjThreshold` lower: (fold – fnew)/(1 + fold) > RelObjThreshold.	`1e-4`
`RootLPAlgorithm`	Algorithm for solving linear programs: `'dual-simplex'` — Dual simplex algorithm `'primal-simplex'` — Primal simplex algorithm	`'dual-simplex'`
`RootLPMaxIter`	Nonnegative integer that is the maximum number of simplex algorithm iterations to solve the initial linear programming problem.	`3e4`
`TolCon`	Real from `1e-9` through `1e-3` that is the maximum discrepancy that linear constraints can have and still be considered satisfied. `TolCon` is not a stopping criterion.	`1e-4`
`TolFunLP`	Nonnegative real where reduced costs must exceed `TolFunLP` for a variable to be taken into the basis.	`1e-7`
`TolGapAbs`	Nonnegative real. `intlinprog` stops if the difference between the internally calculated upper (`U`) and lower (`L`) bounds on the objective function is less than or equal to `TolGapAbs`: `U – L <= TolGapAbs`.	`0`
`TolGapRel`	Real from `0` through `1`. `intlinprog` stops if the relative difference between the internally calculated upper (`U`) and lower (`L`) bounds on the objective function is less than or equal to `TolGapRel`: `(U – L) / (abs(U) + 1) <= TolGapRel`.	`1e-4`
`TolInteger`	Real from `1e-6` through `1e-3`, where the maximum deviation from integer that a component of the solution `x` can have and still be considered an integer. `TolInteger` is not a stopping criterion.	`1e-5`

Example: options = optimoptions('intlinprog','MaxTime',120)

`problem` — Structure encapsulating inputs and optionsstructure

Structure encapsulating the inputs and options, specified with the following fields.

`f`	Vector representing objective `f'*x` (required)
`intcon`	Vector indicating variables that take integer values (required)
`Aineq`	Matrix in linear inequality constraints `Aineq*x` ≤ `bineq`
`bineq`	Vector in linear inequality constraints `Aineq*x` ≤ `bineq`
`Aeq`	Matrix in linear equality constraints `Aeq*x = beq`
`beq`	Vector in linear equality constraints `Aeq*x = beq`
`lb`	Vector of lower bounds
`ub`	Vector of upper bounds
`solver`	`'intlinprog'` (required)
`options`	Options created using `optimoptions` (required)

You must specify at least these fields in the problem structure. Other fields are optional:

f
intcon
solver
options

Example: problem.f = [1,2,3]; problem.intcon = [2,3]; problem.options = optimoptions('intlinprog'); problem.Aineq = [-3,-2,-1]; problem.bineq = -20; problem.lb = [-6.1,-1.2,7.3]; problem.solver = 'intlinprog';

Data Types: struct

Output Arguments

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`x` — Solutionreal vector

Solution, returned as a vector that minimizes f'*x subject to all bounds, integer constraints, and linear constraints.

When a problem is infeasible or unbounded, x is [].

`fval` — Objective valuereal scalar

Objective value, returned as the scalar value f'*x at the solution x.

When a problem is infeasible or unbounded, fval is [].

`exitflag` — Algorithm stopping conditioninteger

Algorithm stopping condition, returned as an integer identifying the reason the algorithm stopped. The following lists the values of exitflag and the corresponding reasons intlinprog stopped.

`2`	`intlinprog` stopped prematurely. Integer feasible point found.
`1`	`intlinprog` converged to the solution `x`.
`0`	`intlinprog` stopped prematurely. No integer feasible point found.
`-2`	No feasible point found.
`-3`	Root LP problem is unbounded.

The exit message can give more detailed information on the reason intlinprog stopped, such as exceeding a tolerance.

`output` — Solution process summarystructure

Solution process summary, returned as a structure containing information about the optimization process.

`relativegap`	Relative difference between upper (`U`) and lower (`L`) bounds of the objective function that `intlinprog` calculates in its branch-and-bound algorithm. `relativegap = (U - L) / (abs(U) + 1)` If `intcon = []`, `relativegap = []`.
`absolutegap`	Difference between upper and lower bounds of the objective function that `intlinprog` calculates in its branch-and-bound algorithm. If `intcon = []`, `absolutegap = []`.
`numfeaspoints`	Number of integer feasible points found. If `intcon = []`, `numfeaspoints = []`. Also, if the initial relaxed problem is infeasible, `numfeaspoints = []`.
`numnodes`	Number of nodes in branch-and-bound algorithm. If the solution was found during preprocessing or during the initial cuts, `numnodes = 0`. If `intcon = []`, `numnodes = []`.
`constrviolation`	Constraint violation that is positive for violated constraints. `constrviolation = max([0; norm(Aeqx-beq, inf); (lb-x); (x-ub); (Aix-bi)])`
`message`	Exit message.

Limitations

Often, some supposedly integer-valued components of the solution x(intCon) are not precisely integers. intlinprog deems as integers all solution values within the TolInteger tolerance of an integer.

To round all supposed integers to be exactly integers, use the round function.

x(intcon) = round(x(intcon));

Caution Rounding solutions can cause the solution to become infeasible. Check feasibility after rounding:

max(A*x - b) % See if entries are not too positive, so have small infeasibility
max(abs(Aeq*x - beq)) % See if entries are near enough to zero
max(x - ub) % Positive entries are violated bounds
max(lb - x) % Positive entries are violated bounds

intlinprog does not enforce that solution components be integer-valued when their absolute values exceed 2.1e9. When your solution has such components, intlinprog warns you. If you receive this warning, check the solution to see whether supposedly integer-valued components of the solution are close to integers.
intlinprog does not allow components of the problem, such as coefficients in f, A, or ub, to exceed 1e25 in absolute value. If you try to run intlinprog with such a problem, intlinprog issues an error.
Currently, you cannot run intlinprog in the Optimization app.

More About

expand all

MILP

Mixed-integer linear programming definition.

MILP means find the minimum of a problem specified by

$\min_{x} f^{T} x subject to {\begin{array}{l} x (intcon) are integers \\ A \cdot x \leq b \\ A e q \cdot x = b e q \\ l b \leq x \leq u b . \end{array}$

f, x, intcon, b, beq, lb, and ub are vectors, and A and Aeq are matrices.

You can specify f, intcon, lb, and ub as vectors or arrays. See Matrix Arguments.

Tips

To specify binary variables, set the variables to be integers in intcon, and give them lower bounds of 0 and upper bounds of 1.
Save memory by specifying sparse linear constraint matrices A and Aeq. However, you cannot use sparse matrices for b and beq.
To provide logical indices for integer components, meaning a binary vector with 1 indicating an integer, convert to intcon form using find. For example,
```
logicalindices = [1,0,0,1,1,0,0];
intcon = find(logicalindices)
```
```
intcon =

     1     4     5
```
intlinprog replaces bintprog. To update old bintprog code to use intlinprog, make the following changes:
- Set intcon to 1:numVars, where numVars is the number of variables in your problem.
- Set lb to zeros(numVars,1).
- Set ub to ones(numVars,1).
- Update any relevant options. Use optimoptions to create options for intlinprog.
- Change your call to bintprog as follows:
  [x,fval,exitflag,output] = bintprog(f,A,b,Aeq,Beq,x0,options) % Change your call to: [x,fval,exitflag,output] = intlinprog(f,intcon,A,b,Aeq,Beq,lb,ub,options)

Documentation

intlinprog

Syntax

Description

Examples

Solve an MILP with Linear Inequalities

Solve an MILP with All Types of Constraints

Solve an MILP with Nondefault Options

Use a Problem Structure

Examine the MILP Solution and Process

Related Examples

Input Arguments

`f` — Coefficient vectorreal vector

`intcon` — Vector of integer constraintsvector of integers

`A` — Linear inequality constraint matrixreal matrix

`b` — Linear inequality constraint vectorreal vector

`Aeq` — Linear equality constraint matrix`[]` (default) | real matrix

`beq` — Linear equality constraint vector`[]` (default) | real vector

`lb` — Lower bounds`[]` (default) | real vector or array

`ub` — Upper bounds`[]` (default) | real vector or array

`options` — Options for `intlinprog`options created using `optimoptions`

`problem` — Structure encapsulating inputs and optionsstructure

Output Arguments

`x` — Solutionreal vector

`fval` — Objective valuereal scalar

`exitflag` — Algorithm stopping conditioninteger

`output` — Solution process summarystructure

Limitations

More About

MILP

Tips

See Also

Introduced in R2014a

Optimization Toolbox Documentation

Other Documentation

Support

intlinprog

Syntax

Description

Examples

Related Examples

Input Arguments

f — Coefficient vectorreal vector

intcon — Vector of integer constraintsvector of integers

A — Linear inequality constraint matrixreal matrix

b — Linear inequality constraint vectorreal vector

Aeq — Linear equality constraint matrix[] (default) | real matrix

beq — Linear equality constraint vector[] (default) | real vector

lb — Lower bounds[] (default) | real vector or array

ub — Upper bounds[] (default) | real vector or array

options — Options for intlinprogoptions created using optimoptions

problem — Structure encapsulating inputs and optionsstructure

Output Arguments

x — Solutionreal vector

fval — Objective valuereal scalar

exitflag — Algorithm stopping conditioninteger

output — Solution process summarystructure

Limitations

More About

See Also

Introduced in R2014a

`f` — Coefficient vectorreal vector

`intcon` — Vector of integer constraintsvector of integers

`A` — Linear inequality constraint matrixreal matrix

`b` — Linear inequality constraint vectorreal vector

`Aeq` — Linear equality constraint matrix`[]` (default) | real matrix

`beq` — Linear equality constraint vector`[]` (default) | real vector

`lb` — Lower bounds`[]` (default) | real vector or array

`ub` — Upper bounds`[]` (default) | real vector or array

`options` — Options for `intlinprog`options created using `optimoptions`

`problem` — Structure encapsulating inputs and optionsstructure

`x` — Solutionreal vector

`fval` — Objective valuereal scalar

`exitflag` — Algorithm stopping conditioninteger

`output` — Solution process summarystructure