OptimizationVariable

Variable for optimization

Description

An OptimizationVariable object contains variables for optimization expressions. Use expressions to represent an objective function, constraints, or equations. Variables are symbolic in nature, and can be arrays of any size.

Tip

For the full workflow, see Problem-Based Optimization Workflow or Problem-Based Workflow for Solving Equations.

Creation

Create an OptimizationVariable object using optimvar.

Properties

Array-Wide Properties

Name — Variable name

Read-only: string | character vector

This property is read-only.

Variable name, specified as a string or character vector.

Name gives the variable label to be displayed, such as in show or write. Name also gives the field names in the solution structure that solve returns.

Tip

To avoid confusion, set name to be the MATLAB^® variable name. For example,

metal = optimvar("metal")

Data Types: char | string

Type — Variable type

'continuous' (default) | 'integer' | 'semi-continuous' | 'semi-integer'

Variable type, specified as 'continuous', 'integer', 'semi-continuous', or 'semi-integer'.

• 'continuous' — Real values.

• 'integer' — Integer values.

• 'semi-continuous' — Zero or real values between the lower and upper bounds, which must be strictly positive and cannot exceed 1e5. This type applies only to mixed-integer linear programming (intlinprog).

• 'semi-integer' — Zero or integer values between the lower and upper bounds, which must be strictly positive and cannot exceed 1e5. This type applies only to mixed-integer linear programming (intlinprog).

Type applies to the entire variable array. To have multiple variable types, create multiple variables.

Tip

To specify a binary variable, use the 'integer' type and specify LowerBound = 0 and UpperBound = 1.

Data Types: char | string

IndexNames — Index names

'' (default) | cell array of strings | cell array of character vectors

Index names, specified as a cell array of strings or character vectors. For information on using index names, see Named Index for Optimization Variables.

Data Types: cell

Element-wise Properties

LowerBound — Lower bound

-Inf (default) | real scalar | real array

Lower bound, specified as a real scalar or as a real array having the same dimensions as the OptimizationVariable object. Scalar values apply to all elements of the variable.

The LowerBound property is always displayed as an array. However, you can set the property as a scalar that applies to all elements. For example,

var.LowerBound = 0

Data Types: double

UpperBound — Upper bound

Inf (default) | real scalar | real array

Upper bound, specified as a real scalar or as a real array having the same dimensions as the OptimizationVariable object. Scalar values apply to all elements of the variable.

The UpperBound property is always displayed as an array. However, you can set the property as a scalar that applies to all elements. For example

var.UpperBound = 1

Data Types: double

Object Functions

show	Display information about optimization object
showbounds	Display variable bounds
write	Save optimization object description
writebounds	Save description of variable bounds

Examples

Create Scalar Optimization Variable

Create a scalar optimization variable named dollars.

dollars = optimvar("dollars")

dollars = 
  OptimizationVariable with properties:

          Name: 'dollars'
          Type: 'continuous'
    IndexNames: {{}  {}}
    LowerBound: -Inf
    UpperBound: Inf

  See variables with show.
  See bounds with showbounds.

Create Optimization Variable Vector

Create a 3-by-1 optimization variable vector named x.

x = optimvar("x",3)

x = 
  3×1 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'x'
          Type: 'continuous'
    IndexNames: {{}  {}}

  Elementwise properties:
    LowerBound: [3×1 double]
    UpperBound: [3×1 double]

  See variables with show.
  See bounds with showbounds.

Create Optimization Variables Indexed by Strings

Create an integer optimization variable vector named bolts that is indexed by the strings "brass", "stainless", and "galvanized". Use the indices of bolts to create an optimization expression, and experiment with creating bolts using character arrays or in a different orientation.

Create bolts using strings in a row orientation.

bnames = ["brass","stainless","galvanized"];
bolts = optimvar("bolts",bnames,Type="integer")

bolts = 
  1×3 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'bolts'
          Type: 'integer'
    IndexNames: {{}  {1×3 cell}}

  Elementwise properties:
    LowerBound: [-Inf -Inf -Inf]
    UpperBound: [Inf Inf Inf]

  See variables with show.
  See bounds with showbounds.

Create an optimization expression using the string indices.

y = bolts("brass") + 2*bolts("stainless") + 4*bolts("galvanized")

y = 
  Linear OptimizationExpression

    bolts('brass') + 2*bolts('stainless') + 4*bolts('galvanized')

Use a cell array of character vectors instead of strings to get a variable with the same indices as before.

bnames = {'brass','stainless','galvanized'};
bolts = optimvar("bolts",bnames,Type="integer")

bolts = 
  1×3 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'bolts'
          Type: 'integer'
    IndexNames: {{}  {1×3 cell}}

  Elementwise properties:
    LowerBound: [-Inf -Inf -Inf]
    UpperBound: [Inf Inf Inf]

  See variables with show.
  See bounds with showbounds.

Use a column-oriented version of bnames, 3-by-1 instead of 1-by-3, and observe that bolts has that orientation as well.

bnames = ["brass";"stainless";"galvanized"];
bolts = optimvar("bolts",bnames,Type="integer")

bolts = 
  3×1 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'bolts'
          Type: 'integer'
    IndexNames: {{1×3 cell}  {}}

  Elementwise properties:
    LowerBound: [3×1 double]
    UpperBound: [3×1 double]

  See variables with show.
  See bounds with showbounds.

Create Multidimensional Optimization Variables

Create a 3-by-4-by-2 array of optimization variables named xarray.

xarray = optimvar("xarray",3,4,2)

xarray = 
  3×4×2 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'xarray'
          Type: 'continuous'
    IndexNames: {{}  {}  {}}

  Elementwise properties:
    LowerBound: [3×4×2 double]
    UpperBound: [3×4×2 double]

  See variables with show.
  See bounds with showbounds.

You can also create multidimensional variables indexed by a mixture of names and numeric indices. For example, create a 3-by-4 array of optimization variables where the first dimension is indexed by the strings 'brass', 'stainless', and 'galvanized', and the second dimension is numerically indexed.

bnames = ["brass","stainless","galvanized"];
bolts = optimvar("bolts",bnames,4)

bolts = 
  3×4 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'bolts'
          Type: 'continuous'
    IndexNames: {{1×3 cell}  {}}

  Elementwise properties:
    LowerBound: [3×4 double]
    UpperBound: [3×4 double]

  See variables with show.
  See bounds with showbounds.

Create Binary Optimization Variables

Create an optimization variable named x of size 3-by-3-by-3 that represents binary variables.

x = optimvar("x",3,3,3,Type="integer",LowerBound=0,UpperBound=1)

x = 
  3×3×3 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'x'
          Type: 'integer'
    IndexNames: {{}  {}  {}}

  Elementwise properties:
    LowerBound: [3×3×3 double]
    UpperBound: [3×3×3 double]

  See variables with show.
  See bounds with showbounds.

Create Semicontinuous and Semi-integer Optimization Variables

Create a semicontinuous optimization variable named x with a lower bound of $$\pi /2$$ and an upper bound of $$2\pi$$.

x = optimvar("x",Type="semi-continuous",...
    LowerBound=pi/2,UpperBound=2*pi)

x = 
  OptimizationVariable with properties:

          Name: 'x'
          Type: 'semi-continuous'
    IndexNames: {{}  {}}
    LowerBound: 1.5708
    UpperBound: 6.2832

  See variables with show.
  See bounds with showbounds.

Create a semi-integer 3-D variable named y with lower bounds of [10,20,30] and upper bounds of [20,40,60].

y = optimvar("y",3,Type="semi-integer",...
    LowerBound=[10,20,30],UpperBound=[20,40,60])

y = 
  3×1 OptimizationVariable array with properties:

  Array-wide properties:
          Name: 'y'
          Type: 'semi-integer'
    IndexNames: {{}  {}}

  Elementwise properties:
    LowerBound: [3×1 double]
    UpperBound: [3×1 double]

  See variables with show.
  See bounds with showbounds.

Semicontinuous and semi-integer variables must have strictly positive bounds that do not exceed 1e5.

More About

Arithmetic Operations

For the list of supported operations on optimization variables, see Supported Operations for Optimization Variables and Expressions.

Reference

An optimization variable reference is an optimization variable that is a subset of another optimization variable. The reference variable points to, meaning it is an alias of, the original variable. The reference variable does not have an independent existence.

For example, suppose x is a 3-element optimization variable:

x = optimvar("x",3,1);

Take y as the last two elements of x.

y = x(2:3);

Then y(1) is an alias of x(2), and y(2) is an alias of x(3). If you use y in an optimization expression, the expression includes x, not y. For example,

expr = [1,2]*y

expr = 

  OptimizationExpression

	x(2) + 2*x(3)

Furthermore, any modification of y produces a modification of x. For example,

y.LowerBound = 2;
showbounds(x)

2 <= x(2, 1) 
2 <= x(3, 1) 

Semicontinuous and Semi-Integer Variables

For mixed-integer linear programming problems, you can specify Type="semi-continuous" or Type="semi-integer" for the variables. These variables must have strictly positive bounds that do not exceed 1e5.

Semicontinuous and semi-integer variables can take the value 0 or any value from the lower bound to the upper bound. Semi-integer variables can take only integer values within the bounds, whereas semicontinuous variables can take any real value within the bounds.

Tips

• OptimizationVariable objects have handle copy behavior. See Handle Object Behavior and Comparison of Handle and Value Classes. Handle copy behavior means that a copy of an OptimizationVariable points to the original and does not have an independent existence. For example, create a variable x, copy it to y, then set a property of y. Note that x takes on the new property value.

  x = optimvar('x','LowerBound',1);
  y = x;
  y.LowerBound = 0;
  showbounds(x)

      0 <= x

Version History

Introduced in R2017b

R2025a: Create 'semi-continuous' and 'semi-integer' variables

For mixed-integer linear programming problems, variables can have the type 'semi-continuous' or 'semi-integer'. See Semicontinuous and Semi-Integer Variables.

R2024a: Create optimization variables using the "like" syntax

You can create optimization variables using the "like" syntax. This capability can simplify the initialization of optimization expressions, as shown in Use "like" Syntax to Initialize the Array.