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Create optimization problem


Use optimproblem to create an optimization problem.


For the full workflow, see Problem-Based Optimization Workflow.


prob = optimproblem creates an optimization problem with default properties.


prob = optimproblem(Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, to specify a maximization problem instead of a minimization problem, use prob = optimproblem('ObjectiveSense','maximize').


All names in an optimization problem must be unique. Specifically, all variable names, objective function names, and constraint function names must be different.


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Create an optimization problem with default properties.

prob = optimproblem
prob = 
  OptimizationProblem with properties:

       Description: ''
    ObjectiveSense: 'minimize'
         Variables: [0x0 struct] containing 0 OptimizationVariables
         Objective: [0x0 OptimizationExpression]
       Constraints: [0x0 struct] containing 0 OptimizationConstraints

  No problem defined.

Create a linear programming problem for maximization. The problem has two positive variables and three linear inequality constraints.

prob = optimproblem('ObjectiveSense','max');

Create positive variables. Include an objective function in the problem.

x = optimvar('x',2,1,'LowerBound',0);
prob.Objective = x(1) + 2*x(2);

Create linear inequality constraints in the problem.

cons1 = x(1) + 5*x(2) <= 100;
cons2 = x(1) + x(2) <= 40;
cons3 = 2*x(1) + x(2)/2 <= 60;
prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;
prob.Constraints.cons3 = cons3;

Review the problem.

  OptimizationProblem : 

	Solve for:

	maximize :
       x(1) + 2*x(2)

	subject to cons1:
       x(1) + 5*x(2) <= 100

	subject to cons2:
       x(1) + x(2) <= 40

	subject to cons3:
       2*x(1) + 0.5*x(2) <= 60

	variable bounds:
       0 <= x(1)
       0 <= x(2)

Solve the problem.

sol = solve(prob);
Solving problem using linprog.

Optimal solution found.
ans = 2×1


Create a problem with two objective functions of a 2-D variable x. Create the objective functions as expressions in x, and place them in the objective as structures.

x = optimvar("x",2,LowerBound=-2,UpperBound=2);
prob = optimproblem;
prob.Objective.first = norm(x)^2;
prob.Objective.second = norm(x - [1;0])^2;

Solve the problem.

rng default % For reproducibility
sol = solve(prob);
Solving problem using gamultiobj.
gamultiobj stopped because the average change in the spread of Pareto solutions is less than options.FunctionTolerance.

Plot the solution.


Examine one point on the Pareto front. To do so, click the figure and click the Data Tips tool:


Then click a point on the Pareto front.


The index of the pictured point is 9. You can find the x value associated with this point as the solution with index 9.

ans = 2×1


Input Arguments

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Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: To specify a maximization problem, use prob = optimproblem('ObjectiveSense','maximize').

Problem constraints, specified as an OptimizationConstraint array or a structure with OptimizationConstraint arrays as fields.

Example: prob = optimproblem('Constraints',sum(x,2) == 1)

Problem label, specified as a string or character vector. The software does not use Description for computation. Description is an arbitrary label that you can use for any reason. For example, you can share, archive, or present a model or problem, and store descriptive information about the model or problem in Description.

Example: "An iterative approach to the Traveling Salesman problem"

Data Types: char | string

Objective function, specified as a scalar OptimizationExpression object, an array of OptimizationExpression objects, or a structure with scalar OptimizationExpression as fields.

  • For a scalar (single-objective) problem, specify the objective function as a scalar optimization expression or as a structure with a scalar optimization expression as the value.

  • For a multiobjective problem, specify the objective functions as a vector-valued optimization expression, as an array of optimization expressions, or as a structure of optimization expressions. For example, this objective is a structure of optimization expressions in a scalar optimization variable x:

    prob = optimproblem;
    prob.Objective.first = x^2;
    prob.Objective.second = (x + 1)^2;

Example: prob = optimproblem('Objective',sum(sum(x))) for a 2-D variable x.

Example: prob = optimproblem('Objective',(x-a).^2) where x and a have size 2-by-1, and x is an optimization variable.

Sense of optimization, specified as 'minimize' or 'maximize'. You can also specify 'min' to obtain 'minimize' or 'max' to obtain 'maximize'. The solve function minimizes an objective when ObjectiveSense is 'minimize' and maximizes an objective when ObjectiveSense is 'maximize'.

ObjectiveSense can be a structure with values 'minimize', 'min', 'maximize', or 'max'. You can use this form when the problem objective is a structure. The Objective and ObjectiveSense structures should have the same field names, so the ObjectiveSense applies to the corresponding Objective. For example,

x = optimvar('x',2,"UpperBound",2,"LowerBound",-2);
prob = optimproblem;
prob.Objective.first = norm(x)^2;
prob.Objective.second = -norm(x - [1;0])^2;
prob.ObjectiveSense.first = "min";
prob.ObjectiveSense.second = "max";

If Objective is a structure, you can specify ObjectiveSense as a name such as 'max'. In this case, all objectives have the same ObjectiveSense.

Example: prob = optimproblem('ObjectiveSense','max')

Data Types: char | string

Output Arguments

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Optimization problem, returned as an OptimizationProblem object. Typically, to complete the problem description, you specify an objective function and constraints. However, you can have a feasibility problem, which has no objective function, or you can have a problem with no constraints. Solve a complete problem by calling solve.


The problem-based approach does not support complex values in the following: an objective function, nonlinear equalities, and nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result might be incorrect.

Version History

Introduced in R2017b