Create optimization problem
optimproblem to create an optimization
For the full workflow, see Problem-Based Optimization Workflow.
uses additional options specified by one or more
prob = optimproblem(
arguments. For example, to specify a maximization problem instead of a minimization
All names in an optimization problem must be unique. Specifically, all variable names, objective function names, and constraint function names must be different.
Create Optimization Problem
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 and Solve Maximization Problem
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: x 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 25.0000 15.0000
Create and Solve Multiobjective Problem
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. Optimization terminated: average change in the spread of Pareto solutions 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 0.5544 -0.0306
Specify optional pairs of arguments as
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
Constraints — Problem constraints
OptimizationConstraint array | structure with
OptimizationConstraint arrays as
Problem constraints, specified as an
OptimizationConstraint array or a structure with
OptimizationConstraint arrays as fields.
prob = optimproblem('Constraints',sum(x,2) ==
Description — Problem label
'' (default) | string | character vector
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
"An iterative approach to the Traveling Salesman problem"
Objective — Objective function
OptimizationExpression | array of
OptimizationExpression | structure with scalar
Objective function, specified as a scalar
OptimizationExpression object, an array of
OptimizationExpression objects, or a structure
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
prob = optimproblem; prob.Objective.first = x^2; prob.Objective.second = (x + 1)^2;
optimproblem('Objective',sum(sum(x))) for a 2-D variable
prob = optimproblem('Objective',(x-a).^2)
a have size 2-by-1,
x is an optimization variable.
ObjectiveSense — Sense of optimization
'minimize' (default) |
'max' | structure with the listed values as fields
Sense of optimization, specified as
'maximize'. You can also specify
'min' to obtain
'max' to obtain
solve function minimizes an objective when
and maximizes an objective when
ObjectiveSense can be a structure with values
'max'. You can
use this form when the problem objective is a structure. The
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";
Objective is a structure, you can specify
ObjectiveSense as a name such as
'max'. In this case, all objectives have the same
prob — Optimization problem
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
The problem-based approach does not support complex values in an objective function, nonlinear equalities, or nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result can be incorrect.