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There are two different Optimization Toolbox™ approaches for solving linear programming, mixed-integer linear programming, and quadratic programming problems. This section describes the problem-based approach. Solver-Based Optimization Problem Setup describes the solver-based approach.

To solve a problem with linear or quadratic objective function and linear constraints, perform the following steps.

Create a problem object by using

`optimproblem`

. A problem object is a container that you fill with objective and constraint expressions. These expressions define the problem, along with bounds that exist in the problem variables.For example, create a maximization problem.

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

Create named variables by using

`optimvar`

. An optimization variable is a symbolic variable that you use to describe the problem objective and constraints. Include any bounds in the variable definitions.For example, create a 15-by-3 array of binary variables named

`'x'`

.x = optimvar('x',15,3,'Type','integer','LowerBound',0,'UpperBound',1);

Define the objective function in the problem object as an expression in the named variables.

For example, assume that you have a real matrix

`f`

of the same size as a matrix of variables`x`

, and the objective is the sum of the entries in`f`

times the corresponding variables`x`

.prob.Objective = sum(sum(f.*x));

Define constraints in the problem object as expressions in the named variables.

For example, assume that the sum of the variables in each row of

`x`

must be one, and the sum of the variables in each column must be no more than one.onesum = sum(x,2) == 1; vertsum = sum(x,1) <= 1; prob.Constraints.onesum = onesum; prob.Constraints.vertsum = vertsum;

Solve the optimization problem by using

`solve`

.sol = solve(prob);

In addition to these basic steps, you can review the problem definition before solving
the problem by using `showproblem`

and related functions. Set options for `solve`

by using `optimoptions`

, as explained in Change Default Solver or Options.

For a basic example, see Mixed-Integer Linear Programming Basics: Problem-Based or the video version Solve a Mixed-Integer Linear Programming Problem using Optimization Modeling. For larger, scalable examples, see Factory, Warehouse, Sales Allocation Model: Problem-Based, Traveling Salesman Problem: Problem-Based, or other examples in Linear Programming and Mixed-Integer Linear Programming or Quadratic Programming.

`optimproblem`

| `optimvar`

| `solve`