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Problem-Based Optimization Algorithms

Internally, the solve function solves optimization problems by calling a solver:

  • linprog for linear objective and linear constraints

  • intlinprog for linear objective and linear constraints and integer constraints

  • quadprog for quadratic objective and linear constraints

  • lsqlin or lsqnonneg for linear least-squares with linear constraints

  • lsqcurvefit or lsqnonlin for nonlinear least-squares with bound constraints

  • fminunc for problems without any constraints (not even variable bounds) and with a general nonlinear objective function

  • fmincon for problems with a nonlinear constraint, or with a general nonlinear objective and at least one constraint

  • fzero for a scalar nonlinear equation

  • lsqlin for systems of linear equations, with or without bounds

  • fsolve for systems of nonlinear equations without constraints

  • lsqnonlin for systems of nonlinear equations with bounds

Before solve can call these functions, the problems must be converted to solver form, either by solve or some other associated functions or objects. This conversion entails, for example, linear constraints having a matrix representation rather than an optimization variable expression.

The first step in the algorithm occurs as you place optimization expressions into the problem. An OptimizationProblem object has an internal list of the variables used in its expressions. Each variable has a linear index in the expression, and a size. Therefore, the problem variables have an implied matrix form. The prob2struct function performs the conversion from problem form to solver form. For an example, see Convert Problem to Structure.

For nonlinear optimization problems, solve uses automatic differentiation to compute the gradients of the objective function and nonlinear constraint functions. These derivatives apply when the objective and constraint functions are composed of Supported Operations on Optimization Variables and Expressions and do not use the fcn2optimexpr function. When automatic differentiation does not apply, solvers estimate derivatives using finite differences. For details of automatic differentiation, see Automatic Differentiation Background.

For the default and allowed solvers that solve calls, depending on the problem objective and constraints, see 'solver'. You can override the default by using the 'solver' name-value pair argument when calling solve.

For the algorithm that intlinprog uses to solve MILP problems, see intlinprog Algorithm. For the algorithms that linprog uses to solve linear programming problems, see Linear Programming Algorithms. For the algorithms that quadprog uses to solve quadratic programming problems, see Quadratic Programming Algorithms. For linear or nonlinear least-squares solver algorithms, see Least-Squares (Model Fitting) Algorithms. For nonlinear solver algorithms, see Unconstrained Nonlinear Optimization Algorithms and Constrained Nonlinear Optimization Algorithms.

For nonlinear equation solving, solve internally represents each equation as the difference between the left and right sides. Then solve attempts to minimize the sum of squares of the equation components. For the algorithms for solving nonlinear systems of equations, see Equation Solving Algorithms. When the problem also has bounds, solve calls lsqnonlin to minimize the sum of squares of equation components. See Least-Squares (Model Fitting) Algorithms.


If your objective function is a sum of squares, and you want solve to recognize it as such, write it as sum(expr.^2), and not as expr'*expr or any other form. The internal parser recognizes only explicit sums of squares. For details, see Write Objective Function for Problem-Based Least Squares. For an example, see Nonnegative Linear Least Squares, Problem-Based.

See Also

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