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Choosing a Solver

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Optimization Decision Table Problems Handled by Optimization Toolbox Functions

Optimization Decision Table

The following table is designed to help you choose a solver. It does not address multiobjective optimization or equation solving. There are more details on all the solvers in Problems Handled by Optimization Toolbox Functions.

Use the table as follows:

1. Identify your objective function as one of five types:

   • Linear

   • Quadratic

   • Sum-of-squares (Least squares)

   • Smooth nonlinear

   • Nonsmooth

2. Identify your constraints as one of five types:

   • None (unconstrained)

   • Bound

   • Linear (including bound)

   • General smooth

   • Discrete (integer)

3. Use the table to identify a relevant solver.

In this table:

• Blank entries means there is no Optimization Toolbox solver specifically designed for this type of problem.

• * means relevant solvers are found in Genetic Algorithm and Direct Search Toolbox functions (licensed separately from Optimization Toolbox solvers).

• fmincon applies to most smooth objective functions with smooth constraints. It is not listed as a preferred solver for least squares or linear or quadratic programming because the listed solvers are usually more efficient.

• The table has suggested functions, but it is not meant to unduly restrict your choices. For example, fmincon is known to be effective on some nonsmooth problems.

• The Genetic Algorithm and Direct Search Toolbox function ga can be programmed to address discrete problems. It is not listed in the table because additional programming is needed to solve discrete problems.

Solvers by Objective and Constraint

Constraint Type	Objective Type
	Linear	Quadratic	Least Squares	Smooth nonlinear	Nonsmooth
None	n/a (f = const, or min = )	quadprog, Theory, Examples	\, lsqcurvefit, lsqnonlin, Theory, Examples	fminsearch, fminunc, Theory, Examples	fminsearch, *
Bound	linprog, Theory, Examples	quadprog, Theory, Examples	lsqcurvefit, lsqlin, lsqnonlin, lsqnonneg, Theory, Examples	fminbnd, fmincon, fseminf, Theory, Examples	*
Linear	linprog, Theory, Examples	quadprog, Theory, Examples	lsqlin, Theory, Examples	fmincon, fseminf, Theory, Examples	*
General smooth	fmincon, Theory, Examples	fmincon, Theory, Examples	fmincon, Theory, Examples	fmincon, fseminf, Theory, Examples	*
Discrete	bintprog, Theory, Example

Note   This table does not list multiobjective solvers nor equation solvers. See Problems Handled by Optimization Toolbox Functions for a complete list of problems addressed by Optimization Toolbox functions.

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Problems Handled by Optimization Toolbox Functions

The following tables show the functions available for minimization, equation solving, multiobjective optimization, and solving least-squares or data-fitting problems.

Minimization Problems

Type	Formulation	Solver
Scalar minimization	such that l < x < u (x is scalar)	fminbnd
Unconstrained minimization		fminunc, fminsearch
Linear programming	such that A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u	linprog
Quadratic programming	such that A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u	quadprog
Constrained minimization	such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u	fmincon
Semi-infinite minimization	such that K(x,w) ≤ 0 for all w, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u	fseminf
Binary integer programming	such that A·x ≤ b, Aeq·x = beq, x binary	bintprog

Multiobjective Problems

Type	Formulation	Solver
Goal attainment	such that F(x) – w·γ ≤ goal, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u	fgoalattain
Minimax	such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u	fminimax

Equation Solving Problems

Type	Formulation	Solver
Linear equations	C·x = d, n equations, n variables	\ (matrix left division)
Nonlinear equation of one variable	f(x) = 0	fzero
Nonlinear equations	F(x) = 0, n equations, n variables	fsolve

Least-Squares (Model-Fitting) Problems

Type	Formulation	Solver
Linear least-squares	m equations, n variables	\ (matrix left division)
Nonnegative linear-least-squares	such that x ≥ 0	lsqnonneg
Constrained linear-least-squares	such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	lsqlin
Nonlinear least-squares	such that lb ≤ x ≤ ub	lsqnonlin
Nonlinear curve fitting	such that lb ≤ x ≤ ub	lsqcurvefit

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Introduction to Optimization Toolbox Solvers

Writing Objective Functions

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