Problems Handled by Optimization Toolbox Functions

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

Minimization Problems

Type	Formulation	Solver
Scalar minimization	$$\underset{x}{\min }f(x)$$ such that lb < x < ub (x is scalar)	fminbnd
Unconstrained minimization	$$\underset{x}{\min }f(x)$$	fminunc, fminsearch
Linear programming	$$\underset{x}{\min }{f}^{T}x$$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	linprog
Mixed-integer linear programming	$$\underset{x}{\min }{f}^{T}x$$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub, x(intcon) is integer-valued	intlinprog
Quadratic programming	$$\underset{x}{\min }\frac{1}{2}{x}^{T}Hx+c^{T}x$$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	quadprog
Cone programming	$$\underset{x}{\min }{f}^{T}x$$ such that $$\Vert A\cdot x-b\Vert \le d^T\cdot x-\gamma$$, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	coneprog
Constrained minimization	$$\underset{x}{\min }f(x)$$ such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	fmincon
Semi-infinite minimization	$$\underset{x}{\min }f(x)$$ such that K(x,w) ≤ 0 for all w, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	fseminf

Multiobjective Optimization Problems

Type	Formulation	Solver
Goal attainment	$$\underset{x,\gamma }{\min }\gamma$$ such that F(x) – w·γ ≤ goal, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	fgoalattain
Minimax	$$\underset{x}{\min }\underset{i}{\max }{F}_i(x)$$ such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	fminimax

Equation Solving Problems

Type	Formulation	Solver
Linear equations	C·x = d, n equations, n variables	mldivide (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	$$\underset{x}{\min }\frac{1}{2}{\Vert C\cdot x-d\Vert }_2^2$$ m equations, n variables	mldivide (matrix left division)
Nonnegative linear least squares	$$\underset{x}{\min }\frac{1}{2}{\Vert C\cdot x-d\Vert }_2^2$$ such that x ≥ 0	lsqnonneg
Constrained linear least squares	$$\underset{x}{\min }\frac{1}{2}{\Vert C\cdot x-d\Vert }_2^2$$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	lsqlin
Nonlinear least squares	$$\underset{x}{\min }{\Vert F(x)\Vert }_2^2=\underset{x}{\min }{\displaystyle \sum _i{F}_i^2(x)}$$ such that lb ≤ x ≤ ub	lsqnonlin
Nonlinear curve fitting	$$\underset{x}{\min }{\Vert F(x,xdata)-ydata\Vert }_2^2$$ such that lb ≤ x ≤ ub	lsqcurvefit

Related Topics

• Solver-Based Optimization Problem Setup