## 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

TypeFormulationSolver

Scalar minimization

$\underset{x}{\mathrm{min}}f\left(x\right)$

such that lb < x < ub (x is scalar)

fminbnd

Unconstrained minimization

$\underset{x}{\mathrm{min}}f\left(x\right)$

Linear programming

$\underset{x}{\mathrm{min}}{f}^{T}x$

such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

linprog

Mixed-integer linear programming

$\underset{x}{\mathrm{min}}{f}^{T}x$

such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub, x(intcon) is integer-valued.

intlinprog

$\underset{x}{\mathrm{min}}\frac{1}{2}{x}^{T}Hx+{c}^{T}x$

such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

Constrained minimization

$\underset{x}{\mathrm{min}}f\left(x\right)$

such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

fmincon

Semi-infinite minimization

$\underset{x}{\mathrm{min}}f\left(x\right)$

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 Problems

TypeFormulationSolver

Goal attainment

$\underset{x,\gamma }{\mathrm{min}}\gamma$

such that F(x) – w·γ ≤ goal, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

fgoalattain

Minimax

$\underset{x}{\mathrm{min}}\underset{i}{\mathrm{max}}{F}_{i}\left(x\right)$

such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

fminimax

Equation Solving Problems

TypeFormulationSolver

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

TypeFormulationSolver

Linear least-squares

$\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}$

m equations, n variables

mldivide (matrix left division)

Nonnegative linear-least-squares

$\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}$

such that x ≥ 0

lsqnonneg

Constrained linear-least-squares

$\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}$

such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

lsqlin

Nonlinear least-squares

$\underset{x}{\mathrm{min}}{‖F\left(x\right)‖}_{2}^{2}=\underset{x}{\mathrm{min}}\sum _{i}{F}_{i}^{2}\left(x\right)$

such that lb ≤ x ≤ ub

lsqnonlin

Nonlinear curve fitting

$\underset{x}{\mathrm{min}}{‖F\left(x,xdata\right)-ydata‖}_{2}^{2}$

such that lb ≤ x ≤ ub

lsqcurvefit