## 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

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

`quadprog`

Cone programming

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

such that $‖A\cdot x-b‖\le {d}^{T}\cdot x-\gamma$, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub

`coneprog`

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 Optimization 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`