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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	$\min_{x} f (x)$ such that lb < x < ub (x is scalar)	`fminbnd`
Unconstrained minimization	$\min_{x} f (x)$	`fminunc`, `fminsearch`
Linear programming	$\min_{x} f^{T} x$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	`linprog`
Mixed-integer linear programming	$\min_{x} f^{T} x$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub, x(intcon) is integer-valued	`intlinprog`
Quadratic programming	$\min_{x} \frac{1}{2} x^{T} H x + c^{T} x$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	`quadprog`
Cone programming	$\min_{x} f^{T} x$ such that $‖ A \cdot x - b ‖ \leq d^{T} \cdot x - γ$ , A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	`coneprog`
Constrained minimization	$\min_{x} f (x)$ such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	`fmincon`
Semi-infinite minimization	$\min_{x} f (x)$ 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

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

Goal attainment

$\min_{x, γ} γ$

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

fgoalattain

Minimax

$\min_{x} \max_{i} F_{i} (x)$

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	$\min_{x} \frac{1}{2} {‖ C \cdot x - d ‖}_{2}^{2}$ m equations, n variables	`mldivide` (matrix left division)
Nonnegative linear least squares	$\min_{x} \frac{1}{2} {‖ C \cdot x - d ‖}_{2}^{2}$ such that x ≥ 0	`lsqnonneg`
Constrained linear least squares	$\min_{x} \frac{1}{2} {‖ C \cdot x - d ‖}_{2}^{2}$ such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub	`lsqlin`
Nonlinear least squares	$\min_{x} {‖ F (x) ‖}_{2}^{2} = \min_{x} \sum_{i} F_{i}^{2} (x)$ such that lb ≤ x ≤ ub	`lsqnonlin`
Nonlinear curve fitting	$\min_{x} {‖ F (x, x d a t a) - y d a t a ‖}_{2}^{2}$ such that lb ≤ x ≤ ub	`lsqcurvefit`