Problems Handled by Optimization Toolbox Functions - MATLAB & Simulink - MathWorks Deutschland

Problems Handled byOptimization ToolboxFunctions

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 thatlb<x<ub(xis scalar)	`fminbnd`
Unconstrained minimization	$\min_{x} f (x)$	`fminunc`, `fminsearch`
Linear programming	$\min_{x} f^{T} x$ such thatA·x≤b,Aeq·x=beq,lb≤x≤ub	`linprog`
Mixed-integer linear programming	$\min_{x} f^{T} x$ such thatA·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 thatA·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 thatc(x) ≤ 0,ceq(x) = 0,A·x≤b,Aeq·x=beq,lb≤x≤ub	`fmincon`
Semi-infinite minimization	$\min_{x} f (x)$ such thatK(x,w) ≤ 0 for allw,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 thatF(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 thatc(x) ≤ 0,ceq(x) = 0,A·x≤b,Aeq·x=beq,lb≤x≤ub	`fminimax`

Goal attainment

$\min_{x, γ} γ$

such thatF(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 thatc(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,nequations,nvariables	`mldivide`(matrix left division)
Nonlinear equation of one variable	f(x) = 0	`fzero`
Nonlinear equations	F(x) = 0,nequations,nvariables	`fsolve`

Least-Squares (Model-Fitting) Problems

Type	Formulation	Solver
Linear least squares	$\min_{x} \frac{1}{2} {‖ C \cdot x - d ‖}_{2}^{2}$ mequations,nvariables	`mldivide`(matrix left division)
Nonnegative linear least squares	$\min_{x} \frac{1}{2} {‖ C \cdot x - d ‖}_{2}^{2}$ such thatx≥ 0	`lsqnonneg`
Constrained linear least squares	$\min_{x} \frac{1}{2} {‖ C \cdot x - d ‖}_{2}^{2}$ such thatA·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 thatlb ≤ x ≤ ub	`lsqnonlin`
Nonlinear curve fitting	$\min_{x} {‖ F (x, x d a t a) - y d a t a ‖}_{2}^{2}$ such thatlb ≤ x ≤ ub	`lsqcurvefit`