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The Hessian for an unconstrained problem is the matrix of second
derivatives of the objective function *f*:

$$\text{Hessian}{H}_{ij}=\frac{{\partial}^{2}f}{\partial {x}_{i}\partial {x}_{j}}.$$

**Quasi-Newton Algorithm**—`fminunc`

returns an estimated Hessian matrix at the solution. It computes the estimate by finite differences.**Trust-Region Algorithm**—`fminunc`

returns a Hessian matrix at the next-to-last iterate.If you supply a Hessian in the objective function,

`fminunc`

returns this Hessian.If you supply a

`HessMult`

function,`fminunc`

returns the`Hinfo`

matrix from the`HessMult`

function. For more information, see`HessMult`

in the`trust-region`

section of the`fminunc`

`options`

table.Otherwise,

`fminunc`

returns an approximation from a sparse finite difference algorithm on the gradients.

This Hessian is accurate for the next-to-last iterate. However, the next-to-last iterate might not be close to the final point.

The reason the

`trust-region`

algorithm returns the Hessian at the next-to-last point is for efficiency.`fminunc`

uses the Hessian internally to compute its next step. When`fminunc`

reaches a stopping condition, it does not need to compute the next step, so does not compute the Hessian.

The Hessian for a constrained problem is the Hessian of the
Lagrangian. For an objective function *f*, nonlinear
inequality constraint vector *c*, and nonlinear equality
constraint vector *ceq*, the Lagrangian is

$$L=f+{\displaystyle \sum _{i}{\lambda}_{i}{c}_{i}}+{\displaystyle \sum _{j}{\lambda}_{j}ce{q}_{j}}.$$

The *λ _{i}* are Lagrange
multipliers; see First-Order Optimality Measure and Lagrange Multiplier Structures. The Hessian of the Lagrangian
is

$$H={\nabla}^{2}L={\nabla}^{2}f+{\displaystyle \sum _{i}{\lambda}_{i}{\nabla}^{2}{c}_{i}}+{\displaystyle \sum _{j}{\lambda}_{j}{\nabla}^{2}ce{q}_{j}}.$$

`fmincon`

has four algorithms,
with several options for Hessians, as described in fmincon Trust Region Reflective Algorithm, fmincon Active Set Algorithm, and fmincon Interior Point Algorithm. `fmincon`

returns
the following for the Hessian:

—`active-set`

or`sqp`

Algorithm`fmincon`

returns the Hessian approximation it computes at the next-to-last iterate.`fmincon`

computes a quasi-Newton approximation of the Hessian matrix at the solution in the course of its iterations. This approximation does not, in general, match the true Hessian in every component, but only in certain subspaces. Therefore the Hessian that`fmincon`

returns can be inaccurate. For more details of the`active-set`

calculation, see SQP Implementation.—`trust-region-reflective`

Algorithm`fmincon`

returns the Hessian it computes at the next-to-last iterate.If you supply a Hessian in the objective function,

`fmincon`

returns this Hessian.If you supply a

`HessMult`

function,`fmincon`

returns the`Hinfo`

matrix from the`HessMult`

function. For more information, see**Trust-Region-Reflective Algorithm**in`fmincon`

`options`

.Otherwise,

`fmincon`

returns an approximation from a sparse finite difference algorithm on the gradients.

This Hessian is accurate for the next-to-last iterate. However, the next-to-last iterate might not be close to the final point.

The reason the

`trust-region-reflective`

algorithm returns the Hessian at the next-to-last point is for efficiency.`fmincon`

uses the Hessian internally to compute its next step. When`fmincon`

reaches a stopping condition, it does not need to compute the next step, so does not compute the Hessian.`interior-point`

AlgorithmIf the

`Hessian`

option is`lbfgs`

or`fin-diff-grads`

, or if you supply a Hessian multiply function (`HessMult`

),`fmincon`

returns`[]`

for the Hessian.If the

`Hessian`

option is`bfgs`

(the default),`fmincon`

returns a quasi-Newton approximation to the Hessian at the final point. This Hessian can be inaccurate, as in the`active-set`

or`sqp`

algorithm Hessian.If the

`Hessian`

option is`user-supplied`

,`fmincon`

returns the user-supplied Hessian at the final point.

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