# Hessian

A Hessian is a matrix (array) of second-order (partial) derivatives of an objective variable. It is sometimes useful to speed up nonlinear optimization.

## Use in the DefineOptimization function

«Hessian» can be specified as an optional parameter to the DefineOptimization function, to speed up non-linear computations by the Optimization Engine.

If you can explicitly (analytically) compute the first or second derivatives of the objective and left-hand side constraint functions, expressions for these can be optionally provided to the optimizer. When these are available, the optimizer can evaluate these expressions rather than estimating the derivatives through finite differencing. This can speed up convergence by reducing the number of evaluations, and by providing a "cleaner" computation because derivatives, and especially second derivates, can be quite inaccurate or noisy when computed through finite differencing.

The first derivative of the objective is called the *gradient* and is specified in the optional parameter «Gradient». It should compute a value dimensioned by the `Vars`

index.

The first derivative of the left-hand side constraint is the *Jacobian*. It should be dimensioned by the `Constraint`

and `Vars`

indexes. If one is provided, both must be (unless there isn't an objective, or are zero constraints).

Analytica allows expressions for the Hessian (second derivative of the objective) and *LhsHessian* (second derivative of the constraint left-hand side) to be specified using the optional parameters «Hessian» and «LhsHessian». Hessian should compute a derivative with the same dimensionality as Jacobian (vars), and LhsHessian should have the same dimensionality as Jacobian (constraints x vars). The Hessian gives the optimizer information about the curvature.

The built-in GRG Nonlinear solver is most likely to make good use of derivatives. To ensure these are used well, set the ObjNl and LhsNl to `N`

(smooth non-linear).

The GRG Nonlinear solver will evaluate Gradient and Jacobian expressions, but not Hessians. The Knitro engine (an optional add-on engine) can make use of Hessian information. The Quadratic and SOCP Barrier engines use Hessian information, but these are based on the supplied matrices of coefficients, not DefineOptimization.

## Use in the OptEngineInfo function

"Hessian" as one of the outputs of the optional «Item» parameter to the OptEngineInfo function, refers to the number of Hessian evaluations that have been performed by an Optimization Engine.

## See Also

- Hessian matrix -- a Wikipedia article
- Derivative -- a Wikipedia article
- Optimization with derivatives
- DefineOptimization
- OptEngineInfo

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