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What Is First-Order Optimality Measure? |

First-order optimality is a measure of how close a point *x* is
to optimal. Most Optimization Toolbox™ solvers use this measure,
though it has different definitions for different algorithms. First-order
optimality is a necessary condition, but it is not a sufficient condition.
In other words:

The first-order optimality measure must be zero at a minimum.

A point with first-order optimality equal to zero is not necessarily a minimum.

For general information about first-order optimality, see Nocedal and Wright [31]. For specifics about the first-order optimality measures for Optimization Toolbox solvers, see Unconstrained Optimality, Constrained Optimality Theory, and Constrained Optimality in Solver Form.

The `TolFun` tolerance relates to the first-order
optimality measure. Typically, if the first-order optimality measure
is less than `TolFun`, solver iterations end.

Some solvers or algorithms use *relative* first-order
optimality as a stopping criterion. Solver iterations end if the first-order
optimality measure is less than *μ* times `TolFun`,
where *μ* is either:

The infinity norm (maximum) of the gradient of the objective function at

`x0`The infinity norm (maximum) of inputs to the solver, such as

`f`or`b`in`linprog`or`H`in`quadprog`

A relative measure attempts to account for the scale of a problem. Multiplying an objective function by a very large or small number does not change the stopping condition for a relative stopping criterion, but does change it for an unscaled one.

Solvers with enhanced exit messages state, in the stopping criteria details, when they use relative first-order optimality.

For a smooth unconstrained problem,

the first-order optimality
measure is the infinity norm (meaning maximum absolute value) of ∇*f*(*x*),
which is:

This measure of optimality is based on the familiar condition for a smooth function to achieve a minimum: its gradient must be zero. For unconstrained problems, when the first-order optimality measure is nearly zero, the objective function has gradient nearly zero, so the objective function could be near a minimum. If the first-order optimality measure is not small, the objective function is not minimal.

This section summarizes the theory behind the definition of first-order optimality measure for constrained problems. The definition as used in Optimization Toolbox functions is in Constrained Optimality in Solver Form.

For a smooth constrained problem, let *g* and *h* be
vector functions representing all inequality and equality constraints
respectively (meaning bound, linear, and nonlinear constraints):

The meaning of first-order optimality in this case is more complex than for unconstrained problems. The definition is based on the Karush-Kuhn-Tucker (KKT) conditions. The KKT conditions are analogous to the condition that the gradient must be zero at a minimum, modified to take constraints into account. The difference is that the KKT conditions hold for constrained problems.

The KKT conditions use the auxiliary Lagrangian function:

(3-1) |

The vector *λ*, which is the concatenation
of *λ _{g}* and

(3-2) |

(3-3) |

(3-4) |

Solvers do not use the three expressions in Equation 3-4 in the calculation of optimality measure.

The optimality measure associated with Equation 3-2 is

(3-5) |

The optimality measure associated with Equation 3-3 is

(3-6) |

where the norm in Equation 3-6 means infinity norm (maximum) of the vector .

The combined optimality measure is the maximum of the values
calculated in Equation 3-5 and Equation 3-6. Solvers that
accept nonlinear constraint functions report constraint violations *g*(*x*) > 0 or |*h*(*x*)| > 0 as `TolCon` tolerance
violations. See Tolerances and Stopping Criteria.

Most constrained toolbox solvers separate their calculation of first-order optimality measure into bounds, linear functions, and nonlinear functions. The measure is the maximum of the following two norms, which correspond to Equation 3-5 and Equation 3-6:

(3-7) |

(3-8) |

where the norm of the
vectors in Equation 3-7 and Equation 3-8 is the infinity
norm (maximum). The subscripts on the Lagrange multipliers correspond
to solver Lagrange multiplier structures. See Lagrange Multiplier Structures. The summations in Equation 3-7 range over all
constraints. If a bound is ±`Inf`, that term
is not constrained, so it is not part of the summation.

For some large-scale problems with only linear equalities, the
first-order optimality measure is the infinity norm of the *projected* gradient.
In other words, the first-order optimality measure is the size of
the gradient projected onto the null space of `Aeq`.

For least-squares solvers and trust-region-reflective algorithms,
in problems with bounds alone, the first-order optimality measure
is the maximum over *i* of |*v _{i}**

If *x _{i}* is at a bound,

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