## First-Order Optimality Measure

### 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.

### Stopping Rules Related to First-Order Optimality

The OptimalityTolerance tolerance relates to the first-order optimality measure. Typically, if the first-order optimality measure is less than OptimalityTolerance, 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 OptimalityTolerance, 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.

### Unconstrained Optimality

For a smooth unconstrained problem,

$\underset{x}{\mathrm{min}}f\left(x\right),$

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.

### Constrained Optimality Theory

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:

 $L\left(x,\lambda \right)=f\left(x\right)+\sum {\lambda }_{g,i}{g}_{i}\left(x\right)+\sum {\lambda }_{h,i}{h}_{i}\left(x\right).$ (1)
The vector λ, which is the concatenation of λg and λh, is the Lagrange multiplier vector. Its length is the total number of constraints.

The KKT conditions are:

 ${\nabla }_{x}L\left(x,\lambda \right)=0,$ (2)
 (3)
 $\left\{\begin{array}{c}g\left(x\right)\le 0,\\ h\left(x\right)=0,\\ {\lambda }_{g,i}\ge 0.\end{array}$ (4)
Solvers do not use the three expressions in Equation 4 in the calculation of optimality measure.

The optimality measure associated with Equation 2 is

 $‖{\nabla }_{x}L\left(x,\lambda \right)‖=‖\nabla f\left(x\right)+\sum {\lambda }_{g,i}\nabla {g}_{i}\left(x\right)+\sum {\lambda }_{h,i}\nabla {h}_{h,i}\left(x\right)‖.$ (5)
The optimality measure associated with Equation 3 is
 $‖\stackrel{\to }{{\lambda }_{g}g}\left(x\right)‖,$ (6)
where the norm in Equation 6 means infinity norm (maximum) of the vector $\stackrel{\to }{{\lambda }_{g,i}{g}_{i}}\left(x\right)$.

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

### Constrained Optimality in Solver Form

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 5 and Equation 6:

 (7)
 $‖\stackrel{\to }{|{l}_{i}-{x}_{i}|{\lambda }_{lower,i}},\stackrel{\to }{|{x}_{i}-{u}_{i}|{\lambda }_{upper,i}},\stackrel{\to }{|{\left(Ax-b\right)}_{i}|{\lambda }_{ineqlin,i}},\stackrel{\to }{|{c}_{i}\left(x\right)|{\lambda }_{ineqnonlin,i}}‖,$ (8)

where the norm of the vectors in Equation 7 and Equation 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 7 range over all constraints. If a bound is ±Inf, that term is not constrained, so it is not part of the summation.

#### Linear Equalities Only

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.

#### Bounded Least-Squares and Trust-Region-Reflective Solvers

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 |vi*gi|. Here gi is the ith component of the gradient, x is the current point, and

If xi is at a bound, vi is zero. If xi is not at a bound, then at a minimizing point the gradient gi should be zero. Therefore the first-order optimality measure should be zero at a minimizing point.