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,

$$\underset{x}{\min }f(x),$$

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

$$\text{first-order optimality measure = }\underset{i}{\max }\left|{\left(\nabla f(x)\right)}_i\right|={\Vert \nabla f(x)\Vert }_{\infty }.$$

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):

$$\underset{x}{\min }f(x)\text{ subject to }g(x)\le 0,h(x)=0.$$

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:

$$L(x,\lambda )=f(x)+{\displaystyle \sum {\lambda }_{g,i}{g}_i(x)}+{\displaystyle \sum {\lambda }_{h,i}{h}_i(x)}.$$

(1)

The vector λ, which is the concatenation of λ_g and λ_h, is the Lagrange multiplier vector. Its length is the total number of constraints.

The KKT conditions are:

$${\nabla }_{x}L(x,\lambda )=0,$$

(2)

$${\lambda }_{g,i}{g}_i(x)=0\forall i,$$

(3)

$$\{\begin{array}{c}g(x)\le 0,\\ h(x)=0,\\ {\lambda }_{g,i}\ge 0.\end{array}$$

(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

$$\Vert {\nabla }_{x}L(x,\lambda )\Vert =\Vert \nabla f(x)+{\displaystyle \sum {\lambda }_{g,i}\nabla g_i(x)+{\displaystyle \sum {\lambda }_{h,i}\nabla h_{h,i}(x)}}\Vert .$$

(5)

The optimality measure associated with Equation 3 is

$$\Vert \overrightarrow{{\lambda }_{g}g}(x)\Vert ,$$

(6)

where the norm in Equation 6 means infinity norm (maximum) of the vector $$\overrightarrow{{\lambda }_{g,i}{g}_i}(x)$$.

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:

$$\begin{array}{l}\Vert {\nabla }_{x}L(x,\lambda )\Vert =\Vert \nabla f(x)+A^T{\lambda }_{ineqlin}+Ae{q}^T{\lambda }_{eqlin}\\ +{\displaystyle \sum {\lambda }_{ineqnonlin,i}\nabla c_i(x)+{\displaystyle \sum {\lambda }_{eqnonlin,i}\nabla ce{q}_i(x)}}\Vert ,\end{array}$$

(7)

$$\Vert \overrightarrow{\left|l_i-x_i\right|{\lambda }_{lower,i}},\overrightarrow{\left|x_i-u_i\right|{\lambda }_{upper,i}},\overrightarrow{\left|{(Ax-b)}_i\right|{\lambda }_{ineqlin,i}},\overrightarrow{\left|c_i(x)\right|{\lambda }_{ineqnonlin,i}}\Vert ,$$

(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 |v_i*g_i|. Here g_i is the ith component of the gradient, x is the current point, and

$$v_i=\{\begin{array}{ll}\left|x_i-b_i\right|\hfill & \text{if the negative gradient points toward bound }{b}_i\hfill \\ 1\hfill & \text{otherwise}\text{.}\hfill \end{array}$$

If x_i is at a bound, v_i is zero. If x_i is not at a bound, then at a minimizing point the gradient g_i should be zero. Therefore the first-order optimality measure should be zero at a minimizing point.

Related Topics

• Solver Outputs and Iterative Display