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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,
$$\underset{x}{\mathrm{min}}f(x),$$
the first-order optimality measure is the infinity norm (meaning maximum absolute value) of ∇f(x), which is:
$$\text{first-orderoptimalitymeasure=}\underset{i}{\mathrm{max}}\left|{\left(\nabla f(x)\right)}_{i}\right|={\Vert \nabla f(x)\Vert}_{\infty}.$$
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.
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):
$$\underset{x}{\mathrm{min}}f(x)\text{subjectto}g(x)\le 0,\text{}h(x)=0.$$
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:
$$L(x,\lambda )=f(x)+{\displaystyle \sum {\lambda}_{g,i}{g}_{i}(x)}+{\displaystyle \sum {\lambda}_{h,i}{h}_{i}(x)}.$$ | (3-1) |
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.
$${\nabla}_{x}L(x,\lambda )=0,$$ | (3-2) |
$${\lambda}_{g,i}{g}_{i}(x)=0\text{}\forall i,$$ | (3-3) |
$$\{\begin{array}{c}g(x)\le 0,\\ h(x)=0,\\ {\lambda}_{g,i}\ge 0.\end{array}$$ | (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
$$\Vert {\nabla}_{x}L(x,\lambda \Vert =\Vert \nabla f(x)+{\displaystyle \sum {\lambda}_{g,i}\nabla {g}_{i}(x)+{\displaystyle \sum {\lambda}_{h,i}\nabla {h}_{h,i}(x)}}\Vert .$$ | (3-5) |
The optimality measure associated with Equation 3-3 is
$$\Vert \overrightarrow{{\lambda}_{g}g}(x)\Vert ,$$ | (3-6) |
where the norm in Equation 3-6 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 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:
$$\begin{array}{l}\Vert {\nabla}_{x}L(x,\lambda \Vert =\Vert \nabla f(x)+{A}^{T}{\lambda}_{ineqlin}+Ae{q}^{T}{\lambda}_{eqlin}\\ \text{}+{\displaystyle \sum {\lambda}_{ineqnonlin,i}\nabla {c}_{i}(x)+{\displaystyle \sum {\lambda}_{eqnonlin,i}\nabla ce{q}_{i}(x)}}\Vert ,\end{array}$$ | (3-7) |
$$\Vert \overrightarrow{\left|{l}_{i}-{x}_{i}\right|{\lambda}_{lower,i}},\overrightarrow{\left|{x}_{i}-{u}_{i}\right|{\lambda}_{upper,i}},\overrightarrow{\left|{(Ax-b)}_{i}\right|{\lambda}_{ineqlin,i}},\overrightarrow{\left|{c}_{i}(x)\right|{\lambda}_{ineqnonlin,i}}\Vert ,$$ | (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}*g_{i}|. Here g_{i} is the ith component of the gradient, x is the current point, and
$${v}_{i}=\{\begin{array}{ll}\left|{x}_{i}-{b}_{i}\right|\hfill & \text{ifthenegativegradientpointstowardbound}{b}_{i}\hfill \\ 1\hfill & \text{otherwise}\text{.}\hfill \end{array}$$
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.