The initial point seems to be a local minimum or solution because the first-order optimality measure is close to 0. You might be unhappy with this result, since the solver did not improve your initial point.
If you are unsure that the initial point is truly a local minimum, try:
Starting from different points — see Change the Initial Point.
Checking that your objective and constraints are defined correctly (for example, do they return the correct values at some points?) — see Check your Objective and Constraint Functions. Check that an infeasible point does not cause an error in your functions; see Iterations Can Violate Constraints.
Changing tolerances, such as
StepTolerance — see Use Appropriate Tolerances.
Scaling your problem so each coordinate has about the same effect — see Rescale the Problem.
The solver might have reached a local minimum, but cannot be
certain because the first-order optimality measure is not less than
OptimalityTolerance tolerance. (To learn more
about first-order optimality measure, see First-Order Optimality Measure.) To see if the reported solution
is reliable, consider the following suggestions.
|1. Nonsmooth Functions|
|2. Rerun Starting At Final Point|
|3. Try a Different Algorithm|
|4. Change Tolerances|
|5. Rescale the Problem|
|6. Check Nearby Points|
|7. Change Finite Differencing Options|
|8. Provide Analytic Gradients or Jacobian|
|9. Provide a Hessian|
If you try to minimize a nonsmooth function, or have nonsmooth constraints, “Local Minimum Possible” can be the best exit message. This is because the first-order optimality conditions do not apply at a nonsmooth point.
To satisfy yourself that the solution is adequate, try to Check Nearby Points.
Restarting an optimization at the final point can lead to a solution with a better first-order optimality measure. A better (lower) first-order optimality measure gives you more reason to believe that the answer is reliable.
For example, consider the following minimization problem, taken from the example Using Symbolic Mathematics with Optimization Toolbox™ Solvers:
options = optimoptions('fminunc','Algorithm','quasi-newton'); funh = @(x)log(1 + (x(1) - 4/3)^2 + 3*(x(2) - (x(1)^3 - x(1)))^2); [xfinal fval exitflag] = fminunc(funh,[-1;2],options) Local minimum possible. fminunc stopped because it cannot decrease the objective function along the current search direction. xfinal = 1.3333 1.0370 fval = 8.5265e-014 exitflag = 5
The exit flag value of
5 indicates that the
first-order optimality measure was above the function tolerance. Run
the minimization again starting from
[xfinal2 fval2 exitflag2] = fminunc(funh,xfinal,options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. xfinal2 = 1.3333 1.0370 fval2 = 6.5281e-014 exitflag2 = 1
The local minimum is “found,” not “possible,”
and the exitflag is
The two solutions are virtually identical. Yet the second run has
a more satisfactory exit message, since the first-order optimality
measure was low enough:
7.5996e-007, instead of
Many solvers give you a choice of algorithm. Different algorithms can lead to the use of different stopping criteria.
For example, Rerun Starting At Final Point returns
5 from the first run. This run uses the
The trust-region algorithm requires a user-supplied gradient.
a calculation of the objective function and gradient:
function [f gradf] = betopt(x) g = 1 + (x(1)-4/3)^2 + 3*(x(2) - (x(1)^3-x(1)))^2; f = log(g); gradf(1) = 2*(x(1)-4/3) + 6*(x(2) - (x(1)^3-x(1)))*(1-3*x(1)^2); gradf(1) = gradf(1)/g; gradf(2) = 6*(x(2) - (x(1)^3 -x(1)))/g;
Running the optimization using the
results in a different exitflag:
options = optimoptions('fminunc','Algorithm','trust-region','SpecifyObjectiveGradient',true); [xfinal3 fval3 exitflag3] = fminunc(@betopt,[-1;2],options) Local minimum possible. fminunc stopped because the final change in function value relative to its initial value is less than the default value of the function tolerance. xfinal3 = 1.3333 1.0370 fval3 = 7.6659e-012 exitflag3 = 3
The exit condition is better than the
though it is still not the best. Rerunning the algorithm from the
final point produces a better point, with extremely small first-order
[xfinal4 fval4 eflag4 output4] = fminunc(@betopt,xfinal3,options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. xfinal4 = 1.3333 1.0370 fval4 = 0 eflag4 = 1 output4 = iterations: 1 funcCount: 2 cgiterations: 1 firstorderopt: 7.5497e-11 algorithm: 'trust-region' message: 'Local minimum found. Optimization completed because the size o...' constrviolation: 
Sometimes tightening or loosening tolerances leads to a more
satisfactory result. For example, choose a smaller value of
a Different Algorithm section:
options = optimoptions('fminunc','Algorithm','trust-region',... 'OptimalityTolerance',1e-8,'SpecifyObjectiveGradient',true); % default=1e-6 [xfinal3 fval3 eflag3 output3] = fminunc(@betopt,[-1;2],options) Local minimum found. Optimization completed because the size of the gradient is less than the selected value of the function tolerance. xfinal3 = 1.3333 1.0370 fval3 = 0 eflag3 = 1 output3 = iterations: 15 funcCount: 16 cgiterations: 12 firstorderopt: 7.5497e-11 algorithm: 'trust-region' message: 'Local minimum found. Optimization completed because the size...' constrviolation: 
fminunc took one more iteration than before,
arriving at a better solution.
Try to have each coordinate give about the same effect on the objective and constraint functions by scaling and centering. For examples, see Center and Scale Your Problem.
Evaluate your objective function and constraints, if they exist, at points near the final point. If the final point is a local minimum, nearby feasible points have larger objective function values. See Check Nearby Points for an example.
If you have a Global Optimization Toolbox license,
try running the
patternsearch (Global Optimization Toolbox) solver
from the final point.
nearby points, and accepts all types of constraints.
Central finite differences are more time-consuming to evaluate, but are much more accurate. Use central differences when your problem can have high curvature.
To choose central differences at the command line, use
instead of the default
If you do not provide gradients or Jacobians, solvers estimate gradients and Jacobians by finite differences. Therefore, providing these derivatives can save computational time, and can lead to increased accuracy. The problem-based approach can provide gradients automatically; see Automatic Differentiation in Optimization Toolbox.
For constrained problems, providing a gradient has another advantage.
A solver can reach a point
x such that
feasible, but finite differences around
lead to an infeasible point. In this case, a solver can fail or halt
prematurely. Providing a gradient allows a solver to proceed.
Provide gradients or Jacobians in the files for your objective function and nonlinear constraint functions. For details of the syntax, see Writing Scalar Objective Functions, Writing Vector and Matrix Objective Functions, and Nonlinear Constraints.
To check that your gradient or Jacobian function is correct,
CheckGradients option, as described in Checking Validity of Gradients or Jacobians.
If you have a Symbolic Math Toolbox™ license, you can calculate gradients and Hessians programmatically. For an example, see Calculate Gradients and Hessians Using Symbolic Math Toolbox™.
For examples using gradients and Jacobians, see Minimization with Gradient and Hessian, Nonlinear Constraints with Gradients, Calculate Gradients and Hessians Using Symbolic Math Toolbox™, Solve Nonlinear System Without and Including Jacobian, and Large Sparse System of Nonlinear Equations with Jacobian. For automatic differentiation in the problem-based approach, see Effect of Automatic Differentiation in Problem-Based Optimization.
Solvers often run more reliably and with fewer iterations when you supply a Hessian.
The following solvers and algorithms accept Hessians:
Write the Hessian as a separate function. For an example, see fmincon Interior-Point Algorithm with Analytic Hessian.
Give the Hessian as the third output of the objective function. For
an example, see Minimization with Dense Structured Hessian, Linear Equalities.
Give the Hessian as the third output of the objective function. For
an example, see Minimization with Gradient and Hessian.
If you have a Symbolic Math Toolbox license, you can calculate gradients and Hessians programmatically. For an example, see Calculate Gradients and Hessians Using Symbolic Math Toolbox™. To provide a Hessian in the problem-based approach, see Supply Derivatives in Problem-Based Workflow.
The example in Calculate Gradients and Hessians Using Symbolic Math Toolbox™ shows
fmincon taking 77 iterations
without a Hessian, but only 19 iterations with a Hessian.