Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

A solver can report that a minimization succeeded, and yet the
reported solution can be incorrect. For a rather trivial example,
consider minimizing the function *f*(*x*) = *x*^{3} for *x* between
–2 and 2, starting from the point `1/3`

:

options = optimoptions('fmincon','Algorithm','active-set'); ffun = @(x)x^3; xfinal = fmincon(ffun,1/3,[],[],[],[],-2,2,[],options) Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the default valueof the function tolerance, and constraints were satisfied to within the default value of the constraint tolerance. No active inequalities. xfinal = -1.5056e-008

The true minimum occurs at `x = -2`

. `fmincon`

gives this
report because the function *f*(*x*)
is so flat near *x* = 0.

Another common problem is that a solver finds a local minimum, but you might want a global minimum. For more information, see Local vs. Global Optima.

Lesson: check your results, even if the solver reports that it “found” a local minimum, or “solved” an equation.

This section gives techniques for verifying results.

The initial point can have a large effect on the solution. If you obtain the same or worse solutions from various initial points, you become more confident in your solution.

For example, minimize *f*(*x*) = *x*^{3} + *x*^{4} starting
from the point 1/4:

ffun = @(x)x^3 + x^4; options = optimoptions('fminunc','Algorithm','quasi-newton'); [xfinal fval] = fminunc(ffun,1/4,options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. x = -1.6764e-008 fval = -4.7111e-024

Change the initial point by a small amount, and the solver finds a better solution:

[xfinal fval] = fminunc(ffun,1/4+.001,options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. xfinal = -0.7500 fval = -0.1055

`x = -0.75`

is the global solution; starting
from other points cannot improve the solution.

For more information, see Local vs. Global Optima.

To see if there are better values than a reported solution, evaluate your objective function and constraints at various nearby points.

For example, with the objective function `ffun`

from What Can Be Wrong If The Solver Succeeds?,
and the final point `xfinal = -1.5056e-008`

,
calculate `ffun(xfinal±Δ)`

for some `Δ`

:

delta = .1; [ffun(xfinal),ffun(xfinal+delta),ffun(xfinal-delta)] ans = -0.0000 0.0011 -0.0009

The objective function is lower at `ffun(xfinal-Δ)`

,
so the solver reported an incorrect solution.

A less trivial example:

options = optimoptions(@fmincon,'Algorithm','active-set'); lb = [0,-1]; ub = [1,1]; ffun = @(x)(x(1)-(x(1)-x(2))^2); [x fval exitflag] = fmincon(ffun,[1/2 1/3],[],[],[],[],... lb,ub,[],options) Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the default valueof the function tolerance, and constraints were satisfied to within the default value of the constraint tolerance. Active inequalities (to within options.ConstraintTolerance = 1e-006): lower upper ineqlin ineqnonlin 1 x = 1.0e-007 * 0 0.1614 fval = -2.6059e-016 exitflag = 1

Evaluating `ffun`

at nearby feasible points
shows that the solution `x`

is not a true minimum:

[ffun([0,.001]),ffun([0,-.001]),... ffun([.001,-.001]),ffun([.001,.001])] ans = 1.0e-003 * -0.0010 -0.0010 0.9960 1.0000

The first two listed values are smaller than the computed minimum `fval`

.

If you have a Global
Optimization Toolbox license,
you can use the `patternsearch`

function
to check nearby points.

Double-check your objective function and constraint functions to ensure that they correspond to the problem you intend to solve. Suggestions:

Check the evaluation of your objective function at a few points.

Check that each inequality constraint has the correct sign.

If you performed a maximization, remember to take the negative of the reported solution. (This advice assumes that you maximized a function by minimizing the negative of the objective.) For example, to maximize

*f*(*x*) =*x*–*x*^{2}, minimize*g*(*x*) = –*x*+*x*^{2}:options = optimoptions('fminunc','Algorithm','quasi-newton'); [x fval] = fminunc(@(x)-x+x^2,0,options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. x = 0.5000 fval = -0.2500

The maximum of

*f*is 0.25, the negative of`fval`

.Check that an infeasible point does not cause an error in your functions; see Iterations Can Violate Constraints.

Was this topic helpful?