Main Content

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`

(Global Optimization Toolbox) 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.