Why did I get two different results in nonlinear programing problems

clear,clc

A = [1 4 5
     4 2 6
     5 6 3];

prob = optimproblem;
x = optimvar('x',3);

con = sum(x.^2) == 1;

prob.Constraints.con = con;
prob.Objective = x'*A*x;
x0.x = zeros(3,1);
% x0.x = rand(3,1);

show(prob)

  OptimizationProblem : 

	Solve for:
       x

	minimize :
       x(1)^2 + 2*x(2)^2 + 3*x(3)^2 + 4*x(2)*x(1) + 4*x(1)*x(2) + 6*x(3)*x(2) + 6*x(2)*x(3) + 5*x(3)*x(1) + 5*x(1)*x(3)


	subject to con:
       sum(x.^2) == 1
     

[sol,fval,flag,out] = solve(prob,x0)

Solving problem using fmincon.

Converged to an infeasible point.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance but constraints are not
satisfied to within the value of the constraint tolerance.


Consider enabling the interior point method feasibility mode.

sol = struct with fields:
    x: [3x1 double]

fval = 0

flag = 
    NoFeasiblePointFound

out = struct with fields:
              iterations: 1
               funcCount: 2
         constrviolation: 1
                stepsize: 0
               algorithm: 'interior-point'
           firstorderopt: 0
            cgiterations: 0
                 message: 'Converged to an infeasible point....'
            bestfeasible: []
     objectivederivative: "closed-form"
    constraintderivative: "closed-form"
                  solver: 'fmincon'

sol.x

ans = 3x1

0 0 0

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when the initial point is zeres(0),fval is 0,sol.x is [0;0;0].

but when it is rand(3,1),fval is -3.66.

I can't understand this reason

Answer 1

[Torsten]

As you can see from the solver message, with x0 = [0 0 0], fmincon converged to an infeasible point. So you didn't get a solution, but "fmincon" failed.

Answer 2

[John D'Errico]

You have a nonlinear problem. You need to understand that given any set of starting values, an optimizer will sometimes find a solution though it need not always find the same solution since nonlinear problems will often have multiple solutions, sometimes it will get stuck at a non-solution but unable to find someplae better to look from that point, and thirdly, sometimes it will fail to go anywhere, being unable to even find a feasible point to begin iterations.

When you started the solver at all zeros, it got stuck, in the last mode. I might point out that all zeros is often the worst possible place to start a nonlinear solver.

Remember that nonlinear solvers are not some god-like computational beings, always able to solve any problem. They are far closer to my oft used example of a blind person placed on the earth at some point, and then asked to find the point of lowest elevation. The only gear this person is given is a cane to determine a direction to look next, and an altimeter to learn the current elevation. (Ok, some scuba gear might be nice too.) But clearly this individual will often fail to find a viable point, and unless your initial point is a good one, they will often fail to find the globally best location, in the depths of the Pacific Ocean.