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Compare lsqnonlin and fmincon for Constrained Nonlinear Least Squares

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This example shows that lsqnonlin generally takes fewer function evaluations than fmincon when solving constrained least-squares problems. Both solvers use the fmincon 'interior-point' algorithm for solving the problem. Yet lsqnonlin typically solves problems in fewer function evaluations. The reason is that lsqnonlin has more information to work with. fmincon minimizes the sum of squares given as ∑iFi2, where F is a vector function. In contrast, lsqnonlin works with the entire vector F, meaning it has access to all the components of the sum. In other words, fmincon can access only the value of the sum, but lsqnonlin can access each component separately.

The runlsqfmincon helper function listed at the end of this example creates a series of scaled Rosenbrock-type problems with nonlinear constraints for N ranging from 1 to 50, where the number of problem variables is 2N. For a description of the Rosenbrock function, see Solve a Constrained Nonlinear Problem, Problem-Based. The function also plots the results, showing:

  • Number of iterations

  • Number of function counts

  • Resulting residuals

The plots show the differences between finite-difference derivative estimates (labeled FD) and derivatives calculated using automatic differentiation. For a description of automatic differentiation, see Automatic Differentiation Background.

runlsqfmincon;

Figure contains an axes object. The axes object with title Iterations, xlabel N, ylabel Iterations contains 4 objects of type line. These objects represent lsqnonlin, lsqnonlin FD, fmincon, fmincon FD.

Figure contains an axes object. The axes object with title Function count, log-scaled, xlabel N, ylabel log(Function count) contains 4 objects of type line. These objects represent lsqnonlin, lsqnonlin FD, fmincon, fmincon FD.

Figure contains an axes object. The axes object with title Residual, xlabel N, ylabel Residual contains 4 objects of type line. These objects represent lsqnonlin, lsqnonlin FD, fmincon, fmincon FD.

The plots show the following results, which are typical.

  • For each N, the number of iterations for fmincon is more than double that of lsqnonlin and increases approximately linearly with N.

  • The number of iterations does not depend on the derivative estimation scheme.

  • The function count for finite difference (FD) estimation is much higher than for automatic differentiation.

  • The function count for lsqnonlin is lower than that for fmincon for the same derivative estimation scheme.

  • The residuals match for all solution methods, meaning the results are independent of the solver and derivative estimation scheme.

The results indicate that lsqnonlin is more efficient than fmincon in terms of both iterations and function counts. However, different problems can have different results, and for some problems fmincon is more efficient than lsqnonlin.

Helper Function

This code creates the runlsqfmincon helper function.

function [lsq,lsqfd,fmin,fminfd] = runlsqfmincon()
optslsq = optimoptions("lsqnonlin",Display="none",...
    MaxFunctionEvaluations=1e5,MaxIterations=1e4); % Allow for many iterations and Fevals
optsfmincon = optimoptions("fmincon",Display="none",...
    MaxFunctionEvaluations=1e5,MaxIterations=1e4);
% Create structures to hold results
z = zeros(1,50);
lsq = struct('Iterations',z,'Fcount',z,'Residual',z);
lsqfd = lsq;
fmin = lsq;
fminfd = lsq;
rng(1) % Reproducible initial points
x00 = -1/2 + randn(50,1);
y00 = 1/2 + randn(50,1);
for N = 1:50
    x = optimvar("x",N,LowerBound=-3,UpperBound=3);
    y = optimvar("y",N,LowerBound=0,UpperBound=9);
    prob = optimproblem("Objective",sum((10*(y - x.^2)).^2 + (1 - x).^2));
    x0.x = x00(1:N);
    x0.y = y00(1:N);
    % Include a set of nonlinear inequality constraints
    cons = optimconstr(N);
    for i = 1:N
        cons(i) = x(i)^2 + y(i)^2 <= 1/2 + 1/8*i;
    end
    prob.Constraints = cons;
    [sol,fval,exitflag,output] = solve(prob,x0,Options=optslsq);
    lsq.Iterations(N) = output.iterations;
    lsq.Fcount(N) = output.funcCount;
    lsq.Residual(N) = fval;

    [sol,fval,exitflag,output] = solve(prob,x0,Options=optslsq,...
        ObjectiveDerivative='finite-differences',ConstraintDerivative='finite-differences');
    lsqfd.Iterations(N) = output.iterations;
    lsqfd.Fcount(N) = output.funcCount;
    lsqfd.Residual(N) = fval;

    [sol,fval,exitflag,output] = solve(prob,x0,Options=optsfmincon,Solver="fmincon");
    fmin.Iterations(N) = output.iterations;
    fmin.Fcount(N) = output.funcCount;
    fmin.Residual(N) = fval;

    [sol,fval,exitflag,output] = solve(prob,x0,Options=optsfmincon,Solver="fmincon",...
        ObjectiveDerivative='finite-differences',ConstraintDerivative='finite-differences');
    fminfd.Iterations(N) = output.iterations;
    fminfd.Fcount(N) = output.funcCount;
    fminfd.Residual(N) = fval;
end

N = 1:50;    
plot(N,lsq.Iterations,'k',N,lsqfd.Iterations,'b--',N,fmin.Iterations,'g',N,fminfd.Iterations,'r--')
legend('lsqnonlin','lsqnonlin FD','fmincon','fmincon FD','Location','northwest')
xlabel('N')
ylabel('Iterations')
title('Iterations')

figure
semilogy(N,lsq.Fcount,'k',N,lsqfd.Fcount,'b--',N,fmin.Fcount,'g',N,fminfd.Fcount,'r--')
legend('lsqnonlin','lsqnonlin FD','fmincon','fmincon FD','Location','northwest')
xlabel('N')
ylabel('log(Function count)')
title('Function count, log-scaled')

figure
plot(N,lsq.Residual,'k',N,lsqfd.Residual,'b--',N,fmin.Residual,'g',N,fminfd.Residual,'r--')
legend('lsqnonlin','lsqnonlin FD','fmincon','fmincon FD','Location','southeast')
xlabel('N')
ylabel('Residual')
ylim([0,0.4])
title('Residual')

end

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