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Ordinarily, minimization routines use numerical gradients calculated by finite-difference approximation. This procedure systematically perturbs each of the variables in order to calculate function and constraint partial derivatives. Alternatively, you can provide a function to compute partial derivatives analytically. Typically, the problem is solved more accurately and efficiently if such a function is provided.
Consider how to solve
$$\underset{x}{\mathrm{min}}f(x)={e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right).$$
subject to the constraints
x_{1}x_{2} – x_{1} – x_{2} ≤
–1.5,
x_{1}x_{2} ≥
–10.
To solve the problem using analytically determined gradients, do the following.
function [f,gradf] = objfungrad(x) f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1); % Gradient of the objective function: if nargout > 1 gradf = [ f + exp(x(1)) * (8*x(1) + 4*x(2)), exp(x(1))*(4*x(1)+4*x(2)+2)]; end
function [c,ceq,DC,DCeq] = confungrad(x) c(1) = 1.5 + x(1) * x(2) - x(1) - x(2); % Inequality constraints c(2) = -x(1) * x(2)-10; % No nonlinear equality constraints ceq=[]; % Gradient of the constraints: if nargout > 2 DC= [x(2)-1, -x(2); x(1)-1, -x(1)]; DCeq = []; end
gradf contains the partial derivatives of the objective function, f, returned by objfungrad(x), with respect to each of the elements in x:
$$\nabla f=\left[\begin{array}{c}{e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right)+{e}^{{x}_{1}}\left(8{x}_{1}+4{x}_{2}\right)\\ {e}^{{x}_{1}}\left(4{x}_{1}+4{x}_{2}+2\right)\end{array}\right].$$ | (6-58) |
The columns of DC contain the partial derivatives for each respective constraint (i.e., the ith column of DC is the partial derivative of the ith constraint with respect to x). So in the above example, DC is
$$\left[\begin{array}{cc}\frac{\partial {c}_{1}}{\partial {x}_{1}}& \frac{\partial {c}_{2}}{\partial {x}_{1}}\\ \frac{\partial {c}_{1}}{\partial {x}_{2}}& \frac{\partial {c}_{2}}{\partial {x}_{2}}\end{array}\right]=\left[\begin{array}{cc}{x}_{2}-1& -{x}_{2}\\ {x}_{1}-1& -{x}_{1}\end{array}\right].$$ | (6-59) |
Since you are providing the gradient of the objective in objfungrad.m and the gradient of the constraints in confungrad.m, you must tell fmincon that these files contain this additional information. Use optimoptions to turn the options GradObj and GradConstr to 'on' in the example's existing options:
options = optimoptions(options,'GradObj','on','GradConstr','on');
If you do not set these options to 'on', fmincon does not use the analytic gradients.
The arguments lb and ub place lower and upper bounds on the independent variables in x. In this example, there are no bound constraints, so set both to [].
x0 = [-1,1]; % Starting guess options = optimoptions(@fmincon,'Algorithm','sqp'); options = optimoptions(options,'GradObj','on','GradConstr','on'); lb = [ ]; ub = [ ]; % No upper or lower bounds [x,fval] = fmincon(@objfungrad,x0,[],[],[],[],lb,ub,... @confungrad,options);
The results:
x,fval x = -9.5474 1.0474 fval = 0.0236 [c,ceq] = confungrad(x) % Check the constraint values at x c = 1.0e-13 * -0.1066 0.1066 ceq = []