What Is Optimization Toolbox?

Examples That Use Standard AlgorithmsNote that the number of function evaluations to find the solution is reducedbecause we further restricted the search space. Fewer function evaluationsare usually taken when a problem has more constraints and bound limitationsbecause the optimization makes better decisions regarding step size andregions of feasibility than in the unconstrained case. It is, therefore, goodpractice to bound and constrain problems, where possible, to promote fastconvergence to a solution.Constrained Example with GradientsOrdinarily the medium-scale minimization routines use numerical gradientscalculated by finite-difference approximation. This procedure systematicallyperturbs each of the variables in order to calculate function and constraintpartial derivatives. Alternatively, you can provide a function to computepartial derivatives analytically. Typically, the problem is solved moreaccurately and efficiently if such a function is provided.To solve Equation 2-2 using analytically determined gradients, do thefollowing.Step 1: Write an M-file for the objective function and gradient.function [f,G] = 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 functiont = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1);G = [ t + exp(x(1)) * (8*x(1) + 4*x(2)),exp(x(1))*(4*x(1)+4*x(2)+2)];Step 2: Write an M-file for the nonlinear constraints and thegradients of the nonlinear constraints.function [c,ceq,DC,DCeq] = confungrad(x)c(1) = 1.5 + x(1) * x(2) - x(1) - x(2); %Inequality constraintsc(2) = -x(1) * x(2)-10;% Gradient of the constraintsDC= [x(2)-1, -x(2);x(1)-1, -x(1)];% No nonlinear equality constraints2-15