Trust-Region-Reflective Solution

Solve the quadratic program using the'信任区域 - 反思'algorithm.

options = optimoptions('quadprog','Algorithm',“信任区域反光”); tic [x1,fval1,exitflag1,output1] =...quadprog(H,f,[],[],[],[],lb,ub,[],options);

Local minimum possible. quadprog stopped because the relative change in function value is less than the function tolerance.

time1 = toc

time1 = 0.1044

Examine the solution.

fval1,exitflag1,output1.iterations,output1.cgiterations

fval1 = -0.9930

exitflag1 = 3

ans = 18

ans = 1682

The algorithm converges in relatively few iterations, but takes over 1000 CG (conjugate gradient) iterations. To avoid the CG iterations, set options to use a direct solver instead.

options = optimoptions(options,'SubproblemAlgorithm','factorization'); tic [x2,fval2,exitflag2,output2] =...quadprog(H,f,[],[],[],[],lb,ub,[],options);

Local minimum possible. quadprog stopped because the relative change in function value is less than the function tolerance.

time2 = toc

time2 = 0.0185

fval2,exitflag2,output2.iterations,output2.cgiterations

fval2 = -0.9930

exitflag2 = 3

ans = 10

ans = 0

This time, the algorithm takes fewer iterations and no CG iterations. The solution time decreases substantially, despite the relatively time-consuming direct factorization steps, because the solver avoids taking many CG steps.

Interior-Point Solution

The default'interior-point-convex'算法可以解决这个问题。

tic [x3,fval3,exitflag3,output3] =...quadprog(H,f,[],[],[],[],lb,ub);% No options means use the default algorithm

Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. 

time3 = toc.

time3 = 0.0402

fval3,exitflag3,output3.iterations

fval3 = -0.9930

exitflag3 = 1

ans = 8

Compare Results

All algorithms give the same objective function value to display precision, –0.9930.

The'interior-point-convex'algorithm takes the fewest iterations. However, the'信任区域 - 反思'algorithm with the direct subproblem solver reaches the solution fastest.

tt = table（[time1; time2; time3]，[output1.ileations; output2.mertations; Output3.Iltations]，...'VariableNames',["Time""Iterations"],'RowNames',["TRR""TRR Direct""IP"])

tt=3×2 table时间迭代________ __________ TRR 0.10443 18 TRR直接0.018544 10 IP 0.040204 8