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This example shows the value of using sparse arithmetic when you have a sparse problem. The matrix has n rows, where you choose n. For some large n, the active-set algorithm runs out of memory, but the interior-point-convex algorithm works fine.
The problem is to minimize x'*H*x/2 + f'*x subject to
x(1) + x(2) + ... + x(n) = 0,
where f = [-1;-2;-3;...;-n].
Create the parameter n and the utility matrix T. The matrix T is a sparse circulant matrix that is simply a helper for creating the sparse positive-definite quadratic matrix H.
n = 30000; % Adjust n to a large value T = spalloc(n,n,n); % make a sparse circulant matrix r = 1:n-1; for m = r T(m,m+1)=1; end T(n,1) = 1;
Create a sparse vector v. Then create the matrix H by shifted versions of v*v'. The matrix T creates shifts of v.
v(n) = 0; v(1) = 1; v(2) = 2; v(4) = 3; v = (sparse(v))'; % Make a banded type of matrix H = spalloc(n,n,7*n); r = 1:n; for m = r H = H + v*v'; v = T*v; end
Take a look at the structure of H:
spy(H)
Create the problem vector f and linear constraint.
f = -r; % linear term A = ones(1,n); b = 0;
Solve the quadratic programming problem with the interior-point-convex algorithm.
options = optimoptions(@quadprog,'Algorithm','interior-point-convex'); [x,fval,exitflag,output,lambda] = ... quadprog(H,f,A,b,[],[],[],[],[],options); Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the selected value of the function tolerance, and constraints are satisfied to within the selected value of the constraint tolerance.
View the solution value, output structure, and Lagrange multiplier:
fval,output,lambda fval = -3.1331e+10 output = message: 'Minimum found that satisfies the constraints. Optimization com...' algorithm: 'interior-point-convex' firstorderopt: 1.1665e-04 constrviolation: 7.7762e-09 iterations: 6 cgiterations: [] lambda = ineqlin: 1.5000e+004 eqlin: [0x1 double] lower: [30000x1 double] upper: [30000x1 double]
Since there are no lower bounds or upper bounds, all the values in lambda.lower and lambda.upper are 0. The inequality constraint is active, since lambda.ineqlin is nonzero.
Notice that quadprog with the active-set algorithm fails with an out-of-memory error:
options = optimoptions(@quadprog,'Algorithm','active-set'); [x fval] = quadprog(H,f,A,b,[],[],[],[],[],options); Warning: Cannot use sparse matrices with active-set algorithm: converting to full. > In quadprog at 409 Error using full Out of memory. Type HELP MEMORY for your options. Error in quadprog (line 410) H = full(H); A = full(A); Aeq = full(Aeq);