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R1252/simulation-code/cvx/examples/cvxbook/Ch05_duality/qcqp.m
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% Section 5.2.4: Solves a simple QCQP
% Boyd & Vandenberghe, "Convex Optimization"
% Jo?le Skaf - 08/23/05
%
% Solved a QCQP with 3 inequalities:
% minimize 1/2 x'*P0*x + q0'*r + r0
% s.t. 1/2 x'*Pi*x + qi'*r + ri <= 0 for i=1,2,3
% and verifies that strong duality holds.
% Input data
randn('state',13);
n = 6;
P0 = randn(n); P0 = P0'*P0 + eps*eye(n);
P1 = randn(n); P1 = P1'*P1;
P2 = randn(n); P2 = P2'*P2;
P3 = randn(n); P3 = P3'*P3;
q0 = randn(n,1); q1 = randn(n,1); q2 = randn(n,1); q3 = randn(n,1);
r0 = randn(1); r1 = randn(1); r2 = randn(1); r3 = randn(1);
fprintf(1,'Computing the optimal value of the QCQP and its dual... ');
cvx_begin
variable x(n)
dual variables lam1 lam2 lam3
minimize( 0.5*quad_form(x,P0) + q0'*x + r0 )
lam1: 0.5*quad_form(x,P1) + q1'*x + r1 <= 0;
lam2: 0.5*quad_form(x,P2) + q2'*x + r2 <= 0;
lam3: 0.5*quad_form(x,P3) + q3'*x + r3 <= 0;
cvx_end
obj1 = cvx_optval;
P_lam = P0 + lam1*P1 + lam2*P2 + lam3*P3;
q_lam = q0 + lam1*q1 + lam2*q2 + lam3*q3;
r_lam = r0 + lam1*r1 + lam2*r2 + lam3*r3;
obj2 = -0.5*q_lam'*inv(P_lam)*q_lam + r_lam;
fprintf(1,'Done! \n');
% Displaying results
disp('------------------------------------------------------------------------');
disp('The duality gap is equal to ');
disp(obj1-obj2)
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