Why the value of the optimization variable is the same for each iteration

cgbs · November 16, 2024, 7:41am

clc;clear;close
H1=[1/6 0;0 1/3;-5/6 0;0 -1/6 ];
H2=[1/3;-1/3];H3=[10 0;0 10];
C=[1 0 ; 0 1];
w1 =[1.2000 -0.0000;-0.0000 3.0000];
x111=[1.3066 0.0004;0.0004 3.9565];
x112=[1.7819 0.0022;0.0022 2.7275];
x121=[1.8890 0.1069;0.1069 4.8945];
x122=[2.1546 0.1028;0.1028 4.0238];
x211=[1.3066 0.0004;0.0004 3.9565];
x212=[1.7766 0.0064;0.0064 2.6821];
x221=[1.8890 0.1069;0.1069 4.8945];
x222=[2.1546 0.1028;0.1028 4.0238];
x311=[1.3368 0.0020;0.0020 3.9032];
x312=[1.7917 0.0033;0.0033 2.6554];
x321=[1.9669 0.0669;0.0669 4.4936];
x322=[2.2040 0.0725;0.0725 3.4930];
x411=[1.3368 0.0020;0.0020 3.9032];
x412=[1.7893 0.0027;0.0027 2.6693];
x421=[1.9669 0.0669;0.0669 4.4936];
x422=[2.2040 0.0725;0.0725 3.4930];

cvx_begin sdp
cvx_solver mosek
variable W(2,2) symmetric;
variable T(2,2) symmetric;
variable N(1,4);
variable g11(2,10);variable g12(2,10);variable g21(2,10);variable g22(2,10);
variable g31(2,10);variable g32(2,10);variable g41(2,10);variable g42(2,10);

variable r111(1,1) nonnegative;variable r112(1,1) nonnegative;variable r121(1,1) nonnegative;variable r122(1,1) nonnegative;
variable r211(1,1) nonnegative;variable r212(1,1) nonnegative;variable r221(1,1) nonnegative;variable r222(1,1) nonnegative;
variable r311(1,1) nonnegative;variable r312(1,1) nonnegative;variable r321(1,1) nonnegative;variable r322(1,1) nonnegative;
variable r411(1,1) nonnegative;variable r412(1,1) nonnegative;variable r421(1,1) nonnegative;variable r422(1,1) nonnegative;

variable a111(40,40) semidefinite diagonal;variable a112(40,40) semidefinite diagonal;variable a121(40,40) semidefinite diagonal;variable a122(40,40) semidefinite diagonal;
variable a211(40,40) semidefinite diagonal;variable a212(40,40) semidefinite diagonal;variable a221(40,40) semidefinite diagonal;variable a222(40,40) semidefinite diagonal;
variable a311(40,40) semidefinite diagonal;variable a312(40,40) semidefinite diagonal;variable a321(40,40) semidefinite diagonal;variable a322(40,40) semidefinite diagonal;
variable a411(40,40) semidefinite diagonal;variable a412(40,40) semidefinite diagonal;variable a421(40,40) semidefinite diagonal;variable a422(40,40) semidefinite diagonal;

variable b111(2,2) semidefinite diagonal;variable b112(2,2) semidefinite diagonal;variable b121(2,2) semidefinite diagonal;variable b122(2,2) semidefinite diagonal;
variable b211(2,2) semidefinite diagonal;variable b212(2,2) semidefinite diagonal;variable b221(2,2) semidefinite diagonal;variable b222(2,2) semidefinite diagonal;
variable b311(2,2) semidefinite diagonal;variable b312(2,2) semidefinite diagonal;variable b321(2,2) semidefinite diagonal;variable b322(2,2) semidefinite diagonal;
variable b411(2,2) semidefinite diagonal;variable b412(2,2) semidefinite diagonal;variable b421(2,2) semidefinite diagonal;variable b422(2,2) semidefinite diagonal;

variable v111(2,2) ;variable v112(2,2) ;variable v121(2,2) ;variable v122(2,2) ;
variable v211(2,2) ;variable v212(2,2) ;variable v221(2,2) ;variable v222(2,2) ;
variable v311(2,2) ;variable v312(2,2) ;variable v321(2,2) ;variable v322(2,2) ;
variable v411(2,2) ;variable v412(2,2) ;variable v421(2,2) ;variable v422(2,2) ;

variable X111(2,2) symmetric;variable X112(2,2) symmetric;variable X121(2,2) symmetric;variable X122(2,2) symmetric;
variable X211(2,2) symmetric;variable X212(2,2) symmetric;variable X221(2,2) symmetric;variable X222(2,2) symmetric;
variable X311(2,2) symmetric;variable X312(2,2) symmetric;variable X321(2,2) symmetric;variable X322(2,2) symmetric;
variable X411(2,2) symmetric;variable X412(2,2) symmetric;variable X421(2,2) symmetric;variable X422(2,2) symmetric;
maximize log_det(W)
subject to
H1W[-1;1]<=ones(4,1);H1W[1;-1]<=ones(4,1);
H1W[-1;-1]<=ones(4,1);H1W[1;1]<=ones(4,1);
H2Nkron([1;0],[-1;1])<=ones(2,1);H2Nkron([1;0],[1;-1])<=ones(2,1);
H2Nkron([1;0],[-1;-1])<=ones(2,1);H2Nkron([1;0],[1;1])<=ones(2,1);
H2Nkron([0;1],[-1;1])<=ones(2,1);H2Nkron([0;1],[1;-1])<=ones(2,1);
H2Nkron([0;1],[-1;-1])<=ones(2,1);H2Nkron([0;1],[1;1])<=ones(2,1);
e1=[1;0];e2=[0;1];
x1=[0.7814;0.2880];x2=[0.9839;0.2897];x3=[2.6751;1.7373];x4=[5.9519;2.7657];x5=[4.7842;-0.1208];x6=[1.5891;-2.6976];x7=[0.7628;-0.5556];x8=[2.7112;1.9480];x9=[4.0016;1.6816];x10=[4.1556;-0.4786];
x11=[5.2454;1.6229];x12=[4.8165;0.3140 ];x13=[3.5436;-1.0243];x14=[3.8763;0.4997];x15=[0.7066;-2.4871];x16=[-4.7929;-5.6726];x17=[-8.7071;-4.3372];x18=[-11.3191;-3.2707];x19=[-11.8039;-2.6239];x20=[-9.3375;0.3954];
x21=[4.4278;-0.7637];

p1=[0.1341;0.8659];p2=[0.2126;0.7874];p3=[0.8949;0.1051];p4=[0.0715; 0.9285 ];p5=[0.2425;0.7575];p6=[0.0538;0.9462];p7=[0.4417;0.5583];p8=[0.0133;0.9867];p9=[0.8972;0.1028];p10=[0.1967;0.8033];
p11=[0.0934;0.9066];p12=[0.3074;0.6926];p13=[0.4561;0.5439];p14=[0.1017;0.8983];p15=[0.9954;0.0046];p16=[0.3321;0.6679];p17=[0.2973;0.7027];p18=[0.0620;0.9380];p19=[0.2982;0.7018];p20=[0.0464;0.9536];
X2=[kron(p1,x1) kron(p2,x2) kron(p3,x3) kron(p4,x4) kron(p5,x5) kron(p6,x6) kron(p7,x7) kron(p8,x8) kron(p9,x9) kron(p10,x10) kron(p11,x11) kron(p12,x12) kron(p13,x13) kron(p14,x14) kron(p15,x15) kron(p16,x16) kron(p17,x17) kron(p18,x18) kron(p19,x19) kron(p20,x20);
    0.0326    1.5686    0.7864   -2.4607   -2.5148    1.6634    2.4308    0.2026   -2.3451    1.9549   -0.9714   -1.2362    1.4779   -2.9380   -2.7093    1.0075    0.6208    0.1566 1.3783    1.2435];
d=[H3*x2;H3*x3;H3*x4;H3*x5;H3*x6;H3*x7;H3*x8;H3*x9;H3*x10;H3*x11;H3*x12;H3*x13;H3*x14;H3*x15;H3*x16;H3*x17;H3*x18;H3*x19;H3*x20;H3*x21];
Z=kron(X2',H3);

g11==kron(([kron(eye(2),W);N]*kron([0;1],[1;-1]))',eye(2)); 
g21==kron(([kron(eye(2),W);N]*kron([0;1],[-1;1]))',eye(2));
g31==kron(([kron(eye(2),W);N]*kron([0;1],[1;1]))',eye(2));
g41==kron(([kron(eye(2),W);N]*kron([0;1],[-1;-1]))',eye(2));

g12==kron(([kron(eye(2),W);N]*kron([1;0],[1;-1]))',eye(2));
g22==kron(([kron(eye(2),W);N]*kron([1;0],[-1;1]))',eye(2));
g32==kron(([kron(eye(2),W);N]*kron([1;0],[1;1]))',eye(2));
g42==kron(([kron(eye(2),W);N]*kron([1;0],[-1;-1]))',eye(2));

[W'*inv(x111)*w1+w1*inv(x111)*W-w1*inv(x111)*(X111)*inv(x111)*w1 (r111)*C'*e1;(r111)*e1'*C r111]==semidefinite(3);
[W'*inv(x112)*w1+w1*inv(x112)*W-w1*inv(x112)*(X112)*inv(x121)*w1 (r112)*C'*e2;(r112)*e2'*C r112]==semidefinite(3);
[W'*inv(x121)*w1+w1*inv(x121)*W-w1*inv(x121)*(X121)*inv(x121)*w1 (r121)*C'*e1;(r121)*e1'*C r121]==semidefinite(3);
[W'*inv(x122)*w1+w1*inv(x122)*W-w1*inv(x122)*(X122)*inv(x122)*w1 (r122)*C'*e2;(r122)*e2'*C r122]==semidefinite(3);
[W'*inv(x211)*w1+w1*inv(x211)*W-w1*inv(x211)*(X211)*inv(x211)*w1 (r211)*C'*e1;(r211)*e1'*C r211]==semidefinite(3);
[W'*inv(x212)*w1+w1*inv(x212)*W-w1*inv(x212)*(X212)*inv(x212)*w1 (r212)*C'*e2;(r212)*e2'*C r212]==semidefinite(3);
[W'*inv(x221)*w1+w1*inv(x221)*W-w1*inv(x221)*(X221)*inv(x221)*w1 (r221)*C'*e1;(r221)*e1'*C r221]==semidefinite(3);
[W'*inv(x222)*w1+w1*inv(x222)*W-w1*inv(x222)*(X222)*inv(x222)*w1 (r222)*C'*e2;(r222)*e2'*C r222]==semidefinite(3);
[W'*inv(x311)*w1+w1*inv(x311)*W-w1*inv(x311)*(X311)*inv(x311)*w1 (r311)*C'*e1;(r311)*e1'*C r311]==semidefinite(3);
[W'*inv(x312)*w1+w1*inv(x312)*W-w1*inv(x312)*(X312)*inv(x312)*w1 (r312)*C'*e2;(r312)*e2'*C r312]==semidefinite(3);
[W'*inv(x321)*w1+w1*inv(x321)*W-w1*inv(x321)*(X321)*inv(x321)*w1 (r321)*C'*e1;(r321)*e1'*C r321]==semidefinite(3);
[W'*inv(x322)*w1+w1*inv(x322)*W-w1*inv(x322)*(X322)*inv(x322)*w1 (r322)*C'*e2;(r322)*e2'*C r322]==semidefinite(3);
[W'*inv(x411)*w1+w1*inv(x411)*W-w1*inv(x411)*(X411)*inv(x411)*w1 (r411)*C'*e1;(r411)*e1'*C r411]==semidefinite(3);
[W'*inv(x412)*w1+w1*inv(x412)*W-w1*inv(x412)*(X412)*inv(x412)*w1 (r412)*C'*e2;(r412)*e2'*C r412]==semidefinite(3);
[W'*inv(x421)*w1+w1*inv(x421)*W-w1*inv(x421)*(X421)*inv(x421)*w1 (r421)*C'*e1;(r421)*e1'*C r421]==semidefinite(3);
[W'*inv(x422)*w1+w1*inv(x422)*W-w1*inv(x422)*(X422)*inv(x422)*w1 (r422)*C'*e2;(r422)*e2'*C r422]==semidefinite(3);

W'*w1+w1'*W-w1'*w1-T==semidefinite(2);
[r111-ones(40,1)'*(a111)*ones(40,1)-[1;1]'*(b111)*[1;1]+d'*(a111)*d -d'*(a111)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a111)*d Z'*(a111)*Z zeros(10,2) (g11)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b111)*H3 eye(2) zeros(2,2);
zeros(2,1) g11 eye(2) (v111)+(v111)' (v111)';
  zeros(2,1) zeros(2,10) zeros(2,2) v111 X111]==semidefinite(17);
[r112-ones(40,1)'*(a112)*ones(40,1)-[1;1]'*(b112)*[1;1]+d'*(a112)*d -d'*(a112)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a112)*d Z'*(a112)*Z zeros(10,2) (g11)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b112)*H3 eye(2) zeros(2,2);
 zeros(2,1) g11 eye(2) (v112)+(v112)' (v112)';
 zeros(2,1) zeros(2,10) zeros(2,2) v112 X112]==semidefinite(17);
[r121-ones(40,1)'*(a121)*ones(40,1)-[1;1]'*(b121)*[1;1]+d'*(a121)*d -d'*(a121)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a121)*d Z'*(a121)*Z zeros(10,2) (g12)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b121)*H3 eye(2) zeros(2,2);
 zeros(2,1) g12 eye(2) (v121)+(v121)' (v121)';
 zeros(2,1) zeros(2,10) zeros(2,2) v121 X121]==semidefinite(17);
[r122-ones(40,1)'*(a122)*ones(40,1)-[1;1]'*(b122)*[1;1]+d'*(a111)*d -d'*(a122)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a122)*d Z'*(a122)*Z zeros(10,2) (g12)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b122)*H3 eye(2) zeros(2,2);
 zeros(2,1) g12 eye(2) (v122)+(v122)' (v122)';
 zeros(2,1) zeros(2,10) zeros(2,2) v122 X122]==semidefinite(17);

%
[r211-ones(40,1)‘(a211)ones(40,1)-[1;1]'(b211)[1;1]+d’(a211)d -d’(a211)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a211)d Z’(a211)Z zeros(10,2) (g21)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b211)H3 eye(2) zeros(2,2);
zeros(2,1) g21 eye(2) (v211)+(v211)’ (v211)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v211 X211]==semidefinite(17);
[r212-ones(40,1)’(a212)ones(40,1)-[1;1]'(b212)[1;1]+d’(a212)d -d’(a212)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a212)d Z’(a212)Z zeros(10,2) (g21)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b112)H3 eye(2) zeros(2,2);
zeros(2,1) g21 eye(2) (v212)+(v212)’ (v212)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v212 X212]==semidefinite(17);
[r221-ones(40,1)’(a221)ones(40,1)-[1;1]'(b221)[1;1]+d’(a221)d -d’(a221)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a221)d Z’(a221)Z zeros(10,2) (g22)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b221)H3 eye(2) zeros(2,2);
zeros(2,1) g22 eye(2) (v221)+(v221)’ (v221)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v221 X221]==semidefinite(17);
[r222-ones(40,1)’(a222)ones(40,1)-[1;1]'(b222)[1;1]+d’(a222)d -d’(a222)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a222)d Z’(a222)Z zeros(10,2) (g22)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b222)H3 eye(2) zeros(2,2);
zeros(2,1) g22 eye(2) (v222)+(v222)’ (v222)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v222 X222]==semidefinite(17);
%
[r311-ones(40,1)’(a311)ones(40,1)-[1;1]'(b311)[1;1]+d’(a311)d -d’(a311)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a311)d Z’(a311)Z zeros(10,2) (g31)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b311)H3 eye(2) zeros(2,2);
zeros(2,1) g31 eye(2) (v311)+(v311)’ (v311)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v311 X311]==semidefinite(17);
[r312-ones(40,1)’(a312)ones(40,1)-[1;1]'(b312)[1;1]+d’(a312)d -d’(a312)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a312)d Z’(a312)Z zeros(10,2) (g31)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b312)H3 eye(2) zeros(2,2);
zeros(2,1) g31 eye(2) (v312)+(v312)’ (v312)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v312 X312]==semidefinite(17);
[r321-ones(40,1)’(a321)ones(40,1)-[1;1]'(b321)[1;1]+d’(a321)d -d’(a321)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a321)d Z’(a321)Z zeros(10,2) (g32)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b321)H3 eye(2) zeros(2,2);
zeros(2,1) g32 eye(2) (v321)+(v321)’ (v321)‘;
zeros(2,1) zeros(2,10) zeros(2,2) v321 X321]==semidefinite(17);
[r322-ones(40,1)’(a322)ones(40,1)-[1;1]'(b322)[1;1]+d’*(a322)d -d’(a322)Z zeros(1,2) zeros(1,2) zeros(1,2);
-Z’(a322)d Z’(a322)Z zeros(10,2) (g32)’ zeros(10,2);
zeros(2,1) zeros(2,10) H3’(b322)*H3 eye(2) zeros(2,2);
zeros(2,1) g32 eye(2) (v322)+(v322)’ (v322)';
zeros(2,1) zeros(2,10) zeros(2,2) v322 X322]==semidefinite(17);

[r411-ones(40,1)'*(a411)*ones(40,1)-[1;1]'*(b411)*[1;1]+d'*(a411)*d -d'*(a411)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a411)*d Z'*(a411)*Z zeros(10,2) (g41)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b411)*H3 eye(2) zeros(2,2);
 zeros(2,1) g41 eye(2) (v411)+(v411)' (v411)';
 zeros(2,1) zeros(2,10) zeros(2,2) v411 X411]==semidefinite(17);
[r412-ones(40,1)'*(a412)*ones(40,1)-[1;1]'*(b412)*[1;1]+d'*(a412)*d -d'*(a412)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a412)*d Z'*(a412)*Z zeros(10,2) (g41)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b412)*H3 eye(2) zeros(2,2);
 zeros(2,1) g41 eye(2) (v112)+(v112)' (v112)';
 zeros(2,1) zeros(2,10) zeros(2,2) v412 X412]==semidefinite(17);
[r421-ones(40,1)'*(a421)*ones(40,1)-[1;1]'*(b421)*[1;1]+d'*(a421)*d -d'*(a421)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a421)*d Z'*(a421)*Z zeros(10,2) (g42)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b421)*H3 eye(2) zeros(2,2);
 zeros(2,1) g42 eye(2) (v421)+(v421)' (v421)';
 zeros(2,1) zeros(2,10) zeros(2,2) v421 X421]==semidefinite(17);
[r422-ones(40,1)'*(a422)*ones(40,1)-[1;1]'*(b422)*[1;1]+d'*(a422)*d -d'*(a422)*Z zeros(1,2) zeros(1,2) zeros(1,2);
 -Z'*(a422)*d Z'*(a422)*Z zeros(10,2) (g42)' zeros(10,2);
 zeros(2,1) zeros(2,10) H3'*(b422)*H3 eye(2) zeros(2,2);
 zeros(2,1) g42 eye(2) (v422)+(v422)' (v422)';
 zeros(2,1) zeros(2,10) zeros(2,2) v422 X422]==semidefinite(17);

cvx_end
cvx_optval

The above iterative algorithm takes different w1, x111, x112, x121, x122 each time, but the value of the optimization variable W is the same. I hope you can come up with some suggestions and some ways to achieve a different initial value, i.e., each iteration, we get a different optimization variable W

Mark_L_Stone · November 16, 2024, 12:48pm

You have a complicated model with a lot of semidefinite constraints… I have no idea what this model us supposed to represent. I don’t have the time or energy to try to understand it.and the roles that those particular input data play. Likely no other forum reader does either. Maybe W is constrained so much by some of the constraints that the values of those input data don’t matter (at least over the range of values you tried). This is your research project, do you are going to have to put in the effort to figure pit what is going on. Perhaps you should try simplifying things to get better understanding what is going on. You could remove the objective or change to different objective and see what happens. You could remove some constraints, etc.

cgbs · November 16, 2024, 1:34pm

Ok,thank you for your reply.