n = 2;
K = 11;
randn('state',0);
P = randn(n,K);
fprintf(1,'Minimizing the L1-norm of the sum of the distances to fixed points...');
cvx_begin
variable x1(2)
minimize ( sum(norms(x1*ones(1,K) - P,1)) )
cvx_end
fprintf(1,'Done! \n');
fprintf(1,'Minimizing the L2-norm of the sum of the distances to fixed points...');
cvx_begin
variable x2(2)
minimize ( sum(norms(x2*ones(1,K) - P,2)) )
cvx_end
fprintf(1,'Done! \n');
disp('------------------------------------------------------------------');
disp('The optimal point location for the L1-norm case is: ');
disp(x1);
disp('The optimal point location for the L2-norm case is: ');
disp(x2);
Minimizing the L1-norm of the sum of the distances to fixed points...
Calling Mosek 9.1.9: 44 variables, 20 equality constraints
------------------------------------------------------------
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 20
Cones : 22
Scalar variables : 44
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 20
Cones : 22
Scalar variables : 44
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the dual
Optimizer - Constraints : 2
Optimizer - Cones : 0
Optimizer - Scalar variables : 22 conic : 0
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 2 after factor : 2
Factor - dense dim. : 0 flops : 4.60e+01
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.8e+00 0.0e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 1.7e+00 1.8e-15 3.4e-01 6.87e-01 7.391774363e+00 7.464633694e+00 4.4e-01 0.01
2 2.8e-01 4.4e-16 2.5e-02 8.41e-01 1.198361653e+01 1.200857301e+01 7.3e-02 0.01
3 4.8e-02 5.6e-16 1.8e-03 9.47e-01 1.350850611e+01 1.351330498e+01 1.2e-02 0.01
4 1.3e-02 6.7e-16 2.4e-04 9.89e-01 1.376190729e+01 1.376316386e+01 3.2e-03 0.01
5 1.2e-03 3.3e-16 7.1e-06 9.97e-01 1.385853821e+01 1.385865862e+01 3.1e-04 0.01
6 8.2e-07 6.7e-16 1.3e-10 1.00e+00 1.386809357e+01 1.386809366e+01 2.1e-07 0.01
7 3.2e-12 2.2e-16 5.1e-16 1.00e+00 1.386809997e+01 1.386809997e+01 8.4e-13 0.01
Optimizer terminated. Time: 0.01
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 1.3868099974e+01 nrm: 4e+00 Viol. con: 1e-16 var: 0e+00 cones: 3e-12
Dual. obj: 1.3868099974e+01 nrm: 1e+00 Viol. con: 0e+00 var: 0e+00 cones: 0e+00
Optimizer summary
Optimizer - time: 0.01
Interior-point - iterations : 7 time: 0.01
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +13.8681
Done!
Minimizing the L2-norm of the sum of the distances to fixed points...
Calling Mosek 9.1.9: 33 variables, 13 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 13
Cones : 11
Scalar variables : 33
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 13
Cones : 11
Scalar variables : 33
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 2
Optimizer - Cones : 11
Optimizer - Scalar variables : 33 conic : 33
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 3 after factor : 3
Factor - dense dim. : 0 flops : 1.15e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 0.0e+00 2.0e+00 1.2e+01 0.00e+00 0.000000000e+00 -1.100000000e+01 1.0e+00 0.00
1 5.6e-16 6.2e-01 3.4e+00 -1.63e-01 -6.996197543e+00 -1.223100824e+01 3.2e-01 0.01
2 4.4e-16 1.1e-01 2.9e-01 6.18e-01 -1.036223969e+01 -1.142407568e+01 5.6e-02 0.01
3 5.6e-16 1.6e-02 1.6e-02 9.08e-01 -1.132123240e+01 -1.147908569e+01 8.0e-03 0.01
4 8.9e-16 9.9e-04 2.6e-04 9.86e-01 -1.147326822e+01 -1.148335935e+01 5.1e-04 0.01
5 1.4e-14 2.2e-05 8.5e-07 9.99e-01 -1.148368982e+01 -1.148391076e+01 1.1e-05 0.01
6 8.4e-14 8.1e-07 6.1e-09 1.00e+00 -1.148391996e+01 -1.148392822e+01 4.2e-07 0.01
7 5.4e-13 3.7e-08 6.0e-11 1.00e+00 -1.148392883e+01 -1.148392920e+01 1.9e-08 0.01
8 2.2e-12 5.0e-09 3.0e-12 1.00e+00 -1.148392919e+01 -1.148392924e+01 2.6e-09 0.01
Optimizer terminated. Time: 0.01
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -1.1483929194e+01 nrm: 1e+00 Viol. con: 4e-12 var: 0e+00 cones: 0e+00
Dual. obj: -1.1483929245e+01 nrm: 2e+00 Viol. con: 0e+00 var: 1e-08 cones: 0e+00
Optimizer summary
Optimizer - time: 0.01
Interior-point - iterations : 8 time: 0.01
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +11.4839
Done!
------------------------------------------------------------------
The optimal point location for the L1-norm case is:
-0.0956
0.1139
The optimal point location for the L2-norm case is:
0.1252
0.1716