randn('state',0);
m = 16; n = 8;
A = randn(m,n);
b = randn(m,1);
M = 2;
disp('Computing the solution of the robust least-squares problem...');
cvx_begin
variable x1(n)
minimize( sum(huber(A*x1-b,M)) )
cvx_end
disp('Computing the solution of the least-squares problem with variable weights...');
cvx_begin
variable x2(n)
variable w(m)
minimize( sum(quad_over_lin(diag(A*x2-b),w'+1)) + M^2*ones(1,m)*w)
w >= 0;
cvx_end
disp('Computing the solution of the quadratic program...');
cvx_begin
variable x3(n)
variable u(m)
variable v(m)
minimize( sum(square(u) + 2*M*v) )
A*x3 - b <= u + v;
A*x3 - b >= -u - v;
u >= 0;
u <= M;
v >= 0;
cvx_end
disp('------------------------------------------------------------------------');
disp('The optimal solutions for problem formulations 1, 2 and 3 are given');
disp('respectively as follows (per column): ');
[x1 x2 x3]
Computing the solution of the robust least-squares problem...
Calling Mosek 9.1.9: 152 variables, 64 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 : 64
Cones : 32
Scalar variables : 152
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 : 64
Cones : 32
Scalar variables : 152
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the dual
Optimizer - Constraints : 24
Optimizer - Cones : 32
Optimizer - Scalar variables : 96 conic : 80
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 : 180 after factor : 180
Factor - dense dim. : 0 flops : 5.64e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.0e+00 2.0e+00 2.5e+01 0.00e+00 1.600000000e+01 -8.000000000e+00 1.0e+00 0.00
1 4.1e-01 4.1e-01 3.9e+00 5.26e-02 1.772761786e+00 -5.760426069e+00 2.1e-01 0.01
2 8.9e-02 8.9e-02 4.1e-01 7.48e-01 4.160217895e+00 2.359199320e+00 4.5e-02 0.01
3 2.3e-02 2.3e-02 5.2e-02 9.67e-01 4.210999257e+00 3.748912815e+00 1.1e-02 0.01
4 7.6e-03 7.6e-03 1.0e-02 1.00e+00 4.222835207e+00 4.069222755e+00 3.8e-03 0.01
5 2.4e-03 2.4e-03 1.8e-03 1.00e+00 4.209102647e+00 4.160168718e+00 1.2e-03 0.01
6 3.4e-04 3.4e-04 9.6e-05 1.00e+00 4.209726646e+00 4.202854232e+00 1.7e-04 0.01
7 5.0e-06 5.0e-06 1.7e-07 1.00e+00 4.209707372e+00 4.209605489e+00 2.5e-06 0.01
8 5.1e-08 5.1e-08 1.7e-10 1.00e+00 4.209705258e+00 4.209704232e+00 2.5e-08 0.01
9 2.6e-09 2.7e-09 2.1e-12 1.00e+00 4.209705215e+00 4.209705162e+00 1.3e-09 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 4.2097052151e+00 nrm: 2e+00 Viol. con: 2e-16 var: 1e-09 cones: 3e-09
Dual. obj: 4.2097051615e+00 nrm: 4e+00 Viol. con: 0e+00 var: 4e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 9 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): +4.20971
Computing the solution of the least-squares problem with variable weights...
Calling Mosek 9.1.9: 304 variables, 40 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 : 40
Cones : 16
Scalar variables : 304
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 16
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 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 : 40
Cones : 16
Scalar variables : 304
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 24
Optimizer - Cones : 16
Optimizer - Scalar variables : 48 conic : 48
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 : 180 after factor : 180
Factor - dense dim. : 0 flops : 5.60e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.0e+00 1.5e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 2.9e-01 2.2e-01 1.5e-01 -2.40e-01 -7.426102505e+00 -6.708055895e+00 1.4e-01 0.01
2 2.2e-02 1.6e-02 4.3e-03 5.64e-01 -1.916059676e+01 -1.903705542e+01 1.1e-02 0.01
3 2.1e-04 1.6e-04 4.1e-06 9.60e-01 -2.019946035e+01 -2.019820045e+01 1.1e-04 0.01
4 5.4e-06 4.1e-06 1.7e-08 1.00e+00 -2.020942828e+01 -2.020939573e+01 2.7e-06 0.01
5 4.5e-07 3.3e-07 4.0e-10 1.00e+00 -2.020968252e+01 -2.020967986e+01 2.2e-07 0.01
6 4.8e-08 3.6e-08 1.4e-11 1.00e+00 -2.020970278e+01 -2.020970249e+01 2.4e-08 0.01
7 1.3e-08 4.9e-09 7.2e-13 1.00e+00 -2.020970488e+01 -2.020970484e+01 3.3e-09 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.0209704882e+01 nrm: 4e+00 Viol. con: 4e-08 var: 0e+00 cones: 0e+00
Dual. obj: -2.0209704843e+01 nrm: 1e+00 Viol. con: 0e+00 var: 1e-08 cones: 0e+00
Optimizer summary
Optimizer - time: 0.02
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): +4.2097
Computing the solution of the quadratic program...
Calling Mosek 9.1.9: 128 variables, 56 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 : 56
Cones : 16
Scalar variables : 128
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 : 56
Cones : 16
Scalar variables : 128
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 40
Optimizer - Cones : 16
Optimizer - Scalar variables : 128 conic : 48
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 : 340 after factor : 340
Factor - dense dim. : 0 flops : 6.76e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 4.0e+00 8.0e+00 2.5e+01 0.00e+00 1.600000000e+01 -8.000000000e+00 1.0e+00 0.00
1 6.8e-01 1.4e+00 4.6e+00 -3.93e-01 -1.940066974e+01 -2.754748703e+01 1.7e-01 0.01
2 1.3e-01 2.7e-01 4.2e-01 6.76e-01 -5.357917364e+00 -7.159690897e+00 3.3e-02 0.01
3 4.3e-02 8.6e-02 7.5e-02 9.85e-01 -4.471826379e+00 -5.054170932e+00 1.1e-02 0.01
4 1.7e-02 3.4e-02 1.8e-02 1.00e+00 -4.360094607e+00 -4.589807682e+00 4.2e-03 0.01
5 6.6e-03 1.3e-02 4.3e-03 1.01e+00 -4.240156456e+00 -4.330398655e+00 1.7e-03 0.01
6 1.8e-03 3.5e-03 5.8e-04 1.00e+00 -4.220572075e+00 -4.244669924e+00 4.4e-04 0.01
7 2.1e-04 4.3e-04 2.4e-05 1.00e+00 -4.210448264e+00 -4.213375005e+00 5.3e-05 0.01
8 2.8e-06 5.5e-06 3.6e-08 1.00e+00 -4.209715109e+00 -4.209753126e+00 6.9e-07 0.01
9 6.2e-08 1.2e-07 1.2e-10 1.00e+00 -4.209705505e+00 -4.209706361e+00 1.6e-08 0.01
10 6.1e-09 1.2e-08 3.7e-12 1.00e+00 -4.209705243e+00 -4.209705326e+00 1.5e-09 0.01
11 6.7e-10 1.3e-09 1.3e-13 1.00e+00 -4.209705215e+00 -4.209705225e+00 1.7e-10 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -4.2097052154e+00 nrm: 4e+00 Viol. con: 2e-09 var: 1e-09 cones: 0e+00
Dual. obj: -4.2097052245e+00 nrm: 2e+00 Viol. con: 0e+00 var: 8e-10 cones: 0e+00
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 11 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): +4.20971
------------------------------------------------------------------------
The optimal solutions for problem formulations 1, 2 and 3 are given
respectively as follows (per column):
ans =
0.3888 0.3888 0.3888
0.1262 0.1262 0.1262
-0.3337 -0.3337 -0.3337
0.1326 0.1326 0.1326
0.5500 0.5500 0.5500
0.3526 0.3526 0.3526
-0.6562 -0.6562 -0.6562
0.8309 0.8309 0.8309