Now I have a constrained least squares problem with two variables to be solved, and I adopt the method of alternating optimization. The first subproblem is an unconstrained least squares problem, and I directly solve the pseudo-inverse to get the optimal solution. The second subproblem has a transverse modulus constraint, resulting in non-convex parts in the constraint term, so I conduct SCA for this part. I found that in the process of iteration, the objective function is constantly decreasing, and finally approaches a value, and the obtained solution also satisfies all constraints. But when I brought it in, it was completely different from the shape I wanted it to be.
For alternate optimization, I used SCA to deal with the non-convex parts of the constraint terms. The final result satisfies the constraint, but does not match the target at all (least squares problem)
Here are the results of the last iteration:
Calling Mosek_2 10.0.27: 1784 variables, 736 equality constraints
For improved efficiency, Mosek_2 is solving the dual problem.
MOSEK Version 10.0.27 (Build date: 2022-11-1 14:42:52)
Copyright © MOSEK ApS, Denmark WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 736
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 248
Scalar variables : 1784
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 : 0.03
Lin. dep. - number : 0
Presolve terminated. Time: 0.06
Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 736
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 248
Scalar variables : 1784
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 32
Optimizer - solved problem : the primal
Optimizer - Constraints : 724
Optimizer - Cones : 248
Optimizer - Scalar variables : 1776 conic : 1296
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 : 5772 after factor : 5772
Factor - dense dim. : 8 flops : 1.18e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 2.0e+00 2.5e+02 0.00e+00 2.440000000e+02 -2.000000000e+00 1.0e+00 0.11
1 1.4e-01 2.8e-01 3.7e+01 -4.20e-01 7.388184814e+01 -3.000214458e+00 1.4e-01 0.28
2 1.8e-02 3.6e-02 1.7e+00 1.03e+00 9.418732738e+00 -5.054525739e-01 1.8e-02 0.28
3 3.4e-03 6.8e-03 1.3e-01 9.84e-01 1.633772986e+00 -2.134601838e-01 3.4e-03 0.30
4 5.5e-04 1.1e-03 7.8e-03 1.01e+00 1.577288589e-01 -1.374867191e-01 5.5e-04 0.30
5 1.1e-05 2.1e-05 2.1e-05 1.00e+00 -1.203515961e-01 -1.260844819e-01 1.1e-05 0.31
6 3.8e-07 7.6e-07 1.5e-07 9.97e-01 -1.257085713e-01 -1.259148595e-01 3.8e-07 0.31
7 9.5e-08 1.9e-07 2.3e-08 7.38e-01 -1.258715685e-01 -1.259326471e-01 9.4e-08 0.33
8 9.5e-08 1.9e-07 2.3e-08 5.92e-01 -1.258715692e-01 -1.259326472e-01 9.4e-08 0.34
9 9.5e-08 1.9e-07 2.3e-08 6.78e-01 -1.258716476e-01 -1.259326678e-01 9.4e-08 0.36
10 9.5e-08 1.9e-07 2.3e-08 6.78e-01 -1.258716476e-01 -1.259326678e-01 9.4e-08 0.38
11 9.5e-08 1.9e-07 2.3e-08 6.78e-01 -1.258716476e-01 -1.259326678e-01 9.4e-08 0.39
12 9.5e-08 1.9e-07 2.3e-08 6.68e-01 -1.258716476e-01 -1.259326679e-01 9.4e-08 0.41
13 9.5e-08 1.9e-07 2.3e-08 6.68e-01 -1.258716476e-01 -1.259326679e-01 9.4e-08 0.42
14 9.5e-08 1.9e-07 2.3e-08 6.68e-01 -1.258716476e-01 -1.259326679e-01 9.4e-08 0.44
15 4.6e-08 9.1e-08 7.4e-09 1.00e+00 -1.259129234e-01 -1.259424194e-01 4.6e-08 0.45
16 2.2e-08 3.7e-08 1.9e-09 1.00e+00 -1.259424664e-01 -1.259544292e-01 1.8e-08 0.45
17 2.2e-08 3.7e-08 1.9e-09 1.00e+00 -1.259424704e-01 -1.259544316e-01 1.8e-08 0.47
18 2.0e-08 2.7e-08 1.2e-09 1.00e+00 -1.259506435e-01 -1.259595199e-01 1.4e-08 0.48
19 2.0e-08 2.7e-08 1.2e-09 1.00e+00 -1.259506440e-01 -1.259595203e-01 1.4e-08 0.50
20 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259511582e-01 -1.259598648e-01 1.4e-08 0.52
21 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259512853e-01 -1.259599501e-01 1.4e-08 0.52
22 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259512855e-01 -1.259599503e-01 1.4e-08 0.53
23 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259512861e-01 -1.259599506e-01 1.4e-08 0.55
24 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259512901e-01 -1.259599533e-01 1.4e-08 0.56
25 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259512940e-01 -1.259599560e-01 1.4e-08 0.58
26 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259513019e-01 -1.259599613e-01 1.4e-08 0.59
27 2.0e-08 2.7e-08 1.1e-09 1.00e+00 -1.259513019e-01 -1.259599613e-01 1.4e-08 0.61
28 1.9e-08 2.6e-08 1.1e-09 1.00e+00 -1.259514285e-01 -1.259600464e-01 1.4e-08 0.62
29 1.9e-08 2.6e-08 1.1e-09 1.00e+00 -1.259514287e-01 -1.259600466e-01 1.4e-08 0.64
30 1.9e-08 2.6e-08 1.1e-09 1.00e+00 -1.259514292e-01 -1.259600469e-01 1.4e-08 0.66
31 1.9e-08 2.6e-08 1.1e-09 1.00e+00 -1.259514293e-01 -1.259600470e-01 1.4e-08 0.67
Optimizer terminated. Time: 0.78
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -1.2595142926e-01 nrm: 7e+00 Viol. con: 1e-07 var: 0e+00 cones: 0e+00
Dual. obj: -1.2596004697e-01 nrm: 1e+00 Viol. con: 0e+00 var: 7e-08 cones: 0e+00
Optimizer summary
Optimizer - time: 0.78
Interior-point - iterations : 32 time: 0.69
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): +0.12596