Tartalomjegyzék

LMI tricks

Teljes Matlab script kiegészítő függvényekkel.

File:   LMI_tricks.m
Author: Peter Polcz (ppolcz@gmail.com)

Created on 2017. November 21.

global SCOPE_DEPTH VERBOSE LATEX_EQNR
SCOPE_DEPTH = 0;
VERBOSE = 1;
LATEX_EQNR = 0;

Preserve a full matrix to have very large values

C = sdpvar(2,3);

sdpopts = sdpsettings('verbose', 0);

mu = 10000;

optimize( [ mu*eye(2) C ; C' eye(3) ] >= 0 , sum(C(:)) , sdpopts);
val_C = value(C)

Output:

val_C =
  -40.8248  -40.8248  -40.8248
  -40.8248  -40.8248  -40.8248

optimize( [ mu*eye(2) C ; C' eye(3) ] >= 0 , -sum(C(:)) , sdpopts);
val_C = value(C)

Output:

val_C =
   40.8248   40.8248   40.8248
   40.8248   40.8248   40.8248

optimize( [ mu*eye(2) C ; C' eye(3) ] >= 0 , C(1) , sdpopts);
val_C = value(C)

Output:

val_C =
 -100.0000         0         0
         0         0         0

optimize( [ mu*eye(2) C ; C' eye(3) ] >= 0 , C(1)+C(3) , sdpopts);
val_C = value(C)

Output:

val_C =
  -70.7107  -70.7107         0
   -0.0000    0.0000         0

a = 5;
b = 3;

A = sdpvar(5);
R = sdpvar(5,3);

M = [
    A R
    R' zeros(b)
    ];

sdpopts = sdpsettings('solver','mosek');

optimize([ A <= -1e-10 , M >= 0 ] , [], sdpopts)

A = value(A);
M = value(M);

eig(A), eig(M)

Output:

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 30              
  Cones                  : 0               
  Scalar variables       : 0               
  Matrix variables       : 2               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries                  : 0                 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            : 30              
  Cones                  : 0               
  Scalar variables       : 0               
  Matrix variables       : 2               
  Integer variables      : 0               

Optimizer  - threads                : 4               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 30
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 0                 conic                  : 0               
Optimizer  - Semi-definite variables: 2                 scalarized             : 51              
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 : 465               after factor           : 465             
Factor     - dense dim.             : 0                 flops                  : 1.44e+04        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   0.0e+00  1.0e+00  1.0e+00  0.00e+00   -5.000000000e-10  0.000000000e+00   1.0e+00  0.00  
Optimizer terminated. Time: 0.01    


MOSEK DUAL INFEASIBILITY REPORT.

Problem status: The problem is dual infeasible

Interior-point solution summary
  Problem status  : DUAL_INFEASIBLE
  Solution status : DUAL_INFEASIBLE_CER
  Primal.  obj: -5.0000000000e-10   nrm: 1e+00    Viol.  con: 0e+00    barvar: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 0         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    

ans = 
  struct with fields:

    yalmiptime: 0.1463
    solvertime: 0.0187
          info: 'Infeasible problem (MOSEK)'
       problem: 1
ans =
     0
     0
     0
     0
     0
ans =
     0
     0
     0
     0
     0
     0
     0
     0

Fontos kérdések

1. kérdés

$$ P \succeq 0 \overset{?}{\Leftrightarrow} \pmqty{P & R \\ R^T & 0} \succeq 0 $$

Válasz: nem, mivel. Attól, hogy $P \succeq 0$ még létezhet $R$ ú.h. $\pmqty{P & R \\ R^T & 0}$ indefinit lesz.

a = 5;
b = 3;

P = randn(a);
P = P*P';
R = randn(a,b);

\begin{align} {\LARGE(1) \quad} \text{Sajátértékek}: \left(\begin{array}{cccccccc} -2.05 & -0.778 & -0.646 & 1.81 & 2.38 & 6.41 & 8.11 & 17.5 \\ \end{array}\right) \end{align}

Fordítva igaz-e? Sejtés: igen.

Published with MATLAB® R2017b, and sanitarized by BeautifulSoup (Python script using BSoup is written by Polcz).