Quickguide BMI Problem

This page is part of the Quickguide Manual. See Quickguide.

The linear semi-definite programming problem with bilinear matrix inequalities (bmi) is defined similarly to SDP but with the matrix inequality

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q^{i}_0 + \Sum{k=1}{n} Q^{i}_{k}x_{k} + \Sum{k=1}{n}\Sum{l=1}{n} x_{k}x_{l}K^{i}_{kl} \preccurlyeq 0 \\ }

The following file defines and solves a problem in TOMLAB.

File: tomlab/quickguide/bmiQG.m

Open the file for viewing, and execute bmiQG in Matlab.

 % bmiQG is a small example problem for defining and solving
 % semi definite programming problems with bilinear matrix 
 % inequalities using the TOMLAB format.
 
 Name='bmi.ps example 3';
 A   = [];
 b_U = [];
 b_L = [];
 
 c   = [ 0 0 1 ];  % cost vector 
 
 % One matrix constraint, set linear part first
 SDP = [];
 
 % The constant matrix is stored as 
 % SDP(i).Q{j} when SDP(i).Qidx(j) ==0
 SDP(1).Q{1}  = [-10 -0.5 -2 ;-0.5 4.5 0 ;-2 0 0 ];
 
 SDP(1).Q{2} = [ 9 0.5 0       ;  0.5  0  -3 ;  0  -3 -1 ];
 SDP(1).Q{3} = [-1.8 -0.1 -0.4 ; -0.1 1.2 -1 ; -0.4 -1 0 ];
 
 % Sparse is fine, too. Eventually, all the matrices are
 % converted to sparse format. 
 SDP(1).Q{4} = -speye(3); 
 
 SDP(1).Qidx = [0; 1; 2; 3];
 
 % Now bilinear part
 
 % K_12 of constraint 1 (of 1) is nonzero, so set in SDP(i).K{1}.
 SDP(1).K{1} = [0 0 2 ; 0 -5.5 3 ; 2 3 0 ];
 SDP(1).Kidx = [1 2];   
 n   = length(c);
 
 x_L = [-5 ; -3 ; -Inf];
 x_U = [ 2 ;  7 ;  Inf];
 x_0 = [ 0 ;  0 ;  0  ];
 
 f_Low = [];
 
 Prob = bmiAssign([], c, SDP, A, b_L, b_U, x_L, x_U, x_0,...
                  Name, f_Low);
 
 Result = tomRun('penbmi', Prob, 1);