semidef - semidefinite programming

Calling Sequence

[x,Z,ul,info]=semidef(x0,Z0,F,blck_szs,c,options)

Parameters

• x0 : m x 1 real column vector (must be strictly primal feasible, see below)
• Z0 : L x 1 real vector (compressed form of a strictly feasible dual matrix, see below)
• F : L x (m+1) real matrix
• blck_szs : p x 2 integer matrix (sizes of the blocks) defining the dimensions of the (square) diagonal blocks size(Fi(j)=blck_szs(j) j=1,...,m+1.
• c : m x 1 real vector
• options : row vector with five entries [nu,abstol,reltol,0,maxiters]
• ul : row vector with two entries

Description

[x,Z,ul,info]=semidef(x0,Z0,F,blck_szs,c,options) solves semidefinite program:

    minimize    c'*x
    subject to  F_0 + x_1*F_1 + ... + x_m*F_m  >= 0

 and its dual
 
    maximize    -Tr F_0 Z
    subject to  Tr F_i Z = c_i, i=1,...,m
                Z >= 0

   

exploiting block structure in the matrices F_i.

It interfaces L. Vandenberghe and S. Boyd sp.c program.

The Fi's matrices are stored columnwise in F in compressed format: if F_i^j, i=0,..,m, j=1,...,L denote the jth (symmetric) diagonal block of F_i, then

    [ pack(F_0^1)  pack(F_1^1) ...  pack(F_m^1) ]
    [ pack(F_0^2)  pack(F_1^2) ...  pack(F_m^2) ]
F=  [   ...       ...          ...              ]
    [ pack(F_0^L)  pack(F_1^L) ...  pack(F_m^L) ]
   

where pack(M), for symmetric M, is the vector [M(1,1);M(1,2);...;M(1,n);M(2,2);M(2,3);...;M(2,n);...;M(n,n)] (obtained by scanning columnwise the lower triangular part of M).

blck_szs gives the size of block j, ie, size(F_i^j)=blck_szs(j).

Z is a block diagonal matrix with L blocks Z^0, ..., Z^{L-1}. Z^j has size blck_szs[j] times blck_szs[j]. Every block is stored using packed storage of the lower triangular part.

The 2 vector ul contains the primal objective value c'*x and the dual objective value -Tr F_0*Z.

The entries of options are respectively: nu = a real parameter which ntrols the rate of convergence. abstol = absolute tolerance. reltol = relative tolerance (has a special meaning when negative). tv target value, only referenced if reltol < 0. iters = on entry: maximum number of iterations >= 0, on exit: the number of iterations taken.

info returns 1 if maxiters exceeded, 2 if absolute accuracy is reached, 3 if relative accuracy is reached, 4 if target value is reached, 5 if target value is not achievable; negative values indicate errors.

Convergence criterion:

 (1) maxiters is exceeded
 (2) duality gap is less than abstol
 (3) primal and dual objective are both positive and
     duality gap is less than (reltol * dual objective)
     or primal and dual objective are both negative and
     duality gap is less than (reltol * minus the primal objective)
 (4) reltol is negative and
     primal objective is less than tv or dual objective is greater
     than tv
   

Examples

F0=[2,1,0,0;
    1,2,0,0;
    0,0,3,1
    0,0,1,3];
F1=[1,2,0,0;
    2,1,0,0;
    0,0,1,3;
    0,0,3,1]
F2=[2,2,0,0;
    2,2,0,0;
    0,0,3,4;
    0,0,4,4];
blck_szs=[2,2];
F01=F0(1:2,1:2);F02=F0(3:4,3:4);
F11=F1(1:2,1:2);F12=F1(3:4,3:4);
F21=F2(1:2,1:2);F22=F2(3:4,3:4);
x0=[0;0]
Z0=2*F0;
Z01=Z0(1:2,1:2);Z02=Z0(3:4,3:4);
FF=[[F01(:);F02(:)],[F11(:);F12(:)],[F21(:);F22(:)]]
ZZ0=[[Z01(:);Z02(:)]];
c=[trace(F1*Z0);trace(F2*Z0)];
options=[10,1.d-10,1.d-10,0,50];
[x,Z,ul,info]=semidef(x0,pack(ZZ0),pack(FF),blck_szs,c,options)
w=vec2list(unpack(Z,blck_szs),[blck_szs;blck_szs]);Z=sysdiag(w(1),w(2))
c'*x+trace(F0*Z)
spec(F0+F1*x(1)+F2*x(2))
trace(F1*Z)-c(1)
trace(F2*Z)-c(2)