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# semidef

Solve semidefinite problems.

### Syntax

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

### Arguments

- x0
m-by-1 real column vector (must be strictly primal feasible, see below)

- Z0
L-by-1 real vector (compressed form of a strictly feasible dual matrix, see below)

- F
L-by-(m+1) real matrix

- blocksizes
p-by-2 integer matrix (sizes of the blocks) defining the dimensions of the (square) diagonal blocks

`size(Fi(j)=blocksizes(j) j=1,...,m+1`

.- c
m-by-1 real vector

- options
a 1-by-5 matrix of doubles

`[nu,abstol,reltol,tv,maxiters]`

- ul
a 1-by-2 matrix of doubles.

### Description

`semidef`

solves the semidefinite program:

and its dual:

exploiting block structure in the matrices
`F_i`

.
Here, `Tr`

is the trace operator, i.e. the sum of the
diagonal entries of the matrix.

The problem data are the vector `c`

and `m+1`

block-diagonal symmetric matrices `F0`

, `F1`

,
..., `Fm`

.
Moreover, we assume that the matrices `Fi`

have
`L`

diagonal blocks.

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

where, for each symmetric block `M`

, the vector `cmp(M)`

is
the compressed representation of `M`

, that is
`[M(1,1);M(1,2);...;M(1,n);M(2,2);M(2,3);...;M(2,n);...;M(n,n)]`

,
obtained by scanning rowwise the upper triangular part of
`M`

.

For example, the matrix

is stored as

[1; 2; 3; 4; 5; 6; 7; 8; 9; 10]

with `blocksizes=[2,3,1]`

.

In order to create the matrix `F`

, the user can combine the
`list2vec`

and `pack`

function, as described in the example below.

`blocksizes`

gives the size of block
`j`

, ie,
`size(F_i^j)=blocksizes(j)`

.

Z is a block diagonal matrix with L blocks ```
Z^0, ...,
Z^{L-1}
```

.`Z^j`

has size ```
blocksizes[j]
times blocksizes[j]
```

.Every block is stored using packed storage of
the lower triangular part.

The 1-by-2 matrix of doubles `ul`

contains the primal objective
value `c'*x`

and the dual objective value
`-trace(F_0*Z`

).

The entries of `options`

are respectively:

`nu`

: a real parameter which controls the rate of convergence.`abstol`

: absolute tolerance. The absolute tolerance cannot be lower than 1.0e-8, that is, the absolute tolerance used in the algorithm is the maximum of the user-defined tolerance and the constant tolerance 1.0e-8.`reltol`

: relative tolerance (has a special meaning when negative).`tv`

: the target value, only referenced if`reltol < 0`

.`maxiters`

: the maximum number of iterations, a positive integer value.

On output, the `info`

variable contains the status of the execution.

`info=1`

if maxiters exceeded,`info=2`

if absolute accuracy is reached,`info=3`

if relative accuracy is reached,`info=4`

if target value is reached,`info=5`

if target value is not achievable;negative values indicate errors.

Convergence criterion is based on the following conditions that is, the algorithm stops if one of the following conditions is true:

maxiters is exceeded

duality gap is less than abstol

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)reltol is negative and primal objective is less than tv or dual objective is greater than

`tv`

.

### Examples

// 1. Define the initial guess x0=[0;0]; // // 2. Create a compressed representation of F // Define 3 symmetric block-diagonal matrices: F0, F1, F2 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] // Define the size of the two blocks: // the first block has size 2, // the second block has size 2. blocksizes=[2,2]; // Extract the two blocks of the matrices. 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); // Create 3 column vectors, containing nonzero entries // in F0, F1, F2. F0nnz = list2vec(list(F01,F02)); F1nnz = list2vec(list(F11,F12)); F2nnz = list2vec(list(F21,F22)); // Create a 16-by-3 matrix, representing the // nonzero entries of the 3 matrices F0, F1, F2. FF=[F0nnz,F1nnz,F2nnz] // Compress FF CFF = pack(FF,blocksizes); // // 3. Create a compressed representation of Z // Create the matrix Z0 Z0=2*F0; // Extract the two blocks of the matrix Z01=Z0(1:2,1:2); Z02=Z0(3:4,3:4); // Create 2 column vectors, containing nonzero entries // in Z0. ZZ0 = [Z01(:);Z02(:)]; // Compress ZZO CZZ0 = pack(ZZ0,blocksizes); // // 4. Create the linear vector c c=[trace(F1*Z0);trace(F2*Z0)]; // // 5. Define the algorithm options options=[10,1.d-10,1.d-10,0,50]; // 6. Solve the problem [x,CZ,ul,info]=semidef(x0,CZZ0,CFF,blocksizes,c,options) // // 7. Check the output // Unpack CZ Z=unpack(CZ,blocksizes); w=vec2list(Z,[blocksizes;blocksizes]); Z=blockdiag(w(1),w(2)) c'*x+trace(F0*Z) // Check that the eigenvalues of the matrix are positive spec(F0+F1*x(1)+F2*x(2)) trace(F1*Z)-c(1) trace(F2*Z)-c(2)

### Implementation notes

This function is based on L. Vandenberghe and S. Boyd sp.c program.

### References

L. Vandenberghe and S. Boyd, " Semidefinite Programming," Informations Systems Laboratory, Stanford University, 1994.

Ju. E. Nesterov and M. J. Todd, "Self-Scaled Cones and Interior-Point Methods in Nonlinear Programming," Working Paper, CORE, Catholic University of Louvain, Louvain-la-Neuve, Belgium, April 1994.

SP: Software for Semidefinite Programming, User's Guide, Beta Version, November 1994, L. Vandenberghe and S. Boyd, 1994 http://www.ee.ucla.edu/~vandenbe/sp.html

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