MATLAB TOMLAB PENSDP PENBMI solvers: Linear Semi-Definite Optimization, Linear Matrix Inequalities, Linear Semi-Definite Programming Problem, Bilinear Matrix Inequalities.

3 TOMLAB /PENOPT Solver Reference

The PENOPT solvers are a set of solvers that were developed by the PENOPT GbR. Table 1 lists the solvers included in TOMLAB /PENOPT. The solvers are called using a set of MEX-file interfaces developed as part of TOMLAB . All functionality of the PENOPT solvers are available and changeable in the TOMLAB framework in Matlab .

Detailed descriptions of the TOMLAB /PENOPT solvers are given in the following sections. Also see the M-file help for each solver.

Additional solvers reference guides for the TOMLAB /PENOPT solvers are available for download from the TOMLAB home page http://tomopt.com. Extensive TOMLAB m-file help is also available, for example help penbmiTL in Matlab will display the features of the PENBMI solver using the TOMLAB format.

TOMLAB /PENOPT solves

The linear semi-definite programming problem with linear matrix inequalities (sdp) is defined as

min

f(x) = c^Tx

s/t

x_L

≤

x_U

b_L

≤

b_U

Q₀ⁱ +

k=1

ⁿ Q_kⁱx_k

0, i=1,…,m.

(1)

where c, x, x_L, x_U

Rⁿ, A

R^{m_l
× n}, b_L,b_U

R^m_l and Q_kⁱ are symmetric matrices of similar dimensions in each constraint i. If there are several LMI constraints, each may have it's own dimension.

The linear semi-definite programming problem with bilinear matrix inequalities (bmi) is defined similarly to (1) but with the matrix following inequality constraints

Q₀ⁱ +

k=1

ⁿ Q_kⁱx_k +

k=1

ⁿ

l=1

ⁿ x_kx_lK_klⁱ

0 (2)

The MEX solvers pensdp and penbmi treat sdp and bmi problems, respectively. These are available in the TOMLAB /PENSDP and TOMLAB /PENBMI toolboxes.

The MEX-file solver pensdp available in the TOMLAB /PENSDP toolbox implements a penalty method aimed at large-scale dense and sparse sdp problems. The interface to the solver allows for data input in the sparse SDPA input format as well as a TOMLAB specific format corresponding to (1).

The MEX-file solver penbmi available in the TOMLAB /PENBMI toolbox is similar to pensdp , with added support for the bilinear matrix inequalities (2).

The add-on toolbox TOMLAB /PENOPT solves linear semi-definite programming problems with linear and bilinear matrix inequalities. The solvers are listed in Table 1. They are written in a combination of Matlab and C code.

Table 1: Solvers in TOMLAB /PENOPT.

Function Description Reference

penbmi Sparse and dense linear semi-definite programming using a penalty algorithm.

penfeas_sdp Feasibility check of systems of linear matrix inequalities, using pensdp .

pensdp Sparse and dense linear semi-definite programming using a penalty algorithm.

penfeas_sdp Feasibility check of systems of linear matrix inequalities, using pensdp .

3.1 PENBMI

TOMLAB /PENBMI was developed in co-operation with PENOPT GbR. The following pages describe the solvers and the feasibility checks.

3.1.1 penfeas_bmi

Purpose
penfeas_bmi checks the feasibility of a system of bilinear matrix inequalities (BMI):

A_i⁰ +

k=1

ⁿ A_k⁽ⁱ⁾ x_k +

k=1

ⁿ

l=1

ⁿ x_k x_l K_kl⁽ⁱ⁾

0, k=1,2,…,m

with symmetric matrices A_kⁱ, K_klⁱ

R^{d_l × d_l}, k,l=1,…,n, i=1,…,m and x

Rⁿ.
Calling Syntax
[ ifeas, feas, xfeas ] = penfeas_bmi(p, x_0, options)
Description of Inputs
p PENBMI format structure, e.g. from sdpa2pen .

x_0 Inital guess of the feasible point. Default = 0.

options Optional 3 × 1 vector. May be omitted, in which case the defaults below are used.

options(1)	Output level of PENBMI solver. Default = 0.
options(2)	Upper bound on x_∞. Default = 1000.
	If <0, no box bounds are applied.
options(3)	Weighting parameter in the objective function. Default 10⁻⁴.

Description of Outputs
ifeas Feasibility of the system:

0	System is strictly feasible
1	System is feasible but not necessarily strictly feasible
-1	System is probably infeasible

feas Value of the minimized maximal eigenvalue of BMI.

xfeas Feasible point

Algorithm
The feasibility of the system of BMI's is checked by solving the following bmi problem:

min

Rⁿ,λ

f(x,λ) = λ + w

₂²

s.t.

− x_bound

≤

x_bound

A₀ⁱ +

k=1

ⁿ A_kⁱ x_k

λ I_{n × n},

i=1,…,m

where x_bound is taken from input argument options(2) , or ∞ if options(2) <0.

When λ < 0, the system is strictly feasible; when λ=0, the system is feasible but has no interior point. When λ>0 for any x_bound, the system may be infeasible.

As the semidefinite problem solved is nonconvex and penbmiQ is a local method, it cannot be guaranteed that the solution λ is a global minimum. Consequently, the (local) optimum λ_loc found by penbmiQ may be a positive number (indicating infeasibility), although the system is feasible (i.e., the global optimum λ_glob<0). In such a case, the user may try various initial points x_0, and also different weighting parameter w (option(3) ).

In practice, the conditions on feasibility are as follows: λ < −10⁻⁶ means strict feasibility, |λ|<10⁻⁶ means feasibility (not strict) and λ > 10⁻⁶ indicates infeasibility. The user can decide by the actual value of the output parameter feas (including the optimal λ).

For the case when the BMI system is unbounded, we have to add artificial bounds on x, otherwise PENBMI might diverge. Of course, it may happen that the feasible point lies outside these bounds. In this case penfeas_bmi claims infeasibility, although the system is actually feasible. When in doubts, the user may try to gradually increase x_bound (parameter options(2) ).

On the other hand, for very ill-conditioned and unbounded systems, the default bound 1000 may be too large and PENBMI may not converge. In this case, the user is advised either to increase the weighting parameter w (to 0.01 and bigger) or to decrease the bound (parameter options(2) ) to a smaller number (100–1).

M-files Used
pen.m

MEX-files Used
penbmiQ

See Also
penfeas_sdp.m

3.1.2 penbmiTL

Purpose
Solve (linear) semi-definite programming problems.

penbmiTL solves problems of the form

min

f(x) = c^T x

s/t

x_L

≤

x_U

b_L

≤

b_U

Q_i⁽⁰⁾ +

k=1

ⁿ Q_k⁽ⁱ⁾ x_k +

k=1

ⁿ

l=1

ⁿ x_k x_l K_kl⁽ⁱ⁾

0, k=1,2,…,m

(3)

where x,x_L,x_U

Rⁿ, A

R^{m_l × n}, b_L,b_U

R^m_l and Q_k⁽ⁱ⁾ are sparse or dense symmetric matrices. The matrix sizes may vary between different matrix inequalities but must be the same in each particular constraint.
Calling Syntax
Result = tomRun('penbmi',Prob,...)
Description of Inputs

Prob	Problem description structure. The following fields are used:

	QP.c	Vector with coefficients for linear objective function.

	A	Linear constraints matrix.
	b_L	Lower bound for linear constraints.
	b_U	Upper bound for linear constraints.

	x_L	Lower bound on variables.
	x_U	Upper bound on variables.

	x_0	Starting point.

	PENOPT	Structure with special fields for SDP parameters. Fields used are:

	LMI	Structure array with matrices for the linear terms of the matrix inequalities. See Examples in § 3.2.2 for a discussion of how to set this correctly.

	ioptions	8× 1 vector with options, defaults in (). Any element set to a value less than zero will be replaced by a default value, in some cases fetched from standard Tomlab parameters.

	ioptions(1)	0/1: use default/user defined values for options.
	ioptions(2)	Maximum number of iterations for overall algorithm (50). If not given, Prob.optParam.MaxIter is used.
	ioptions(3)	Maximum number of iterations in unconstrained optimization (100). If not given, Prob.optParam.MinorIter is used.
	ioptions(4)	Output level: 0/(1)/2/3 = silent / summary / brief / full.
	ioptions(5)	(0)/1: Check density of Hessian / Assume dense.
	ioptions(6)	(0)/1: (Do not) use linesearch in unconstrained minimization.
	ioptions(7)	(0)/1: (Do not) write solution vector to output file.
	ioptions(8)	(0)/1: (Do not) write computed multipliers to output file.

	foptions	1 × 7 vector with options, defaults in ().

	foptions(1)	Scaling factor linear constraints; must be positive. (1.0).
	foptions(2)	Restriction for multiplier update; linear constraints (0.7).
	foptions(3)	Restriction for multiplier update; matrix constraints (0.1).
	foptions(4)	Stopping criterium for overall algorithm (10⁻⁷). Tomlab equivalent: Prob.optParam.eps_f .
	foptions(5)	Lower bound for the penalty parameters (10⁻⁶).
	foptions(6)	Lower bound for the multipliers (10⁻¹⁴).
	foptions(7)	Stopping criterium for unconstrained minimization (10⁻²).

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:

	x_k	Optimal point.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum, c.
	v_k	Lagrange multipliers.

	x_0	Starting point.
	f_0	Function value at start.

	xState	State of each variable, described in the TOMLAB User's Guide.

	Iter	Number of iterations.
	ExitFlag	0: OK.
		1: Maximal number of iterations reached.
		2: Unbounded feasible region.
		3: Rank problems. Can not find any solution point.
		4: Illegal x_0 .
		5: No feasible point x_0 found.

	Inform	If ExitFlag > 0, Inform=ExitFlag.
	QP.B	Optimal active set. See input variable QP.B .

	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	FuncEv	Number of function evaluations. Equal to Iter .
	ConstrEv	Number of constraint evaluations. Equal to Iter .
	Prob	Problem structure used.

Description
pensdp implements a penalty algorithm based on the PBM method of Ben-Tal and Zibulevsky. It is possible to give input data in three different formats:

Standard sparse SPDA format
PENSDP Structural format
Tomlab Quick format

In all three cases, problem setup is done via sdpAssign .

See Also
sdpAssign.m , sdpa2pen.m , sdpDemo.m , tomlab/docs/penbmi.pdf

Warnings
Currently penbmi does not work well solving problems of sdp type.

Examples
Setting the LMI constraints is best described by an example. Assume 3 variables x=(x₁,x₂,x₃) and 2 linear matrix inequalities of sizes 3× 3 and 2× 2 respectively, here given on block-diagonal form:

0
	0
		0

0
	1

2	−1	0
	2	0
		2

1
	−1

x₁

0
	0
		0

3
	−3

x₂ +

2	0	−1
	2	0
		2

0
	0

x₃

The LMI structure could then be initialized with the following commands:

%  Constraint 1
>> LMI(1,1).Q0 = [];
>> LMI(1,1).Q  = [2 -1  0 ; ...
                  0  2  0 ; ...
                  0  0  2];
>> LMI(1,2).Q  = [];
>> LMI(1,3).Q  = [2  0 -1 ; ...
                  0  2  0 ; ...
                  0  0  2];

%  Constraint 2, diagonal matrices only
>> LMI(2,1).Q0 = diag( [0, 1] );
>> LMI(2,1).Q  = diag( [1,-1] );
>> LMI(2,2).Q  = diag( [3,-3] );
>> LMI(2,3).Q  = [ ];

%  Use LMI in call to sdpAssign:
>> Prob=sdpAssign(c,LMI,...)

%  ... or set directly into Prob.PENSDP.LMI field:
>> Prob.PENSDP.LMI = LMI;

Some points of interest:

The LMI structure must be of correct size. This is important if a LMI constraint has zero matrices for the highest numbered variables. If the above example had zero coefficient matrices for x₃, the user would have to set LMI(1,3).Q = [] explicitly, so that the LMI structure array is really 2 × 3. (LMI(2,3).Q would automatically become empty in this case, unless set otherwise by the user).
MATLAB sparse format is allowed and encouraged.
Only the upper triangular part of each matrix is used (symmetry is assumed).

Input in Sparse SDPA Format is handled by the conversion routine sdpa2pen . For example, the problem defined in tomlab/examples/arch0.dat-s can be solved using the following statements:

  >> p = sdpa2pen('arch0.dat-s')
  p =
     vars: 174
     fobj: [1x174 double]
     constr: 174
     ci: [1x174 double]
     bi_dim: [1x174 double]
     bi_idx: [1x174 double]
     bi_val: [1x174 double]
     mconstr: 1
     ai_dim: 175
     ai_row: [1x2874 double]
     ai_col: [1x2874 double]
     ai_val: [1x2874 double]
     msizes: 161
     ai_idx: [175x1 double]
     ai_nzs: [175x1 double]
     x0: [1x174 double]
     ioptions: 0
     foptions: []

  >> Prob   = sdpAssign(p);          % Can call sdpAssign with only 'p' structure
  >> Result = tomRun('pensdp',Prob); % Call tomRun to solve problem

3.2 PENSDP

TOMLAB /PENSDP was developed in co-operation with PENOPT GbR. The following pages describe the solvers and the feasibility checks.

3.2.1 penfeas_sdp

Purpose
penfeas_sdp checks the feasibility of a system of linear matrix inequalities (LMI):

A₀ⁱ +

k=1

ⁿ A_kⁱ x_k

0, i=1,…,m

with symmetric matrices A_kⁱ

R^{d_l × d_l}, k=1,…,n, i=1,…,m and x

Rⁿ.
Calling Syntax
[ifeas,feas,xfeas] = penfeas_sdp(p, options )
Description of Inputs
p PENSDP format structure, e.g. from sdpa2pen .

options Optional 2 × 1 vector. May be omitted, in which case the defaults below are used.

options(1)	Output level of PENSDP solver. Default = 0.
options(2)	Upper bound on x_∞. Default = 1000.
	If <0, no box bounds are applied.

Description of Outputs
ifeas Feasibility of the system:

0	System is strictly feasible
1	System is feasible but not necessarily strictly feasible
-1	System is probably infeasible

feas Value of the minimized maximal eigenvalue of LMI.

xfeas Feasible point

Algorithm
The feasibility of the system of LMI's is checked by solving the following sdp problem:

min

Rⁿ,λ

f(x,λ) = λ

s.t.

− x_bound

≤

x_bound

A₀ⁱ +

k=1

ⁿ A_kⁱ x_k

λ I_{n × n},

i=1,…,m

where x_bound is taken from input argument options(2) , or ∞ if options(2) <0.

When λ < 0, the system is strictly feasible; when λ=0, the system is feasible but has no interior point. When λ>0 for any x_bound, the system is infeasible.

In practice, the conditions on feasibility are as follows: λ < −10⁻⁶ means strict feasibility, |λ|<10⁻⁶ means feasibility (not strict) and λ > 10⁻⁶ indicates infeasibility. The user can decide by the actual value of the output parameter feas (including the optimal λ).

For the case when the LMI system is unbounded, we have to add artificial bounds on x, otherwise PENSDP might diverge. Of course, it may happen that the feasible point lies outside these bounds. In this case penfeas_sdp claims infeasibility, although the system is actually feasible. When in doubts, the user may try to gradually increase x_bound (parameter options(2) ). On the other hand, for very ill-conditioned and unbounded systems, the default bound 1000 may be too large and PENSDP may not converge. In this case, the user is advised to decrease parameter options(2) to a smaller number (100–1).

M-files Used
pen.m

MEX-files Used
pensdp

See Also
penfeas_bmi.m

3.2.2 pensdpTL

Purpose
Solve (linear) semi-definite programming problems.

pensdpTL solves problems of the form

min

f(x) = c^T x

s/t

x_L

≤

x_U

b_L

≤

b_U

Q_i⁽⁰⁾ +

k=1

Q_k⁽ⁱ⁾ x_k

k=1,2,…,m_LMI

(4)

where x,x_L,x_U

Rⁿ, A

R^{m_l × n}, b_L,b_U

R^m_l and Q_k⁽ⁱ⁾ are sparse or dense symmetric matrices. The matrix sizes may vary between different linear matrix inequalities (LMI) but must be the same in each particular constraint.
Calling Syntax
Result = tomRun('pensdp',Prob,...)
Description of Inputs

Prob	Problem description structure. The following fields are used:

	QP.c	Vector with coefficients for linear objective function.

	A	Linear constraints matrix.
	b_L	Lower bound for linear constraints.
	b_U	Upper bound for linear constraints.

	x_L	Lower bound on variables.
	x_U	Upper bound on variables.

	x_0	Starting point.

	PriLevOpt	Print level in pensdpTL and MEX interface. The print level in the solver is controlled by PENOPT.ioptions(4) .
	PENOPT	Structure with special fields for SDP parameters. Fields used are:

	LMI	Structure array with matrices for the linear matrix inequalities. See Examples in § 3.2.2 for a discussion of how to set this correctly.

	ioptions	8× 1 vector with options, defaults in (). Any element set to a value less than zero will be replaced by a default value, in some cases fetched from standard Tomlab parameters.

	ioptions(1)	0/1: use default/user defined values for options.
	ioptions(2)	Maximum number of iterations for overall algorithm (50). If not given, Prob.optParam.MaxIter is used.
	ioptions(3)	Maximum number of iterations in unconstrained optimization (100). If not given, Prob.optParam.MinorIter is used.
	ioptions(4)	Output level: 0/(1)/2/3 = silent/summary/brief/full. Tomlab parameter: Prob.PriLevOpt .
	ioptions(5)	(0)/1: Check density of Hessian / Assume dense.
	ioptions(6)	(0)/1: (Do not) use linesearch in unconstrained minimization.
	ioptions(7)	(0)/1: (Do not) write solution vector to output file.
	ioptions(8)	(0)/1: (Do not) write computed multipliers to output file.

	foptions	7 × 1 vector with optimization parameters, defaults in ():
	foptions(1)	Scaling factor linear constraints; must be positive. (1.0).
	foptions(2)	Restriction for multiplier update; linear constraints (0.7).
	foptions(3)	Restriction for multiplier update; matrix constraints (0.1).
	foptions(4)	Stopping criterium for overall algorithm (10⁻⁷). Tomlab equivalent: Prob.optParam.eps_f .
	foptions(5)	Lower bound for the penalty parameters (10⁻⁶).
	foptions(6)	Lower bound for the multipliers (10⁻¹⁴).
	foptions(7)	Stopping criterium for unconstrained minimization (10⁻²).

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:

	x_k	Optimal point.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum, c.
	v_k	Lagrange multipliers.

	x_0	Starting point.
	f_0	Function value at start.

	xState	State of each variable, described in the TOMLAB User's Guide.

	Iter	Number of iterations.
	ExitFlag	0: OK.
		1: Maximal number of iterations reached.
		2: Unbounded feasible region.
		3: Rank problems. Can not find any solution point.
		4: Illegal x_0 .
		5: No feasible point x_0 found.

	Inform	If ExitFlag > 0, Inform=ExitFlag.
	QP.B	Optimal active set. See input variable QP.B .

	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	FuncEv	Number of function evaluations. Equal to Iter .
	ConstrEv	Number of constraint evaluations. Equal to Iter .
	Prob	Problem structure used.

Description
pensdp implements a penalty algorithm based on the PBM method of Ben-Tal and Zibulevsky. It is possible to give input data in three different formats:

Standard sparse SPDA format
PENSDP Structural format
Tomlab Quick format

In all three cases, problem setup is done via sdpAssign .

See Also
sdpAssign.m , sdpa2pen.m , sdpDemo.m , tomlab/docs/pensdp.pdf, penfeas_sdp.m

Examples
Setting the LMI constraints is best described by an example. Assume 3 variables x=(x₁,x₂,x₃) and 2 linear matrix inequalities of sizes 3× 3 and 2× 2 respectively, here given on block-diagonal form:

0
	0
		0

0
	1

2	−1	0
	2	0
		2

1
	−1

x₁

0
	0
		0

3
	−3

x₂ +

2	0	−1
	2	0
		2

0
	0

x₃

The LMI structure should then be initialized with the following commands:

%  Constraint 1
>> LMI(1,1).Q0 = [ ];
>> LMI(1,1).Q  = [2 -1  0 ; ...
                  0  2  0 ; ...
                  0  0  2 ];
>> LMI(1,2).Q  = [ ];
>> LMI(1,3).Q  = [ 2  0 -1 ; ...
                   0  2  0 ; ...
                   0  0  2 ];

%  Constraint 2, diagonal matrices only
>> LMI(2,1).Q0 = diag( [0, 1] );
>> LMI(2,1).Q  = diag( [1,-1] );
>> LMI(2,2).Q  = diag( [3,-3] );
>> LMI(2,3).Q  = [ ];

%  Use LMI in call to sdpAssign:
>> Prob=sdpAssign(c,LMI,...)

Some points of interest:

The LMI structure must be of correct size. This is important if a LMI constraint has zero matrices for the highest numbered variables. If the above example had zero coefficient matrices for x₃, the user would have to set LMI(1,3).Q = [] explicitly, so that the LMI structure array is really 2 × 3. (LMI(2,3).Q would automatically become empty in this case, unless set otherwise by the user).
MATLAB sparse format is allowed and encouraged.
Only the upper triangular part of each matrix is used (symmetry is assumed).

Input in Sparse SDPA Format is handled by the conversion routine sdpa2pen . For example, the problem defined in tomlab/examples/arch0.dat-s can be solved using the following statements:

  >> p = sdpa2pen('arch0.dat-s')
  p =
     vars: 174
     fobj: [1x174 double]
     constr: 174
     ci: [1x174 double]
     bi_dim: [1x174 double]
     bi_idx: [1x174 double]
     bi_val: [1x174 double]
     mconstr: 1
     ai_dim: 175
     ai_row: [1x2874 double]
     ai_col: [1x2874 double]
     ai_val: [1x2874 double]
     msizes: 161
     ai_idx: [175x1 double]
     ai_nzs: [175x1 double]
     x0: [1x174 double]
     ioptions: 0
     foptions: []

  >> Prob=sdpAssign(p);            % Can call sdpAssign with only 'p' structure
  >> Result=tomRun('pensdp',Prob); % Call tomRun to solve problem

« Previous « Start