Seminar on Optimization Techniques: LP, QP, SOCP, SDP, LMI, BMI

Seminar:
LP, QP, SOCP, SDP, LMI, BMI and other UFOs
(not-Dr.-yet) Mikhail V. Konnik
School of Electrical Engineering and Computer Science
The University of Newcastle
Australia

December 13, 2013

(not-Dr.-yet) Mikhail V. Konnik School of Electrical Engineering and Computer ScienceThe University of Newcastle / 42
Seminar: LP, QP, SOCP, SDP, LMI, BMI and other UFOs 13, 2013
December
1 Austra

Disclaimer
the Author is not an expert (yet) in LMI/BMI/SDP (but he is doing his
best to become one), and therefore:
the whole presentation is just an overview of LMI/BMI from the point of
view of the Author's (severely incomplete) understanding of optimisation;
the Author is not ready to solve complicated BMI/UFO problems that
the people in the audience surely have :-)

the problem of solving even LMI (let alone BMI) is much more complicated
than many people in the audience think;
software packages for LMI/BMI are not as mature as for e.g. QP;
the Author is known for his anity to reinvent the wheel to better
understand how stu works.
The wheel that is being inventing by the Author is still in the blueprint
stage, is not round, and cannot be used for anything remotely serious.

the Author is not liable for any loss, damage, illness, injury, headache,
crash, collapse of the Universe or anything else caused by this presentation.
proceed at your own risk!

December
2 Austra

Introduction to Linear Matrix Inequalities

Part I ::
A short introduction to Linear Matrix Inequalities
(LMI) and Semidenite Programming (SDP)

December
3 Austra


LMI and Control

Linear Matrix Inequalities and Control

The story of LMI begins in about 1890, when Lyapunov published his seminal
work, where he showed that the dierential equation
d
dx

x (t )

= Ax (t )

is stable (i.e., all trajectories converge to zero) if and only if there exists a
positive-denite matrix P such that[1]
A
The requirement P

T P + PA

0

0 is what we now call a Lyapunov inequality on P .

December
4 Austra


LMI and Control

Linear Matrix Inequalities and Control
The Lyapunov inequality A

T P + PA

0 is not the only example of how one

can convert control problems into LMI: Riccati inequality can be converted
into an LMI, too:
A

T P + PA + PBR −1 B T P + Q

0,

is equivalent to the following LMI (via Schur complement):

−AT P − PA − Q
B

The requirement P

TP

PB
R

0

(1)

0 is what we now call a Lyapunov inequality on P .

December
5 Austra


What is an LMI?

What is an LMI?
A (strict) linear matrix inequality (LMI) is a convex

constraint in the form:
N

F (x )

= F0 + x1 F1 + · · · + xN FN = F0 +

i =1

xi Fi

0,

(2)

where:
x

= [x1 , . . . , xN ]

is a vector of

unknown scalars (the decision

or optimisation variables) x1 , x2 and so on;
F0 , F1 , F2 , . . . , FN are n
The sign F

×n

symmetric matrices, they are given.

0 is a generalised inequality meaning F is a negative denite

matrix (i.e., the largest eigenvalue of F (x ) is negative), F
denite, and F

0 is positive

0 is positive semi-denite.

December
6 Austra


What is an LMI?

What is an LMI? - A set of CONVEX constraints!
Note that the LMI:
F (x )

= F0 + x1 F1 + . . .
N

+xN FN = F0 +

i =1

0,

xi Fi

is just a convex constraint
on x , and therefore:
Its solution set, called
the

feasible set,

convex subset of

is a

RN

Finding a solution x of
LMI, if any, is a

convex

Figure 1: Supporting hyperplanes for LMI.

problem.
Convexity has an important consequence: even though LMI has no analytical
solution in general, it can be solved numerically.

December
7 Austra


Solving Linear Matrix Inequalities in a brute-force way

Solving LMI in a brute-force way

December
8 Austra


Solving Linear Matrix Inequalities in a brute-force way

Solving LMI in a brute-force way
The LMI species a convex

constraint

on

x,

and

to

solve an LMI means that
we can nd x that satises
the LMI.
Brute-force: we can try the
variables in
x1 x2
X =
x2 x3
compute
Z

= AT X + XA

Figure 2:

The corresponding set from the LMI

F (x ) := x1 F1 + x2 F2

I.

check eigenvalues
E

= eig (Z )

December
9 Austra

Semidenite Programming

We can do better: solve LMI via SDP
A problem which subsumes linear, quadratic,
geometric and second-order cone programming
is called a

semidenite program (SDP):
TX

minimize

C

subject to

F0

X

- - this is optimality metric!

N

Ax

+
i =1

xi Fi

0,

- - this is LMI!

=b

where the matrices F0 , F1 , . . . Fn

∈ Sk ,

and A

∈ Rp×n .

That is: LMI cuts a set of solutions via convex set of constraints, and SDP
allows to pick up an optimal solution using some optimality metric.

To solve LMI ⇔ solve a Semidenite Programming (SDP) problem.

Seminar: LP, QP, SOCP, SDP, LMI, BMI and other UFOs 2013
December 13,
10 Austra


What is Semidenite?
The set of real symmetric n × n matrices
is denoted

Sn .

A matrix Z

semidenite
Rn .

∈ Sn
if

is called

T
x Zx ≥

positive

0 for all x

∈

On the other hand, a matrix Z is

called

positive denite

for all nonzero x

∈ Rn .

if x

T Zx

0

The set of positive semidenite matrices
is denoted

Sn .
+

In other words, the eigenvalues

λ1 . . . λn

of a positive

semidenite matrix Z+ are nonnegative (i.e., can be zeros or less than
machine epsilon
The set

Sn
+

).

is a convex cone.

December 13,
11 Austra


What is Semidenite? (continued)
A semidenite program (SDP) is a generalization of a linear program (LP),

where the inequality constraints are replaced by matrix inequalities corresponding
to the cone of positive semidenite matrices.
An SDP in the pure primal form is dened as:
minimise

trace(CX )

subject to

trace(Ai X ) = bi ∀i = 1, . . . , m

X

X

(3)

0.

∈ Sn is the decision variable (can be a vector or a matrix), b ∈ Rm
+
n
C , A1 . . . Am ∈ S+ are given symmetric matrices. There are m ane

where X
and

constraints among the entries of the

positive semidenite matrix X .

feasible sets of SDPs
can have curved faces, together with sharp corners where faces intersect.
Unlike LP, where the feasible sets are polyhedra, the

December 13,
12 Austra


Scope of SDP

TX

minimise

C

subject to

F0

X

SDP:

N

minimise

SOCP:

c

x

+

QP:

x

subject to

LP:

2

x

Ax

0

- - - LMI

+ bi

2

ciT x + di

=g

T Hx + c T x + d

b

Tx + d

minimise

c

subject to

Ax

x

xi Fi

Tx

Fx
1

i =1

Ai x

subject to

minimise

Scope of Semidenite Programming

b

December 13,
13 Austra


SDP: One Ring to Rule them All

Semidenite Programming SDP: One Ring to Rule them All
Using the power of the Sauron's One Ring SDP, re-cast a QP problem into
an SDP problem. Use Cholesky factor
x

subject to

Ax

minimise

X

subject to

C

TX

F (x )

0

:= blockdiag (FQ (x ), Ai x − bi ),

Ai is the ith row of A, and we use Schur complement r
FQ (x )

=

H.

≥b

Equivalent SDP:

where F (x )

of the Hessian matrix

T Q T Qx + c T x + r

minimise

x

H = QT Q

r

− cT x
Qx

x

T QT
I

≥ x T Q T Qx + c T x ,

0.

December 13,
14 Austra

Summary for SDP and LMI

Summary for SDP and LMI
1

A (strict) linear matrix inequality (LMI) is a convex
form:

N
F0

+
i =1

Finding a solution x of LMI, if any, is a

3

To solve LMI

⇔

convex problem.

solve a Semidenite Programming (SDP):

TX

minimize

C

subject to

x1 F1

x

0.

xi Fi

2

Ax

- - this is an optimality metric!

+ · · · + xn Fn + F0

0

- - this is LMI!

=b

4

constraint in the

∈ Sk ,

and A

∈ Rp×n .

LMI cuts a set of solutions via convex set of constraints, and SDP allows
to pick up an optimal solution using some optimality metric.

December 13,
15 Austra

Solving convex SDPs and LMIs

Part II ::
Solving Semidenite Problems (SDP)

December 13,
16 Austra


Analytical techniques and numerical algorithms

Solving convex SDPs and LMIs : analytic techniques and
numerical algorithms
Even small Semidenite problems (and LMIs) are too dicult to solve
analytically - use numerical methods!
1

Primal-dual Interior point methods

2

Logarithmic barrier methods

3

Augmented Lagrangian

There are some analytical methods that can help/simplify/relax the problem:
1

Schur complement - convert an LMI into a matrix

2

S-Procedure - a Lagrange relaxation technique for problems with quadratic
constraints (solves a system of quadratic inequalities via LMI).

3

The Elimination Lemma - eliminates some variables in LMI.

December 13,
17 Austra


Schur complement

Analytical techniques: Schur complement
Schur complement is an indispensable tool for transforming non-linear
constraints into convex LMI. A symmetric matrix X can be decomposed:
X

The matrix S

C

− B T A− 1 B

In other words, for all X

∈

=

A
B

T

B

is called the

n

S , Y

(4)

C

Schur complement of A in X .

∈ Rm×n

, Z

∈

S

m

, the following

statements are equivalent:
0; X

a) Z
b)

− Y T Z −1 Y

X

T
Y

Y

Z

0.

0.

a) Z
b) Z

0; X
0 ;

− Y T Z −1 Y
X
Y

Y

T

Z

0.
0.

December 13,
18 Austra


S-procedure

Analytical techniques: S-procedure for Quadratic Forms and
Strict Inequalities
In some problems, we nd that some quadratic function must be negative
whenever some other quadratic functions are all negative.

With the S-procedure, we can replace this problem by one inequality
to be satised by introducing some positive scalars to be determined.
Motivation:

The fundamental question of the theory of the S-lemma is the

following: ([2], page: 371).
When is a quadratic inequality a consequence of other quadratic
inequalities?

In short terms, the

S-procedure is a Lagrange relaxation method; it tries

to solve a system of quadratic inequalities via LMI.

December 13,
19 Austra


S-procedure

Analytical techniques: S-procedure for Quadratic Forms and
Strict Inequalities
. . . Tp ∈ Rn×n be
condition on T0 . . . Tp :

Let T0

ξ T T0 ξ 0

symmetric matrices.

for all

It is obvious that if there exists

ξ=0

We consider the following

such that

τ1 ≥ 0, . . . τp ≥ 0

ξ T Ti ξ ≥ 0

(5)

such that:

p
T0

−
i =1

τi Ti

0,

then (5) holds.
For further details and bibliography comments, the reader is referred to [3].
*Please do NOT embarrass the Author with questions :-)

December 13,
20 Austra


S-procedure

Analytical techniques: S-procedure, continued - an
illustrative example
The following constraint on the variable P : ([3], page: 36)

∀ξ = 0
ξ
π

and

T

π

satisfying

T P + PA
T
B P

A

π T π ≤ ξ T C T C ξ,
ξ
π

PB
0

(6)

0

Applying the S-procedure, (6) is equivalent to:
A

T P + PA + τ C T C
T
B P

Thus the problem of nding P

0

PB

−τ I

0

such that (6) holds can be expressed as

an LMI in P and the scalar variable

τ.

([3], page: 36)

*Please do NOT embarrass the Author with questions here ;-)

December 13,
21 Austra


The Elimination Lemma

Analytical techniques: S-procedure, continued - an
illustrative example
The Elimination Lemma [3] allows to eliminate some variables appearing in
LMI leading to inequalities without the eliminated variable. Consider:
G (z )

+ U (z )XV (z )T + V (z )X T U (z )T 0,

(7)

ˆ
ˆ
Suppose that for every z , U (z ) and V (z ) are orthogonal complements of
U (z ) and V (z ) respectively. Then (7) holds for some X and z

=

z0 if and

only if the inequalities

T G (z )U (z ) 0,
ˆ

ˆ
U (z )
hold with z

T G (z )V (z ) 0,
ˆ

ˆ
V (z )

(8)

= z0 .

In other words, feasibility of the matrix inequality (7) with variables X and
z is equivalent to the feasibility of (8) with variable z .

December 13,
22 Austra


Numerical algorithms for convex SDP and LMI

Numerical algorithms for convex SDP and LMI

Schur complement, S-procedure, Elimination Lemma and other analytical
tricks will

NOT solve LMIs - use numerical methods!

1

Primal-dual Interior point methods

2

Logarithmic barrier methods

3

Augmented Lagrangian

LMIs are usually solved as SDP problems; therefore it is benecial to study
how the SDP solvers work.
*The Author promises NOT to embarrass the Audience with details here ;-)

December 13,
23 Austra


Canonical and matrix forms of LMIs

Numerical algorithms for convex SDP and LMI: Canonical
and matrix forms of LMIs
Generally, we do not encounter the LMI in the canonical form in control
theory but rather in the form of matrix variables. Thus, before the solution
of LMI commences via SDP, the LMI must be pre-processed, or

parsed.

For example, the Lyapunov's inequality:
A

T P + PA

0 P

= PT

0

(9)

can be written in the canonical form:

m
F (x )

where F0

=0

and Fi

= F0 +

i =1

= −AT Bi − Bi A,

xi Fi

0,

and where Bi , i

(10)

= 1, . . . , n(n + 1)/2

are matrix bases for symmetric matrices of size n.

December 13,
24 Austra


Canonical and matrix forms of LMIs

Canonical form of LMI
Consider the Lyapunov inequality in
where A

−1
0

=

2

−2

and X

matrix form:

=

x1

x2

x2

x3

A

T X + XA

0

. The decision variables are

scalars x1 , x2 , and x3 of the matrix X .
Convert to the

A

canonical form that is F (x ) = F0 + x1 F1 + · · · + xN FN

T X + XA → · · · →

−2x1
2x1 − 3x2

− 3x2
4x2 − 4x3
2x1

:

0

Now we extract the coecients for each variable x1 , x2 and x3 :

−2x1
2x1 − 3x2

− 3x2
4x2 − 4x3

2x1

→ x1 ·

−2

2

2

0

matrix

F1

+x2 ·

0

−3

−3

4

matrix

Therefore, we converted the problem into LMI x1 F1

F2

0

0

0

+x3 ·

−4

matrix

+ x2 F2 + x3 F3

F3

0.

December 13,
25 Austra


Software toolboxes for numerical solution of SDP and LMI

Solvers:

Parsers:
1

2

YALMIP - yet another
LMI parser
CVX - supports two
solvers:

SeDuMi

1

2

CVX

are

3

parsers,

SDP3 and PENSDP.

infeasible

-

infeasible

path-

SeDuMi

-

Self-Dualself-dual

embedding technique, Interior

is, they don't solve the actual
other solvers such as SeDuMi,

SDPT3

Minimization,

that

problem; instead, they rely on

Mehrotra-type

following algorithm

Please note that both YALMIP
and

-

primal-dual interior-point

and

SDPT3

SDPA

predictor-corrector

Point
4

LMI Control Toolbox
Lab,

Nemirovskii's

- LMI

projective

algorithm

December 13,
26 Austra



Author's success story: Kalman gains via LMI
˜
Denote the state estimation error as x
e

ˆ
= z − z.

=

x

ˆ
−x

and the estimation error

The overall system's dynamics:

˜
x (k

+ 1) = (A − KCy )˜(k ) +
x
e (k )

Bw

− KDyw

w (k )

(11)

˜
= Cz x (k ) + Dzw w (k )

Write the SDP for the state estimator synthesis:
min

Y ,X ,P

trace (X )


subject to

P

 (PA − YCy )T
(PB − YDyw )T
X

T
Cz
P

0

Cz
P
and Y

PA

− YCy

PB

− YDyw

P

0

0



Im



0

0

= PK ,

X

= P −1

December 13,
27 Austra



Author's success story: Kalman gains via LMI
An LMI for Kalman gains for a small 2

×2

system was solved using the

SeDuMi solver and the YALMIP parser interface:

X = sdpvar(2,2); Dyw = [0 0 1 0; 0 0 0 1];
Y = sdpvar(2,2); Cz = [1 0; 0 1];
P = sdpvar(2,2); Dzw = 0; Cy = C_k;
Bw = [0 0
0 0; ...
0 0.01 0 0];
Fset=set(P0) + set( [P, P*A_k-Y*Cy, P*Bw - Y*Dyw;...
(P*A_k-Y*Cy)', P, zeros(2,4);...
(P*Bw - Y*Dyw)', zeros(4,2), eye(4)]0 ) +...
set([X,Cz; Cz', P]0);
sol = solvesdp(Fset, trace(X));
The matrix of gains found via LMI is, of course, the same (up to numerical
errors) as the one found via Riccati equations.

K = new_X*new_Y

December 13,
28 Austra

Summary for methods of solving SDPs

Summary for methods of solving SDPs
1

There are analytical tools that help to simplify an SDP/LMI problem:
Schur complement, S-Procedure, Elimination Lemma.

2

Even small Semidenite problems (and LMIs) are too dicult to solve
analytically - use numerical methods!

3

Good news:

convex SDPs (LMIs) are solvable within reasonable time

using numerical methods (Interior point, Logarithmic barrier, Augmented
Lagrangian).
4

convert an LMI from the matrix form into semidenite (canonical)
parsers (YALMIP, CVX).
To solve an LMI/SDP, we need solvers (SDPA, SDPT3, SeDuMi, LMI
To

form, we need

5

Control Toolbox).
6

Author's success story: using YALMIP and SeDuMi, he solved an LMI
for Kalman gains for a small 2

×2

system.

December 13,
29 Austra

Solving non-convex SDPs and BMIs

Part III ::
Solving non-convex Semidenite Problems and
Bilinear Matrix Inequalities (BMIs)

December 13,
30 Austra

Introduction to Bilinear Matrix Inequalities

Bilinear Matrix Inequalities
While a Linear matrix inequality (LMI) denotes a constraint of the form:

n
F (x )

= F0 +

i =1

where Fi are xed symmetric matrices and x
a

0,

xi Fi

∈

(12)

n

is the decision variable,

R

Bilinear Matrix Inequality constraints are in the form:
n
F (x )

= F0 +

i =1

n
xi Fi

n

+
j =1 j =1

Fi ,j xi xj

0,

(13)

are denoted BMIs (bilinear matrix inequalities).
Bilinear Matrix Inequalities are in the realm of

non-convex

Semidenite

Programming.

December 13,
31 Austra


Bilinear Matrix Inequalities: a Challenge

non-convex and
NP-hard in general[4], hence intractable in theory.
BMIs correspond to non-convex and (possibly) non-smooth Semidenite
Optimisation problems with BMIs are known to be

Programming - this is global optimisation (i.e., exponential time, local
solutions...);
no reliable solver exists for non-convex SDP problems:

branch-and-

bound, genetic algorithms, spectral bundle method, barrier methods no guarantee to nd a (global) solution within reasonable time.
numerical algorithms are inhumanly dicult to devise: only few exist.

December 13,
32 Austra


Bilinear Matrix Inequalities: an Illustration
Below is an illustration of the

non-convex

objective

is

function,

max

3

an

example

of

a

BMI

problem.

x2

subject to

which

quadratic problem with linear

x

2
2
− 2x2 − x1 − x2 ≥ 0

− x1 − x2 − x1 x2 ≥ 0
1

+ x2 x1 ≥ 0

Figure 3:

Illustration of a BMI: a non-

convex feasible set delimited by circular
and hyperbolic arcs (adapted from[5]).

December 13,
33 Austra


Numerical algorithms for non-convex SDPs and BMIs

Numerical algorithms for non-convex SDPs and BMIs
Don't even try to solve BMIs analytically - it is dicult even for numerical
methods of global optimisation (non-convex SDPs):
1

Genetic algorithms:

Read-coded Genetic Algorithms, Extrapolation-

directed Crossover (EDX), Minimal Generation Gap (MGG);
2

Branch-and-Bound, Branch-and-Cut methods (have troubles with solving
medium/large problems due to loose lower-bound approximations);

3

Coordinate-descent method (BMI problem is solved independently for
each coordinate at each step using a LMI optimisation)

4

spectral Bundle method;

5

attempts to use Augmented Lagrangian / Barrier methods.

*The Author promises NOT to embarrass the Audience with details here ;-)

December 13,
34 Austra

Software toolboxes for numerical solution of non-convex
Introduction to Bilinear Matrix Inequalities SDPs and BMIs

Software toolboxes for numerical solution of non-convex
SDPs and BMIs

Solvers:
1

Parsers:
1

YALMIP LMI parser

PENBMI

-

penalty/barrier

function
yet

another

(Augmented

Lagrangian?)
2

HIFOO

-

hybrid

(quasi-Newton
bundling

and

method
updating,
gradient

sampling)

December 13,
35 Austra

Summary for methods of solving non-convex SDPs and
BMIs

Summary for methods of solving non-convex
SDPs and BMIs
1

Bilinear Matrix Inequalities are in the realm of

non-convex Semidenite

Programming = Global Optimisation.
2

BMIs are known to be

non-convex

and

NP-hard in general,

hence

intractable in theory.
3

no reliable solver exists for non-convex SDP problems:

branch-and-

bound, simulated annealing, genetic algorithms, spectral bundle method,
barrier methods - no guarantee to nd a (global) solution within reasonable
time.
4

numerical algorithms are inhumanly dicult to devise, only few exist:
PENBMI and HIFOO (current on 2013).

December 13,
36 Austra

Conclusion and Summary

Part IV ::
Conclusion and Summary

December 13,
37 Austra

Conclusion: Linear Matrix Inequalities / convex SDPs

Conclusion: Bilinear Matrix Inequalities /
non-convex SDPs
1

a

Linear Matrix Inequality is a convex constraint in the form:
N
F0

2

+
i =1

0.

xi Fi

In mathematical programming terminology, to solve LMI means to
solve a Semidenite Programming (SDP) problem:

TX

minimize

C

subject to

x1 F1

x

Ax

- - this is an optimality metric!

+ · · · + xn Fn + F0

0

- - this is LMI!

=b


∈ Sk ,

and A

∈ Rp×n .

December 13,
38 Austra

Conclusion: Linear Matrix Inequalities / convex SDPs

non-convex SDPs, continued
1

There are analytical tools that help to simplify an SDP/LMI problem:
Schur complement (most useful), S-Procedure, Elimination Lemma.

2

convex SDPs (LMIs) can be solved within reasonable time using welldeveloped numerical methods (Primal-dual Interior point, Logarithmic
barrier, Augmented Lagrangian).

3

To convert an LMI from the matrix form into semidenite (canonical)
form, we need

4

parsers (YALMIP, CVX).

To solve an LMI/SDP, we need solvers (SDPA, SDPT3, SeDuMi, LMI
Control Toolbox).

5

Author's success story: using YALMIP and SeDuMi, he solved an LMI
for Kalman gains for a small 2

×2

system.

December 13,
39 Austra

Conclusion: Bilinear Matrix Inequalities / non-convex SDPs

non-convex SDPs
1

a

Bilinear Matrix Inequality constraints are in the form:
n
F (x )

2
3

= F0 +

i =1

n
xi Fi

n

+
j =1 j =1

Fi ,j xi xj

0,

non-convex Semidenite Programming.
BMIs = non-convex SDPs, which are known to be NP-hard in general,
BMIs are in the realm of

hence intractable in theory.
4

no reliable solver exists for non-convex SDPs: branch-and-bound, genetic
algorithms, barrier methods - no guarantee to nd a (global) solution
within reasonable time.

5

numerical algorithms are dicult to devise, only few toolboxes exist.

December 13,
40 Austra


The End

- [Frodo:] I will take the Ring to Mordor!
[pause]
- [Frodo:] Though... I do not know the way.

Frodo Baggins
from The Lord of the Rings: The Fellowship of the Ring

December 13,
41 Austra


Convex set of LMI constraints

Let us denote by

X

the set of points x

∈

Rm

that satisfy:

X :=

m
x

The set

: F0 +

X

i =1

xi Fi

0

.

is convex

since we have
F (x )
if

∀ :

0 if and only
z

T
z F (x )z ≥

∈ Rn :
0.

December 13,
42 Austra



Stephen Boyd, V Balakrishnan, E Feron, and Laurent El Ghaoui.
History of linear matrix inequalities in control theory.
In American Control Conference, 1994, volume 1, pages 3134. IEEE, 1994.
Imre Pólik and Tamás Terlaky.
A survey of the s-lemma.
SIAM review, 49(3):371418, 2007.
Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan.
Linear matrix inequalities in system and control theory, volume 15.
SIAM, 1994.
Onur Toker and Hitay Ozbay.
On the np-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output
feedback.
In American Control Conference, 1995. Proceedings of the, volume 4, pages 25252526. IEEE, 1995.
Didier Henrion.
Course on lmi: What is an lmi?
Technical report, www.laas.fr/~henrion, October 2006.

December 13,
42 Austra

Seminar on Optimization Techniques: LP, QP, SOCP, SDP, LMI, BMI

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Ähnlich wie Seminar on Optimization Techniques: LP, QP, SOCP, SDP, LMI, BMI

Ähnlich wie Seminar on Optimization Techniques: LP, QP, SOCP, SDP, LMI, BMI (20)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

Seminar on Optimization Techniques: LP, QP, SOCP, SDP, LMI, BMI