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Semidefinite Optimization
J. Oravec
Slovak University of Technology in Bratislava
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Semidefinite Optimization
– super class of convex optimization problems
SDP
SOCP
QCQP
QP
LP
Convex Optimization
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Linear Programming
– standard LP:
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Linear Programming
– standard LP:
– feasibility set:
– polyhedron
x1
x2
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Linear Programming
– standard LP:
– feasibility set:
– polyhedron
– polytope
x1
x2
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Linear Programming
– standard LP:
– feasibility set:
– polyhedron
– polytope
x1
x2
x*
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Linear Programming
– standard LP:
– feasibility set:
– polyhedron
– polytope
x1
x2
x*
degeneracy:infinitely many solutions
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Linear Programming
– standard LP:
– feasibility set:
– polyhedron:
– polytope:
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Linear Programming
– primal LP:
– dual LP:
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– optimization of a linear function subject to constraints of linear matrix inequality:
Semidefinite Programming
– decision variables are symmetric matrices:
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Semidefinite Programming
– standard SDP:
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Semidefinite Programming
– standard SDP:
– feasibility set:
– spectrahedron
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Semidefinite Programming
– standard SDP:
– feasibility set:
– spectrahedron
semialgebric set
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Semidefinite Programming
– standard SDP:
– feasibility set:
– spectrahedron
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Semidefinite Programming
– primal SDP:
– dual SDP:
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– Lyapunov function:
– quadratic Lyapunov function:
Lyapunov Stability
– Lyapunov stability (discrete time domain)
– Lyapunov stability (continuous time domain)
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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Lyapunov Stability
A. Lyapunov: The general problem of the stability of motion, 1892
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– Lyapunov function:
– quadratic Lyapunov function:
Lyapunov Stability
– Lyapunov stability (discrete time domain)
– Lyapunov stability (continuous time domain)
A. Lyapunov: The general problem of the stability of motion, 1892
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YALMIP
– originally: Yet Another LMI Parser
– since 2003, by Johan Löfberg, Linköping University
– language for modeling and solution of the (non-)convex
optimization problems in MATALB
– freely available: https://yalmip.github.io
– YALMIP : A Toolbox for Modeling and Optimization in MATLAB,
In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.
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YALMIP: How to
– define decision variable:
>>P = sdpvar(N,N,’symmetric’) % symmetric matrix NxN
>>P = sdpvar(N,N) % symmetric matrix NxN
>>P = sdpvar(N) % symmetric matrix NxN
>>P = sdpvar(N,N,’full’) % non-symmetric matrix NxN
Linear matrix variable 2x2 (symmetric, real, 3 variables)
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YALMIP: How to
– define constraints of optimization problem:
>>constr = [ P >= 0 ] % LMI
>>constr = constr + [[ A'*P*A – P ] <= 0] %LMI
++++++++++++++++++++++++++++++++
| ID| Constraint|
++++++++++++++++++++++++++++++++
| #1| Matrix inequality 2x2|
| #2| Matrix inequality 2x2|
++++++++++++++++++++++++++++++++
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YALMIP: How to
– define objective of optimization problem:
>>obj = trace(P)
Linear scalar (real, 2 variables)
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YALMIP: How to
– define settings of optimization problem:
>>opt = sdpsettings('solver','mosek’) % SDP solver
>>opt = sdpsettings(opt,'verbose’,0) % silent mode
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YALMIP: How to
– solve optimization problem:
>>sol = optimize(constr,obj,opt)
yalmiptime: 0.5157
solvertime: 0.1963
info: 'Successfully solved (MOSEK)'
problem: 0
– parser time
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YALMIP: How to
– solve optimization problem:
>>sol = optimize(constr,obj,opt)
yalmiptime: 0.5157
solvertime: 0.1963
info: 'Successfully solved (MOSEK)'
problem: 0
– optimization time
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YALMIP: How to
– solve optimization problem:
>>sol = optimize(constr,obj,opt)
yalmiptime: 0.5157
solvertime: 0.1963
info: 'Successfully solved (MOSEK)'
problem: 0
– no problems detected
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YALMIP: How to
– solve optimization problem:
>>sol = optimize(constr,obj,opt)
yalmiptime: 0.5157
solvertime: 0.1963
info: 'Infeasible problem (MOSEK)'
problem: 1
– solution does not exist
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YALMIP: How to
– solve optimization problem:
>>sol = optimize(constr,obj,opt)
yalmiptime: 0.5157
solvertime: 0.1963
info: 'Unbounded objective function (MOSEK)'
problem: 2
– do not trust the solution and bound the objective function
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YALMIP: How to
– solve optimization problem:
>>sol = optimize(constr,obj,opt)
yalmiptime: 0.5157
solvertime: 0.1963
info: 'Numerical problems (MOSEK)'
problem: 4
– returned solution may not be correct – double-check feasibility
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YALMIP: How to
– recover solution of optimization problem:
>>P_opt = value(P)
>>eigP = eig(P_opt)
>>M = [ A'*P_opt*A - P_opt ]
>>eigM = eig(M)
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YALMIP and SDP solvers
– free: CSDP (OPTI Toolbox), DSDP, LOGDETPPA,
PENLAB (nonline SDPs), SDPA (high-precision),
SDPLR, SDPT3 (logdet), SDPNAL, SEDUMI (default)
– free for academia: MOSEK, PENSDP
– commercial: LMILAB (not recommended), PENBMI
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CP
LP
Linear programming:
Convex Optimization
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CP
QP
LP
Quadratic programming:
Convex Optimization
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CP
QCQP
QP
LP
Quadratically constrained quadratic
programming:
Convex Optimization
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CP
SOCP
QCQP
QP
LP
Second-order cone programming:
Convex Optimization
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CP
Second-order cone programming:
SOCP
QCQP
QP
LP
LP:
S. Boyd – L. Vandenberghe: Convex Optimization, 2004
Convex Optimization
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CP
Second-order cone programming:
SOCP
QCQP
QP
LP
QP:
S. Boyd – L. Vandenberghe: Convex Optimization, 2004
Convex Optimization
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CP QCQP:
SOCP
QCQP
QP
LP
Second-order cone programming:
S. Boyd – L. Vandenberghe: Convex Optimization, 2004
Convex Optimization
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CP QCQP:
SOCP
QCQP
QP
LP
Second-order cone programming:
S. Boyd – L. Vandenberghe: Convex Optimization, 2004
Convex Optimization
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SDP
SOCP
QCQP
QP
LP
CP
MALTAB/YALMIP solvers: MOSEK, PENBMI, CSDP, DSDP, SDPLR, SDPT3, SDPNAL, SEDUMI
Semidefinite programming:
Convex Optimization
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SDP
SOCP
QCQP
QP
LP
CP
SOCP:
Semidefinite programming:
S. Boyd – L. Vandenberghe: Convex Optimization, 2004
Convex Optimization
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References
– G. Blekherman , P. A. Parrilo and R. R. Thomas (2013): Semidefinite Optimization
and Convex Algebraic Geometry
– S. Boyd, L. Vandenberghe (2004): Convex Optimization. Cambridge University Press.
– S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan (1994): Linear Matrix Inequalities in
System and Control Theory. SIAM.
– J. Löfberg, YALMIP : A Toolbox for Modeling and Optimization in MATLAB, In
Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.
![Page 55: Prezentácia programu PowerPointoravec/pedagogy/sdp... · References –G. Blekherman , P. A. Parrilo and R. R. Thomas (2013): Semidefinite Optimization and Convex Algebraic Geometry](https://reader033.vdocuments.site/reader033/viewer/2022050203/5f57029e94193a05ef20d38c/html5/thumbnails/55.jpg)