Automatic Control Laboratory, ETH Zürich www.control.ethz.ch

Automatic dualization

Johan Löfberg

Outline

• YALMIP?• Recent developments• Automatic dualization

– Primal-dual conic problems– What is the problem?– Implementation

• Applications of automatic dualization• Conclusions

What is YALMIP?

Free MATLAB toolbox for rapid definition, manipulation and solution of optimization problems.

Originally aimed towards linear semidefinite programming.

Supports linear, quadratic, second-order cone, semidefinite, bilinear, geometric, parametric and mixed integer programming.

Interfaces most state-of-the-art solvers.

Supported solversYALMIP

Solvers

SeDuMi SDPA

CSDP DSDP

SDPT3 LMILAB

PENSDP

MAXDET CDD

NAG OOQP

LINPROG

MOSEK

QUADPROG

QSOPT

SDPLR

PENBMI CPLEX

LMIRANK GLPK

Modules

External Internal

KYPD MPT MIP GLOBAL

SOS MOMENT

LOGIC

Recent developments1. Strengthened modelling framework for non-convex problems

Automatic derivation of LP Automatic derivation of MILP

2. Improved integration with MPT for multi-parametric optimization

3. Performance improvements in automatic dualization

Automatic dualization

Primal-dual conic pairs

Everything revolves around the primal-dual conic pair

We work with mixed linear, quadratic and semidefinite cones

Primal-dual conic pairs

Much more convenient in practice to allow free variables (equality constraints)

But this is where the problems arise...

Primal or dual?

Example: Silly LP

Dimension of the problem?

Primal:

Dual:

Primal or dual?

Example: MAXCUT

Dimension of the problem?

Primal:

Dual: Ouch! variables

YALMIP always interprets in dual form

Problem: Data (C,A,b,F,f) not unique (including dimensions)

Primal or dual?

Problem: YALMIP always interprets in dual form

Wanted:

Solution:

Automatic conversion from default dual form interpretation to a symbolic primal form interpretation

Detect and extract primal form numerical model and return symbolic dual of this model.

Implementation

Where are we?

External solver(SeDuMi, SDPA, CPLEX...)

Solver interfaceYALMIP to solver format conversion

YALMIP coreParsing, classification, conversion,

solution treatment etc

Symbolic modeling level User experience in MATLAB

Solver specific format

Internal numerical format

Symbolic model

Internal solvers(MIP, BMIBNB, CUTSDP, MPMIQP)

FeaturesSOS, Moment,Block-diagonalization,

Dualization,...

Input: Symbolic YALMIP model (with constraints )

Implementation

Output: Symbolic YALMIP model

Notation:

Primal cone: Variable X & a constraint of the typeTranslated primal cone: Constraint of the typeDual cone: Anything else (except equality constraints)

Implementation

First step: Find simple primal cones

Initialize: Detected cones and offsets X={}, H = {}

ImplementationSecond step : Find translated primal cones

ImplementationThird step: Introduce slacks for dual cones

Status:All primal cone variables are detected, remaining variables are free and remaining constraints are equality constraints.

ImplementationFourth step: Extract numerical data from remaining constraints

Intermediate result: Uniquely defined model

Return: Symbolic dual form model of detected primal problem

Original variables are duals to constraints in the resulting symbolic model. YALMIP automatically tracks this information and updates numerical data accordingly when the new model is solved.

Applications

MAXCUT

(Computations using SDPT3 3.1)

Sum-of-squares

Find s such that a polynomial p(x,s) is non-negative in x

Can be addressed using sum-of-squares which leads to

YALMIP formulates all SOS problems in this primal form, but solving them like this would be inefficient (since it interprets the model in dual form)

Hence, dualization needed

Stupid! Why not derive the dual form directly?

Sum-of-squares

Generality!

1. Additional constraints on s does not complicate dualization

2.

3. ...or derive an image model (QR or sparse basis, one line of code)

For non-convex (non-linear/integer/rank constrained) SOS problems, we can solve the problem as stated...

Sum-of-squares

Stability analysis

Piecewise affine system (simplified)

SDP for piecewise quadratic Lyapunov certificate

Dualize if some H is very tall compared to state dimension

Outlook & conclusion

• Automatic decision to dualize?• Dualize quadratic objective functions?• Partial dualization?

• Performance for large-scale problems...

• Very simple but very useful• Commands:

Download

http://control.ee.ethz.ch/~joloef/yalmip.php