hal-00674608
https://hal-ensta-bretagne.archives-ouvertes.fr/hal-00674608
[CNRS] CNRS - Centre national de la recherche scientifique
[INSTITUT-TELECOM] Institut Télécom
[ENSTA-BRETAGNE-STIC] Département STIC
[ENSIETA-E3I2] Exploitation de l'Information en Environnements Incertains
[UNIV-BREST] Université de Bretagne occidentale - Brest (UBO)
[ENSTA-BRETAGNE] ENSTA Bretagne
[UNIV-UBS] Université de Bretagne Sud
[LAB-STICC] Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
[TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes
[ENIB] Ecole Nationale d'Ingénieurs de Brest
[LAB-STICC_ENIB] Laboratoire des Sciences et Techniques de l'Information, de la Communication et de la Connaissance, site ENIB Brest
[INSTITUTS-TELECOM] composantes instituts telecom
Interval Tools and Convex Optimization For Robust Constraint Feasibility
Clement, Benoît
[SPI.AUTO] Engineering Sciences [physics]/Automatic
COMM
Convex optimization
interval analysis
Let consider the basic optimization problem "find all p such that the constraintsf(x,p) demi definite positive is feasible". A SIVIA algorithm will easily answer the question but with a crippling computational time if the size of the problem is high. If we assume that the constraint function is convex in x and non convex in p, one can imagine that we can reduce the computational time... Then we propose a method that combines Convex Optimization and Interval Tools for robust optimization. The main objective is to reduce the computational time due to the convex optimization phase. During the last decade, major developments in convex optimization were focusing on conic programming, a natural non-smooth extension of linear programming. Conic programming is in fact a universal form of convex programming, since each convex set is the intersection of a hyperplane with an appropriate cone in a higher-dimensional space. The advantage of the conic form is that nearly all convex problems can be reformulated as conic problems solvable in polynomial time, although they are non-smooth. In control framework, a large range of applications are covered by linear programming and semidefinite programming (SDP) but the robust approach often leads to nonlinear problem which are solved with constraints relaxations. The proposed methodology takes advantage of the polynomial computational time of convex optimization and also takes advantage of the exhautive representation of nonlinear optimization by interval analysis. Our presentation will focus on robust control and SDP. Then we will present a combining SDP and Interval tools optimization on a trivial example.
2010-06-15
en
3rd Small Workshop on Interval Methods
Nantes, France