09h45 - 10h10
Best uniform solutions of inconsistent linear inequality systems via LSIP
Best least squares solutions to linear systems of equations and to linear inequality systems have been widely used, in astronomy (since the early 1800s) and in signal and image processing (during the last decades), respectively. For both types of linear systems, the corresponding (quadratic) optimization problem is solvable, the difference being that best least squares solutions can be computed by applying a closed expression when the constraints are given by equations while iterative numerical methods are required for inequality systems. When one deals with inconsistent linear inequality systems with infinitely many constraints, as those arising in robust linear programming, a difficulty arises: the corresponding optimization problem is not always solvable. Then, the linear semi-infinite programming formulation of the best uniform approximation problem becomes a useful tool both in theory (as it allows to characterize existence and uniqueness of solutions in terms of the data) and practice (at least when the index set are low dimensional). The talk is based on recent joint research with J.-B. Hiriart-Urruty and M.A. López.
10h10 - 10h35
On finite linear systems containing strict inequalities
This talk deals with linear systems containing finitely many weak and/or strict inequalities whose solution sets are referred to as evenly convex polyhedral sets. The classical Motzkin Theorem states that every (closed convex) polyhedron is the Minkowski sum of a convex hull of finitely many points and a finitely generated cone. In this sense, similar representations for evenly convex polyhedra have been recently given by using the standard version for classical polyhedra. In this work, we provide a new dual tool that completely characterizes finite linear systems containing strict inequalities and it constitutes the key for obtaining a generalization of Motzkin Theorem for evenly convex polyhedra.
10h35 - 11h00
One special SIP problem arising in parametric convex SIP Programming
We consider a SIP problem of a special form possessing some special properties. We show how optimal properties of this SIP problem permit to conclude about existence of optimal parameters in a family of parametric NLP problems that arise in study of parametric SIP problems. The results obtained can be used for different applications and in the future work dedicated to parametric SIP.