1,258 research outputs found
Stability and Well-Posedness in Linear Semi-Infinite Programming
This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.This research was partially supported by grants PB95-0687 and SAB 95-0311 from DGES and by grants GV-2219/94 and GV-C-CN-10-067-96 from Generalitat Valenciana
Quantitative Stability of Linear Infinite Inequality Systems under Block Perturbations with Applications to Convex Systems
The original motivation for this paper was to provide an efficient
quantitative analysis of convex infinite (or semi-infinite) inequality systems
whose decision variables run over general infinite-dimensional (resp.
finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed
set . Parameter perturbations on the right-hand side of the inequalities are
required to be merely bounded, and thus the natural parameter space is
. Our basic strategy consists of linearizing the parameterized
convex system via splitting convex inequalities into linear ones by using the
Fenchel-Legendre conjugate. This approach yields that arbitrary bounded
right-hand side perturbations of the convex system turn on constant-by-blocks
perturbations in the linearized system. Based on advanced variational analysis,
we derive a precise formula for computing the exact Lipschitzian bound of the
feasible solution map of block-perturbed linear systems, which involves only
the system's data, and then show that this exact bound agrees with the
coderivative norm of the aforementioned mapping. In this way we extend to the
convex setting the results of [3] developed for arbitrary perturbations with no
block structure in the linear framework under the boundedness assumption on the
system's coefficients. The latter boundedness assumption is removed in this
paper when the decision space is reflexive. The last section provides the aimed
application to the convex case
The radius of robust feasibility of uncertain mathematical programs: A Survey and recent developments
The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees feasibility of a robust counterpart of a mathematical program with uncertain constraints. The objective of this review of the state-of-the-art in this field is to present this useful tool of robust optimization to its potential users and to avoid undesirable overlapping of research works on the topic as those we have recently detected. In this paper we overview the existing literature on the radius of robust feasibility in continuous and mixed-integer linearly constrained programs, linearly constrained semi-infinite programs, convexly constrained programs, and conic linearly constrained programs. We also analyze the connection between the radius of robust feasibility and the distance to ill-posedness for different types of uncertain mathematical programs.This research was partially supported by the Australian Research Council, Discovery Project grant and the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22
Contracting Nonlinear Observers: Convex Optimization and Learning from Data
A new approach to design of nonlinear observers (state estimators) is
proposed. The main idea is to (i) construct a convex set of dynamical systems
which are contracting observers for a particular system, and (ii) optimize over
this set for one which minimizes a bound on state-estimation error on a
simulated noisy data set. We construct convex sets of continuous-time and
discrete-time observers, as well as contracting sampled-data observers for
continuous-time systems. Convex bounds for learning are constructed using
Lagrangian relaxation. The utility of the proposed methods are verified using
numerical simulation.Comment: conference submissio
Path-dependent equations and viscosity solutions in infinite dimension
Path-dependent PDEs (PPDEs) are natural objects to study when one deals with
non Markovian models. Recently, after the introduction of the so-called
pathwise (or functional or Dupire) calculus (see [15]), in the case of
finite-dimensional underlying space various papers have been devoted to
studying the well-posedness of such kind of equations, both from the point of
view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g.
[16]). In this paper, motivated by the study of models driven by path-dependent
stochastic PDEs, we give a first well-posedness result for viscosity solutions
of PPDEs when the underlying space is a separable Hilbert space. We also
observe that, in contrast with the finite-dimensional case, our well-posedness
result, even in the Markovian case, applies to equations which cannot be
treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
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