Set-Membership Error-in-Variables Identification Through Convex Relaxation Techniques

2012 ◽  
Vol 57 (2) ◽  
pp. 517-522 ◽  
Author(s):  
V. Cerone ◽  
D. Piga ◽  
D. Regruto
Keyword(s):  
Convex Relaxation ◽  
Relaxation Techniques ◽  
Error In Variables ◽  
Set Membership

scholarly journals The Non-Tightness of a Convex Relaxation to Rotation Recovery

Sensors ◽  
2021 ◽  
Vol 21 (21) ◽  
pp. 7358
Author(s):  
Yuval Alfassi ◽  
Daniel Keren ◽  
Bruce Reznick
Keyword(s):  
Algebraic Geometry ◽  
Polynomial Optimization ◽  
Convex Relaxation ◽  
Real Life ◽  
Probability Of Failure ◽  
Relaxation Techniques ◽  
3D Vision ◽  
Theoretical Contribution ◽  
Common Solution ◽  
Lasserre Hierarchy

We study the Perspective-n-Point (PNP) problem, which is fundamental in 3D vision, for the recovery of camera translation and rotation. A common solution applies polynomial sum-of-squares (SOS) relaxation techniques via semidefinite programming. Our main result is that the polynomials which should be optimized can be non-negative but not SOS, hence the resulting convex relaxation is not tight; specifically, we present an example of real-life configurations for which the convex relaxation in the Lasserre Hierarchy fails, in both the second and third levels. In addition to the theoretical contribution, the conclusion for practitioners is that this commonly-used approach can fail; our experiments suggest that using higher levels of the Lasserre Hierarchy reduces the probability of failure. The methods we use are mostly drawn from the area of polynomial optimization and convex relaxation; we also use some results from real algebraic geometry, as well as Matlab optimization packages for PNP.

Convex underestimating relaxation techniques for nonconvex polynomial programming problems: computational overview

2015 ◽  
Vol 24 (3-4) ◽  
pp. 129-143
Author(s):  
André A. Keller
Keyword(s):  
System Analysis ◽  
Convex Set ◽  
Optimization Problems ◽  
Convex Relaxation ◽  
Engineering Application ◽  
Global Optimum ◽  
Linear Matrix ◽  
Quadratic Term ◽  
Relaxation Techniques ◽  
Convex Envelopes

AbstractThis paper introduces constructing convex-relaxed programs for nonconvex optimization problems. Branch-and-bound algorithms are convex-relaxation-based techniques. The convex envelopes are important, as they represent the uniformly best convex underestimators for nonconvex polynomials over some region. The reformulation-linearization technique (RLT) generates linear programming (LP) relaxations of a quadratic problem. RLT operates in two steps: a reformulation step and a linearization (or convexification) step. In the reformulation phase, the constraint and bound inequalities are replaced by new numerous pairwise products of the constraints. In the linearization phase, each distinct quadratic term is replaced by a single new RLT variable. This RLT process produces an LP relaxation. The LP-RLT yieds a lower bound on the global minimum. LMI formulations (linear matrix inequalities) have been proposed to treat efficiently with nonconvex sets. An LMI is equivalent to a system of polynomial inequalities. A semialgebraic convex set describes the system. The feasible sets are spectrahedra with curved faces, contrary to the LP case with polyhedra. Successive LMI relaxations of increasing size yield the global optimum. Nonlinear inequalities are converted to an LMI form using Schur complements. Optimizing a nonconvex polynomial is equivalent to the LP over a convex set. Engineering application interests include system analysis, control theory, combinatorial optimization, statistics, and structural design optimization.

scholarly journals A convex relaxation approach to set-membership identification of LPV systems

Automatica ◽  
2013 ◽  
Vol 49 (9) ◽  
pp. 2853-2859 ◽  
Author(s):  
Vito Cerone ◽  
Dario Piga ◽  
Diego Regruto
Keyword(s):  
Convex Relaxation ◽  
Lpv Systems ◽  
Set Membership
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