On Constraint Dropping Schemes and Optimality Functions for a Class of Outer Approximations Algorithms

1979 ◽  
Vol 17 (4) ◽  
pp. 477-493 ◽  
Author(s):  
C. Gonzaga ◽  
E. Polak
Keyword(s):  
Optimality Functions ◽  
Outer Approximations

scholarly journals Tight Two-Dimensional Outer-Approximations of Feasible Sets in Wireless Sensor Networks

2016 ◽  
Vol 20 (3) ◽  
pp. 570-573 ◽  
Author(s):  
Siamak Yousefi ◽  
Henk Wymeersch ◽  
Xiao-Wen Chang ◽  
Benoit Champagne
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Wireless Sensor ◽  
Two Dimensional ◽  
Outer Approximations ◽  
Feasible Sets

scholarly journals Bounding sets of sequential quantum correlations and device-independent randomness certification

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 344 ◽  
Author(s):  
Joseph Bowles ◽  
Flavio Baccari ◽  
Alexia Salavrakos
Keyword(s):  
Quantum Information ◽  
Arbitrary Dimension ◽  
Quantum Correlations ◽  
Entangled State ◽  
Outer Approximations ◽  
Resource Requirements ◽  
Local Measurements ◽  
Local Randomness ◽  
Test Scenarios ◽  
Sequential Measurements

An important problem in quantum information theory is that of bounding sets of correlations that arise from making local measurements on entangled states of arbitrary dimension. Currently, the best-known method to tackle this problem is the NPA hierarchy; an infinite sequence of semidefinite programs that provides increasingly tighter outer approximations to the desired set of correlations. In this work we consider a more general scenario in which one performs sequences of local measurements on an entangled state of arbitrary dimension. We show that a simple adaptation of the original NPA hierarchy provides an analogous hierarchy for this scenario, with comparable resource requirements and convergence properties. We then use the method to tackle some problems in device-independent quantum information. First, we show how one can robustly certify over 2.3 bits of device-independent local randomness from a two-quibt state using a sequence of measurements, going beyond the theoretical maximum of two bits that can be achieved with non-sequential measurements. Finally, we show tight upper bounds to two previously defined tasks in sequential Bell test scenarios.

Efficient Singularity-Free Workspace Approximations Using Sum-of-Squares Programming

2020 ◽  
Vol 12 (6) ◽  
Author(s):  
Wankun Sirichotiyakul ◽  
Volkan Patoglu ◽  
Aykut C. Satici
Keyword(s):  
Quadratic Forms ◽  
Optimization Technique ◽  
Robotic Manipulators ◽  
Additional Constraint ◽  
Sum Of Squares ◽  
Type I ◽  
Polynomial Functions ◽  
Type Ii ◽  
Trigonometric Functions ◽  
Outer Approximations

Abstract In this paper, we provide a general framework to determine inner and outer approximations to the singularity-free workspace of fully actuated robotic manipulators, subject to Type-I and Type-II singularities. This framework utilizes the sum-of-squares optimization technique, which is numerically implemented by semidefinite programming. In order to apply the sum-of-squares optimization technique, we convert the trigonometric functions in the kinematics of the manipulator to polynomial functions with an additional constraint. We define two quadratic forms, describing two ellipsoids, whose volumes are optimized to yield inner and outer approximations of the singularity-free workspace.

INNER AND OUTER APPROXIMATION OF BELIEF STRUCTURES USING A HIERARCHICAL CLUSTERING APPROACH

2001 ◽  
Vol 09 (04) ◽  
pp. 437-460 ◽  
Author(s):  
THIERRY DENŒUX
Keyword(s):  
Hierarchical Clustering ◽  
Numerical Experiments ◽  
Belief Function ◽  
Outer Approximation ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Belief Structures ◽  
Outer Approximations ◽  
Clustering Approach

A hierarchical clustering approach is proposed for reducing the number of focal elements in a crisp or fuzzy belief function, yielding strong inner and outer approximations. At each step of the proposed algorithm, two focal elements are merged, and the mass is transfered to their intersection or their union. The resulting approximations allow the calculation of lower and upper bounds on the belief and plausibility degrees induced by the conjunctive or disjunctive sum of any number of belief structures. Numerical experiments demonstrate the effectiveness of this approach.

On tangential approximations of the solution set of set-valued inclusions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amos Uderzo
Keyword(s):  
Crucial Role ◽  
Variational Analysis ◽  
Necessary Optimality Conditions ◽  
Solution Set ◽  
Inclusion Problem ◽  
Feasible Region ◽  
Contingent Cone ◽  
Constrained Problems ◽  
Outer Approximations ◽  
Set Valued Mappings

Abstract In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.

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