Convex Pair Research Articles

A Linearly Convergent Douglas-Rachford Splitting Solver for Markovian Information-Theoretic Optimization Problems

In this work, we propose solving the Information Bottleneck (IB) and Privacy Funnel (PF) problems with Douglas-Rachford Splitting methods (DRS). We study a general Markovian information-theoretic Lagrangian that includes IB and PF into a unified framework. We prove the linear convergence of the proposed solvers using the Kurdyka-Łojasiewicz inequality. Moreover, our analysis is beyond IB and PF and applies to any convex-weakly convex pair objectives. Based on the results, we develop two types of linearly convergent IB solvers, with one improves the performance of convergence over existing solvers while the other can be independent to the relevance-compression trade-off. Moreover, our results apply to PF, yielding a new class of linearly convergent PF solvers. Empirically, the proposed IB solvers IB obtain solutions that are comparable to the Blahut-Arimoto-based benchmark and is convergent for a wider range of the penalty coefficients than existing solvers. For PF, our non-greedy solvers can characterize the privacy-utility trade-off better than the clustering-based greedy solvers.

Noncyclic contractions and relatively nonexpansive mappings in strictly convex fuzzy metric spaces

<abstract><p>A concept of fuzzy projection operator is introduced and use to investigate the non-emptiness of the fuzzy proximal pairs. We then consider the classes of noncyclic contractions and noncyclic relatively nonexpansive mappings and survey the existence of best proximity pairs for such mappings. In the case that the considered mapping is noncyclic relatively nonexpansive, we need a geometric notion of fuzzy proximal normal structure defined on a nonempty and convex pair in a convex fuzzy metric space. We also prove that every nonempty, compact and convex pair of subsets of a strictly convex fuzzy metric space has the fuzzy proximal normal structure.</p></abstract>

Relatively Nonexpansive Mappings in k-Uniformly Convex Banach Spaces

Let A and B be two nonempty subsets of a normed linear space X. A mapping is said to be noncyclic if and In the current paper, we consider the problem of finding the best proximity pair for the noncyclic mapping T, that is, two fixed points of T which achieve the minimum distance between the sets A and B. We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping T. At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair (A, B) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial’s property of the Banach space and adding another assumption on the mapping T, called condition We also show that the same result is true when X is a 2-uniformly convex Banach space. In the setting of k-uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces.

Ground state criteria in one-dimensional antiferromagnetic Ising model with long range interaction

One-dimensional long-range Ising models with antiferromagnetic, convex pair interactions are investigated. A new criterion characterizing ground states is given. The criterion and a new transformation yield short proofs identifying and characterizing the ground states. The uniqueness of periodic ground states up to shifts is shown.

Strong and weak convergence of Ishikawa iterations for best proximity pairs

Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

Common best proximity pairs via the concept of complete proximal normal structure

We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces.

Approximate best proximity point sequences for C*λ mappings in strictly convex Banach spaces

Let A and B be nonempty subsets of a Banach space X and T : A → B be a non-self mapping. An approximate sequence of best proximity points for the mapping T is a sequence {xn } in A such that lim n →∞ || xn − T xn || → dist(A, B). In the current paper, we survey the existence of approximate best proximity point sequences for single and multivalued non-self mappings in strictly convex Banach spaces. We also introduce a geometric notion on a nonempty and convex pair of subsets of a Banach space, called semi-Opial condition, and establish some new best proximity point theorems.

Condensing Operators of Integral Type in Busemann Reflexive Convex Spaces

Let (A, B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space X, and \((E,F)\subseteq (A,B)\) be a nonempty and proximinal pair in X such that \(\mathrm{dist}(E,F)=\mathrm{dist}(A,B)\). We prove that the pair \((\overline{\mathrm{con}}(E),\overline{\mathrm{con}}(F))\) is also proximinal, where \(\overline{\mathrm{con}}(E)\) denotes the closed convex hull of the set E. Moreover, we introduce a new notion of cyclic (noncyclic) mappings involving measure of noncompactness and obtain some new existence results of best proximity points (pairs). As an application of our main conclusions, we study the existence of an optimal solution for a system of integrodifferential equations under new sufficient conditions.

Reliable Graphs for SLAM

Estimation-over-graphs (EoG) is a class of estimation problems that admit a natural graphical representation. Several key problems in robotics and sensor networks, including sensor network localization, synchronization over a group, and simultaneous localization and mapping (SLAM) fall into this category. We pursue two main goals in this work. First, we aim to characterize the impact of the graphical structure of SLAM and related problems on estimation reliability. We draw connections between several notions of graph connectivity and various properties of the underlying estimation problem. In particular, we establish results on the impact of the weighted number of spanning trees on the D-optimality criterion in 2D SLAM. These results enable agents to evaluate estimation reliability based only on the graphical representation of the EoG problem. We then use our findings and study the problem of designing sparse SLAM problems that lead to reliable maximum likelihood estimates through the synthesis of sparse graphs with the maximum weighted tree connectivity. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, we establish several new theoretical results, including the monotone log-submodularity of the weighted number of spanning trees. We exploit these structures and design a complementary greedy–convex pair of efficient approximation algorithms with provable guarantees. The proposed synthesis framework is applied to various forms of the measurement selection problem in resource-constrained SLAM. Our algorithms and theoretical findings are validated using random graphs, existing and new synthetic SLAM benchmarks, and publicly available real pose-graph SLAM datasets.

Homogeneity for Shared Control in the Presence of Disturbances

This paper considers the shared control of a second order plant by a single machine and one person in the presence of exogenous disturbances. A mixed-initiative blending law that takes a passive measure of human intent as the input to a convex pair of exponential functions is proposed. The blending law is combined with a disturbance observer and homogeneous trajectory-tracking control law, which dynamically rescales to ensure that actuator magnitude constraints are respected. The globally uniform ultimate boundedness of the closed loop automatic controller is proved. The shared control system is tested in a simulation where a human user can impose a control input using a joystick. Even if large tracking errors are commanded by the user, the automatic controller is able to quickly return to tracking the desired path when the joystick is released.

A characterization of proximal normal structure via proximal diametral sequences

We introduce a concept of pointwise cyclic relatively nonexpansive mapping involving orbits to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure. Examples are given to support our main conclusions. We also introduce a notion of proximal diametral sequence and establish a characterization of proximal normal structure and show that every nonempty and convex pair in uniformly convex in every direction Banach spaces has weak proximal normal structure. As an application, we give a new existence theorem for cyclic contractions in reflexive Banach spaces without strictly convexity condition.

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

ABSTRACTIn this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.

Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

LetAandBbe two nonempty subsets of a Banach spaceX. A mappingT:A∪B→A∪Bis said to be cyclic relatively nonexpansive ifT(A)⊆BandT(B)⊆AandTx-Ty≤x-yfor all (x,y)∈A×B. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach spaceX. It is shown that if (A,B) is a nonempty, weakly compact, and convex pair and (A,B) has seminormal structure, then a cyclic relatively nonexpansive mappingT:A∪B→A∪Bhas a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.Erratum to “Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings”

Semi-normal structure and best proximity pair results in convex metric spaces

A new geometric notion on a nonempty and convex pair of subsets of a convex metric space $X$, called semi-normal structure, is introduced and used to investigate the existence of best proximity pairs for a new class of mappings, called strongly noncyclic relatively C-nonexpansive. We also study the structure of minimal sets of strongly noncyclic relatively C-nonexpansive mappings in the setting of convex metric spaces.

Equilibrium existence under generalized convexity

Abstract We introduce, in the first part, the notion of weakly convex pair of correspondences, we give its economic interpretation, we state a fixed point and a selection theorem. Then, by using a tehnique based on a continuous selection, we prove existence theorems of quilibrium for an abstract economy. In the second part, we define the weakly biconvex correspondences, we prove a selection theorem and we also demonstrate the existence of equilibrium for a generalized quasi-game (2003 Kim's model). In the last part of the paper, we give other applications in the game theory, finding equilibrium for abstract economies having correspondences with weakly convex graph. We show that the equilibrium exists without continuity assumptions.

Unusual ground states via monotonic convex pair potentials

We have previously shown that inverse statistical-mechanical techniques allow the determination of optimized isotropic pair interactions that self-assemble into low-coordinated crystal configurations in the d-dimensional Euclidean space R(d). In some of these studies, pair interactions with multiple extrema were optimized. In the present work, we attempt to find pair potentials that might be easier to realize experimentally by requiring them to be monotonic and convex. Encoding information in monotonic convex potentials to yield low-coordinated ground-state configurations in Euclidean spaces is highly nontrivial. We adapt a linear programming method and apply it to optimize two repulsive monotonic convex pair potentials, whose classical ground states are counterintuitively the square and honeycomb crystals in R(2). We demonstrate that our optimized pair potentials belong to two wide classes of monotonic convex potentials whose ground states are also the square and honeycomb crystal. We show that these unexpected ground states are stable over a nonzero number density range by checking their (i) phonon spectra, (ii) defect energies and (iii) self assembly by numerically annealing liquid-state configurations to their zero-temperature ground states.

Contributing vertices-based Minkowski sum of a nonconvex--convex pair of polyhedra

The exact Minkowski sum of polyhedra is of particular interest in many applications, ranging from image analysis and processing to computer-aided design and robotics. Its computation and implementation is a difficult and complicated task when nonconvex polyhedra are involved. We present the NCC-CVMS algorithm, an exact and efficient contributing vertices-based Minkowski sum algorithm for the computation of the Minkowski sum of a nonconvex--convex pair of polyhedra, which handles nonmanifold situations and extracts eventual polyhedral holes inside the Minkowski sum outer boundary. Our algorithm does not output boundaries that degenerate into a polyline or a single point. First, we generate a superset of the Minkowski sum facets through the use of the contributing vertices concept and by summing only the features (facets, edges, and vertices) of the input polyhedra which have coincident orientations. Secondly, we compute the 2D arrangements induced by the superset triangles intersections. Finally, we obtain the Minkowski sum through the use of two simple properties of the input polyhedra and the Minkowski sum polyhedron itself, that is, the closeness and the two-manifoldness properties. The NCC-CVMS algorithm is efficient because of the simplifications induced by the use of the contributing vertices concept, the use of 2D arrangements instead of 3D arrangements which are difficult to maintain, and the use of simple properties to recover the Minkowski sum mesh. We implemented our NCC-CVMS algorithm on the base of CGAL and used exact number types. More examples and results of the NCC-CVMS algorithm can be found at: http://liris.cnrs.fr/hichem.barki/mksum/NCC-CVMS

Optimized monotonic convex pair potentials stabilize low-coordinated crystals

We have previously used inverse statistical-mechanical methods to optimize isotropic pair interactions with multiple extrema to yield low-coordinated crystal classical ground states (e.g., honeycomb and diamond structures) in d-dimensional Euclidean space d. Here we demonstrate the counterintuitive result that no extrema are required to produce such low-coordinated classical ground states. Specifically, we show that monotonic convex pair potentials can be optimized to yield classical ground states that are the square and honeycomb crystals in 2 over a non-zero number density range. Such interactions may be feasible to achieve experimentally using colloids and polymers.

The ratio field of values

The ratio field of values, a generalization of the classical field of values to a pair of n-by- n matrices, is defined and studied, primarily from a geometric point of view. Basic functional properties of the ratio field are developed and used. A decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation. Primary theoretical results include the following. It is shown (1) for which denominator matrices the ratio field is always convex, (2) certain other cases of convex pairs are given, and (3) that, at least for n = 2 , the ratio field obeys a near convexity property that the intersection with any line segment has at most n components. Generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated. It is shown that generally such extensions need not be convex or even simply connected.

909 内接ローラ対のトラクション特性に及ぼす潤滑油の影響 : トラクションオイルTD10, TD22の場合(模索潤滑設計II)

909 内接ローラ対のトラクション特性に及ぼす潤滑油の影響 : トラクションオイルTD10, TD22の場合(模索潤滑設計II)