scholarly journals Global convergence of the alternating projection method for the Max-Cut relaxation problem

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 737-746
Author(s):  
Suliman Al-Homidana
Keyword(s):  
Convex Sets ◽  
Alternating Projections ◽  
Correlation Matrices ◽  
Von Neumann ◽  
Relaxation Problem ◽  
Max Cut Problem ◽  
Np Hard Problem ◽  
Convergent Method ◽  
Alternating Projection Method ◽  
Alternating Projections Method

The Max-Cut problem is an NP-hard problem [15]. Extensions of von Neumann?s alternating projections method permit the computation of proximity projections onto convex sets. The present paper exploits this fact by constructing a globally convergent method for the Max-Cut relaxation problem. The feasible set of this relaxed Max-Cut problem is the set of correlation matrices.

scholarly journals A variational approach to the alternating projections method

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Convex Sets ◽  
Nonempty Intersection ◽  
Feasibility Problem ◽  
Alternating Projections ◽  
Limit Sets ◽  
Von Neumann ◽  
Convex Feasibility ◽  
Starting Point ◽  
Method Of Alternating Projections ◽  
Alternating Projections Method

AbstractThe 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

Alternating projections and interpolation of stationary processes

1992 ◽  
Vol 29 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Mohsen Pourahmadi
Keyword(s):  
Missing Values ◽  
Stationary Process ◽  
Dimensional Space ◽  
Stationary Processes ◽  
Finite Dimensional Space ◽  
Alternating Projections ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Explicit Formulae ◽  
Linear Interpolator

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

scholarly journals Are Stable Instances Easy?

2012 ◽  
Vol 21 (5) ◽  
pp. 643-660 ◽  
Author(s):  
YONATAN BILU ◽  
NATHAN LINIAL
Keyword(s):  
Polynomial Time ◽  
Discrete Optimization ◽  
Optimization Problem ◽  
Discrete Optimization Problem ◽  
Hard Problem ◽  
Np Hard ◽  
Max Cut Problem ◽  
Np Hard Problem ◽  
Hard Problems ◽  
Np Hard Problems

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

scholarly journals Projective algorithms for solving complementarity problems

2002 ◽  
Vol 29 (2) ◽  
pp. 99-113
Author(s):  
Caroline N. Haddad ◽  
George J. Habetler
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Convex Sets ◽  
Complementarity Problems ◽  
Linear Complementarity ◽  
Von Neumann ◽  
Nonconvex Sets ◽  
Neumann Type ◽  
Successive Over Relaxation ◽  
Projective Algorithms

We present robust projective algorithms of the von Neumann type for the linear complementarity problem and for the generalized linear complementarity problem. The methods, an extension of Projections Onto Convex Sets (POCS) are applied to a class of problems consisting of finding the intersection of closed nonconvex sets. We give conditions under which convergence occurs (always in2dimensions, and in practice, in higher dimensions) when the matrices areP-matrices (though not necessarily symmetric or positive definite). We provide numerical results with comparisons to Projective Successive Over Relaxation (PSOR).

Signal and Image Synthesis: Alternating Projections Onto Convex Sets

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Speech Processing ◽  
Video Processing ◽  
Convex Sets ◽  
Image Synthesis ◽  
Three Dimensions ◽  
Signal Recovery ◽  
Alternating Projections ◽  
Cellular Radio ◽  
Complex Signals ◽  
Bandlimited Signals

Alternating projections onto convex sets (POCS) [319, 918, 1324, 1333] is a powerful tool for signal and image restoration and synthesis. The desirable properties of a reconstructed signal may be defined by a convex set of constraint parameters. Iteratively projecting onto these convex constraint sets can result in a signal which contains all desired properties. Convex signal sets are frequently encountered in practice and include the sets of bandlimited signals, duration limited signals, causal signals, signals that are the same (e.g., zero) on some given interval, bounded signals, signals of a given area and complex signals with a specified phase. POCS was initially introduced by Bregman [156] and Gubin et al. [558] and was later popularized by Youla & Webb [1550] and Sezan & Stark [1253]. POCS has been applied to such topics as acoustics [300, 1381], beamforming [426], bioinformatics [484], cellular radio control [1148], communications systems [29, 769, 1433], deconvolution and extrapolation [718, 907, 1216], diffraction [421], geophysics [4], image compression [1091, 1473], image processing [311, 321, 470, 471, 672, 736, 834, 1065, 1069, 1093, 1473, 1535, 1547, 1596], holography [880, 1381], interpolation [358, 559, 1266], neural networks [1254, 1543, 909, 913, 1039], pattern recognition [1444, 1588], optimization [598, 1359, 1435], radiotherapy [298, 814, 1385], remote sensing [1223], robotics [740], sampling theory [399, 1334, 1542], signal recovery [320, 737, 1104, 1428, 1594], speech processing [1450], superresolution [399, 633, 654, 834, 1393, 1521], television [736, 786], time-frequency analysis [1037, 1043], tomography [1103, 713, 1212, 1213, 1275, 916, 1322, 1060, 1040], video processing [560, 786, 1092], and watermarking [19, 1470]. Although signal processing applications ofPOCS use sets of signals,POCSis best visualized viewing the operations on sets of points. In this section, POCS is introduced geometrically in two and three dimensions. Such visualization of POCS is invaluable in application of the theory.

scholarly journals On central traces and groups of symmetries of order unit Banach spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 159-166
Author(s):  
Cho-Ho Chu
Keyword(s):  
Convex Sets ◽  
Von Neumann Algebras ◽  
The Other ◽  
Order Unit ◽  
Approximation Process ◽  
Von Neumann ◽  
Easy Consequence ◽  
Finite Dimensional ◽  
Module Homomorphism ◽  
Group Of Symmetries

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).

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