Fourier's Method of Linear Programming and Its Dual

Williams, H. Paul

doi:10.2307/2322281

Cited by 57 publications

(53 citation statements)

References 7 publications

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“…This is achieved via a Fourier-Motzkin (FM) elimination [31]. The final set of inequalities obtained via the FM elimination (and after eliminating over redundant inequalities) gives all the facets of the Shannon cone in the observable subspace.…”

Section: Step 3: Marginalizationmentioning

confidence: 99%

Device-Independent Tests of Entropy

2015

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We show that the entropy of a message can be tested in a device-independent way. Specifically, we consider a prepare-and-measure scenario with classical or quantum communication, and develop two different methods for placing lower bounds on the communication entropy, given observable data. The first method is based on the framework of causal inference networks. The second technique, based on convex optimization, shows that quantum communication provides an advantage over classical, in the sense of requiring a lower entropy to reproduce given data. These ideas may serve as a basis for novel applications in device-independent quantum information processing.The development of device-independent (DI) quantum information processing has attracted growing attention recently. The main idea behind this new paradigm is to achieve quantum information tasks, and guarantee their secure implementation, based on observed data alone. Thus no assumption about the internal working of the devices used in the protocol is in principle required. Notably, realistic protocols for DI quantum cryptography [1] and randomness generation [2,3] were presented, with proof-of-concept experiments for the second [3,4].The strong security of DI protocols finds its origin in a more fundamental aspect of physics, namely the fact that certain physical quantities admit a modelindependent description and can thus be certified in a DI way. The most striking example is Bell nonlocality [5,6], which can be certified (via Bell inequality violation) by observing strong correlations between the results of distant measurements. Notably, this is possible in quantum theory, by performing well-chosen local measurements on distant entangled particles. More recently, it was shown that the dimension of an uncharacterized physical system (loosely speaking, the number of relevant degrees of freedom) can also be tested in a DI way [7][8][9][10]. Conceptually, this allows us to study quantum theory inside a larger framework of physical theories , which already brought insight to quantum foundations [11][12][13][14]. From a more applied point of view, this allows for DI protocols and for black-box characterization of quantum systems [15][16][17][18][19][20].In this context, it is natural to ask whether there exist other physical quantities which admit a DI characterization. Here we show that this is the case by demonstrating that the entropy of a message can be tested in a DI way. Specifically, we present simple and efficient methods for placing lower bounds on the entropy of a classical (or quantum) communication based on observable data alone. We construct such "entropy witnesses" following two different approaches, first using the framework of causal inference networks [21], and second using convex optimization techniques. The first construction is very general, but usually gives suboptimal bounds. The second construction allows us to place tight bounds on the entropy of classical messages for given data. Moreover, it shows that quantum systems provide an advantage ...

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Section: Step 3: Marginalizationmentioning

confidence: 99%

Device-Independent Tests of Entropy

2015

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“…A polyhedron can be projected onto a subspace using Fourier-Motzkin elimination [2,6]. We will suppose the polyhedron is described by the constraint set of an LP in the following form, where A is an m × n integral matrix and b is integral:…”

Section: Lp Projectionmentioning

confidence: 99%

Integer programming as projection

Williams

Hooker

2016

Discrete Optimization

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We generalise polyhedral projection (Fourier-Motzkin elimination) to integer programming (IP) and derive from this an alternative perspective on IP that parallels the classical theory. We first observe that projection of an IP yields an IP augmented with linear congruence relations and finite-domain variables, which we term a generalised IP. The projection algorithm can be converted to a branch-and-bound algorithm for generalised IP in which the search tree has bounded depth (as opposed to conventional branching, in which there is no bound). It also leads to valid inequalities that are analogous to Chvátal-Gomory cuts but are derived from congruences rather than rounding, and whose rank is bounded by the number of variables. Finally, projection provides an alternative approach to IP duality. It yields a value function that consists of nested roundings as in the classical case, but in which ordinary rounding is replaced by rounding to the nearest multiple of an appropriate modulus, and the depth of nesting is again bounded by the number of variables. January 2014Abstract We generalise polyhedral projection (Fourier-Motzkin elimination) to integer programming (IP) and derive from this an alternative perspective on IP that parallels the classical theory. We first observe that projection of an IP yields an IP augmented with linear congruence relations and finite-domain variables, which we term a generalised IP. The projection algorithm can be converted to a branch-and-bound algorithm for generalised IP in which the search tree has bounded depth (as opposed to conventional branching, in which there is no bound). It also leads to valid inequalities that are analogous to Chvátal-Gomory cuts but are derived from congruences rather than rounding, and whose rank is bounded by the number of variables. Finally, projection provides an alternative approach to IP duality. It yields a value function that consists of nested roundings as in the classical case, but in which ordinary rounding is replaced by rounding to the nearest multiple of an appropriate modulus, and the depth of nesting is again bounded by the number of variables.

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“…Projection converts LPs into LPs, in a lower dimension, and offers a method of solving LPs (see Williams [31]) as well as deriving the dual. For IP this is not the case.…”

Section: Figure 5 a Lattice Within A Polytopementioning

confidence: 99%

The problem with integer programming

Williams

2010

IMA Journal of Management Mathematics

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Integer Programming (IP), also known as Discrete Optimisation, is a way of modelling a very wide range of problems involving indivisibilities (eg. Yes/No investment decisions) and non-convexities (eg. economies of scale and fixed cost allocation). Such problems arise in many areas which will be mentioned. However IP demands ingenuity in both building models and solving them. A lot is still not properly understood. This paper investigates the question. 'Is IP like Linear Programming (LP)?' The mathematical and economic properties of IP will be contrasted with LP. It will be suggested that the mathematics and economics of IP are still not properly understood. Many of the results which apply to LP do not apply to IP. It will be asserted that this lack of understanding reveals inadequacies in both the mathematics and economics.It will be shown that in many (but not all) situations the rounding of an LP solution does not produce satisfactory (feasible or optimal) solutions to an IP. A topical example will be given of political apportionment leading to the Alabama Paradox.IP is essentially concerned with the intersection of two structures: (i) Linear inequalities giving rise to polytopes.(ii) Lattices of integer points.Mathematical and computational methods and results exist for both these structures on their own. However mixing them is like mixing oil and water. Problems arise in both the computation of optimal solutions and the economic interpretation of the results. It will be suggested that the appropriate mathematical structure is an integer monoid. This structure will be explained. Connected with this structure are Chvátal functions which are made up of non-negative combinations of the arguments together with the (nested) integer round-up operation. The use of these functions, in place of the conventional non-negative linear combinations of LP allows one to capture many of the classical LP, results eg. The Weyl-Minkowski theorems, 'pricing' of indivisible resources and the closing of the 'duality gap' leading to the optimal IP solution. It will be shown how optimal Chvátal functions can be interpretated as the (non-marginal) valuation of indivisible resources. However a major problem remains as to how to represent them in a transparent and compact way.

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Fourier's Method of Linear Programming and Its Dual

Cited by 57 publications

References 7 publications

Device-Independent Tests of Entropy

Device-Independent Tests of Entropy

Integer programming as projection

The problem with integer programming

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