Nonnegative Linear Function Research Articles

Happiness maximizing sets under group fairness constraints

Finding a happiness maximizing set (HMS) from a database, i.e., selecting a small subset of tuples that preserves the best score with respect to any nonnegative linear utility function, is an important problem in multi-criteria decision-making. When an HMS is extracted from a set of individuals to assist data-driven algorithmic decisions such as hiring and admission, it is crucial to ensure that the HMS can fairly represent different groups of candidates without bias and discrimination. However, although the HMS problem was extensively studied in the database community, existing algorithms do not take group fairness into account and may provide solutions that under-represent some groups. In this paper, we propose and investigate a fair variant of HMS (FairHMS) that not only maximizes the minimum happiness ratio but also guarantees that the number of tuples chosen from each group falls within predefined lower and upper bounds. Similar to the vanilla HMS problem, we show that FairHMS is NP-hard in three and higher dimensions. Therefore, we first propose an exact interval cover-based algorithm called IntCov for FairHMS on two-dimensional databases. Then, we propose a bicriteria approximation algorithm called BiGreedy for FairHMS on multi-dimensional databases by transforming it into a submodular maximization problem under a matroid constraint. We also design an adaptive sampling strategy to improve the practical efficiency of BiGreedy. Extensive experiments on real-world and synthetic datasets confirm the efficacy and efficiency of our proposal.

Approximation Schemes for Two Non-Linear Knapsack Problems and Their Applications in Alternating Current Electric Systems

The purpose of this paper is to study the approximability of two non-linear Knapsack problems, which are motivated by important applications in alternating current electrical systems. The first problem is to maximize a nonnegative linear objective function subject to a quadratic constraint, whilst the second problem is a dual version of the first one, where an objective function is minimized. Both problems are $\np$-hard since they generalize the unbounded Knapsack problem, and it is unlikely to obtain polynomial-time algorithms for them, unless $\p=\np$. It is therefore of great interest to know whether or not there exist efficient approximation algorithms which can return an approximate solution in polynomial time with a reasonable approximation factor. Our contribution of this paper is to present polynomial-time approximation schemes (PTASs) and this is the best possible result one can hope for the studied problems. Our technique is based on the linear-programming approach which seems to be more simple and efficient than the previous one.

Cut Dominants and Forbidden Minors

The cut dominant of a graph is the unbounded polyhedron whose points are all those that dominate some convex combination of proper cuts. Minimizing a nonnegative linear function over the cut dominant is equivalent to finding a minimum weight cut in the graph. We give a forbidden-minor characterization of the graphs whose cut dominant can be defined by inequalities with integer coefficients and right-hand side at most 2. Our result is related to the forbidden-minor characterization of TSP-perfect graphs by Fonlupt and Naddef [Math. Program, 53 (1992), pp. 147--172]. We show how to derive each of the results from the other. Furthermore, we establish general properties of forbidden minors for right-hand sides larger than 2.

Expansion over a system of shifts of pyramidal functions with a square base

A method for approximation of functions of two variables by a linear combination of non-negative piecewise linear functions with a compact support is presented. Two quadratic pyramids are used as generating functions for the system of shifts. The accuracy of this local method is proved to have the same order as the best approximation by piecewise linear functions.

Approximating a class of combinatorial problems with rational objective function

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0 + a 1 x 1 + . . . + a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form (a 0 + a 1 x 1 + . . . + a n x n )/(b 0 + b 1 x 1 + . . . + b n x n ) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo’s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo’s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an α-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an α-approximation (1/α-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.

Expansion over a system of shifts of pyramidal functions

A method for approximation of functions of two variables by a linear combination of nonnegative piecewise linear functions with a compact support is presented. The crucial idea of this method consists in an “integral” calculation method for the coefficients. The accuracy of the approximation in the spaces of continuous and integrable functions is proved to have the same order as the best approximation by piecewise linear functions.

An FPTAS for minimizing the product of two non-negative linear cost functions

We consider a quadratic programming (QP) problem (Π) of the form min x T C x subject to Ax ≥ b, x ≥ 0 where $${C\in {\mathbb R}^{n \times n}_+, {\rm rank}(C)=1}$$ and $${A\in {\mathbb R}^{m \times n}, b\in {\mathbb R}^m}$$ . We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP (Π) as a parameterized LP and “rounding” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0–1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and sub-modular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, t-shortest paths and s, t-min-cuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions.

Dual-bounded generating problems: Efficient and inefficient points for discrete probability distributions and sparse boxes for multidimensional data

We show that ∣ X ∣ ≤ n ∣ Y ∣ must hold for two finite sets X , Y ⊂ R n whenever they can be separated by a nonnegative linear function such that X is above Y and the componentwise minimum of any two distinct points in X is dominated by some point in Y . As a consequence, we obtain an incremental quasi-polynomial time algorithm for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p -inefficient points of a given discrete probability distribution, and for generating all maximal hyper-rectangles which contain a specified fraction of points of a given set in R n . This provides a substantial improvement over previously known exponential time algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, implying that for discrete probability distributions with independent coordinates, both p -efficient and p -inefficient points can be separately generated in incremental polynomial time.

Solving Multicommodity Flow Problems by an Approximation Scheme

We study an approximation scheme to solve minimum cost multicommodity flow problems to a relative accuracy of $\varepsilon\in (0,1]$. The proposed scheme, which we shall call Algorithm ${\cal A}$, is a bisection-based procedure, so it maintains an interval that contains the optimal value. At each iteration, Algorithm ${\cal A}$ defines a block angular sharing problem that is solved to a relative accuracy of $O(\varepsilon)$. The computed solution defines the current approximation to an optimal solution and it is used by Algorithm ${\cal A}$ to throw away half of the current interval. It is shown that when Algorithm ${\cal A}$ no longer shrinks the current interval, the current solution solves the minimum cost multicommodity flow problem to the given accuracy $\varepsilon$. To compute approximate solutions to the sharing problems that Algorithm ${\cal A}$ defines, we propose using Algorithm ${\cal L}$ from [J. Villavicencio and M. D. Grigoriadis, Approximate Lagrangian decomposition with a modified Karmarkar logarithmic potential, in Network Optimization, Lecture Notes in Econom. and Math. Systems 450, Springer, Berlin, 1997, pp. 471--485], but we replace its fixed small step size by one computed using inexact line searches. This modification allows large steps in Algorithm ${\cal L}$. We show that the coordination complexities of Algorithm ${\cal L}$ and Algorithm ${\cal L}$ with inexact line searches are the same if these algorithms are applied to the block angular sharing problems defined by Algorithm ${\cal A}$. This result is also true for block angular sharing problems with convex blocks and with coupling constraints given by nonnegative linear functions.

Nonlinear potentials in function spaces

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

Daniell-Loomis Integrals

In [2] and [3] for arbitrary nonnegative linear functionals on functions vector lattices an integral extension of Lebesgue power has been discussed. Here we generalize this extension process, prove convergence theorems using a suitable “local convergence in measure,” discuss measurability and give characterizations by equality of upper and lower integrals. Riemann-μ, abstract Riemann-Loomis and Bourbaki integrals are subsumed. 0. Introduction. For a semi-ring Ω of sets from an arbitrary set X and μ : Ω → [0,∞[, only finitely additive, an analogue R1(μ,R) to the space L(μ,R) of Lebesgue-μ-integrable functions was introduced by Loomis [11]; this has been extended to Banach space-valued functions by Dunford-Schwartz [4], and in more general form in [6, 7]. Analogues to the Daniell extension process, but without or with weaker continuity assumptions on the elementary integral, have been treated by Aumann [1], Loomis [11] and Gould [5]. The Daniell-Bourbaki integral extension has been generalized with the integral I : B → R introduced in [2], starting with any nonnegative linear functional I on a vector lattice B of real-valued functions on X. If Ω is a δ-ring, μ σ-additive, I = ∫ .dμ on B = step functions over Ω, then R1, L and B coincide modulo null functions [3, 9]. In Sections 2 and 3 we generalize the extension I|B → I|B to I|B → J |L by “localization,” using an appropriate local convergence in measure, which is very useful to obtain convergence theorems in a form analogous to the classical ones (some of which are not true for B). In Section 4 we give various descriptions of the set L of integrable functions, in particular a Darboux-type characterization on L is proved. Always R1 ⊂ L (not true for B), in general B has infinite codimension in L, even modulo null functions. We recall that the abstract space of integrable functions L is constructed similar to the Daniell L and which coincides with L in the Received by the editors on April 21, 1996. AMS Mathematics Subject Classification. 28C05, 26A42. Copyright c ©1997 Rocky Mountain Mathematics Consortium

Local integral metrics and Daniell-Loomis integrals

Recently Guerrero and the first author (Diaz Carrillo) proved an anologue to Daniell's extension process which works for arbitrary nonnegative linear functionals, without any continuity conditions. With the aid of Schäfke's local integral metrics we generalise this extension process and prove convergence theorems using a suitable local mean convergence, which can be traced back to Loomis.

A Generalization of Volterra Models with Continuous Time Delay in Population Dynamics: Boundedness and Global Asymptotic stability

We introduce a generalization of Volterra models with continuous time delay by adding a nonnegative linear vector function of the species. Sufficient conditions for boundedness of solutions and global asymptotic stability of an equilibrium are provided. Applications from population dynamics are presented. One concerns a predator-prey Volterra model with prey-refuges and continuous time delay, and another deals with a Volterra system with currents of immigration for some species. Furthermore, for this kind of model, by a suitable homotopy function, we provide a sufficient condition for the existence of the globally asymptotically stable equilibrium. Finally an example of a simple model in which two predators are competing for one prey which can take shelter is presented.

Dominated extension of nonnegative linear functionals

Dominated extension of nonnegative linear functionals

A Short Proof of an Existence Theorem in Constructive Measure Theory

The most important example of an integration space in the constructive measure theory of Bishop and Cheng is the couple $(X,\mu )$, where $X$ is a locally compact metric space and $\mu$ is a nonnegative linear function on the space of continuous functions of compact support on $X$. Bishop and Cheng’s proof that $(X,\mu )$ is indeed an integration space is rather involved. In this paper a much simpler proof is given.

A short proof of an existence theorem in constructive measure theory

The most important example of an integration space in the constructive measure theory of Bishop and Cheng is the couple $(X,\mu )$, where $X$ is a locally compact metric space and $\mu$ is a nonnegative linear function on the space of continuous functions of compact support on $X$. Bishop and Cheng’s proof that $(X,\mu )$ is indeed an integration space is rather involved. In this paper a much simpler proof is given.

Uniform approximation of nonnegative continuous linear functionals

Uniform approximation of nonnegative continuous linear functionals

Endomorphisms of Partially Ordered Vector Spaces

A vector space V over the real field R is said to be partially ordered if a non-empty subset F + is specified which satisfies the following axioms: (i) if x and y are in V and oc 0, then x + y and VLX are in V) (ii) if x and — x are in T then x = 0. We write, as usual, x ^ y (or y 0 is an or^er MWI if for each x e V there exists f e R with fs ^ #; for such an e we define the functional pe by p6(x) = inf [f e R : fs ^ #]. A linear transformation T of V into itself is called an endomorphism if 7# ^ 0 whenever # ^ 0. A positive linear functional is a non-zero linear functional cp such that 99 (#) ^ 0 whenever x ^ 0. We prove the following theorem. Let V be a partially ordered vector space with an order unit e and let A be an endomorphism of V. Then there exists a positive linear functional (p with i_ A*cp = Qcp [i.e. with cp(Ax) = £9?(#) for all # e F], where £ = lim {pe(A e)}n. n->oo A theorem of this type, but less general and without the determination of Q, has been proved by M. G. Krein using the fixed point theorem of J. Schauder. Our proof does not use any fixed point theorem but depends on some elementary results in the theory of Banach algebras. Many related results have been proved by M. G. Krein and M. A. Rutman [Uspehi Matem. Nauk (N.S.) 3, no 1 (23), 3—95 (1948)]. Their principal results on linear operators leaving invariant a cone with interior may be deduced from our theorem with some gain in simplicity and generality. For example, if V is a reflexive Banach space and F + is a closed cone K such that K and K* (the set of non-negative linear functionals) have interior points, and if A is an endomorphism with spectral radius Q, then there are non-zero vectors ueK, cpeK* with Au = QU and A*cp = QCp.

A generalized notion of boundary

This paper reports some recent results in the theory of generalized parametric surfaces. The object of this theory, introduced by Young in [18], is to generalize the elementary notions of parametric surface in such a way that a general principle of minimum holds for surface integral variational problems in parametric form. Since nonregular problems are not excluded, the notion of generalized parametric surface is of necessity more general than the usual concepts of surface, involving in many instances both a track (surface in the ordinary sense, possibly mildly discontinuous) and a linear average over a space of Jacobian matrices at each point of the track (for ordinary surfaces the average is the trivial one in which the normal has weight one and all others weight zero) (2). In this paper, we first introduce abstract notions of boundary and track, defined for all non-negative linear functionals L on a certain linear space. Since, in particular, every generalized parametric surface is such a linear functional, these definitions apply to all generalized parametric surfaces. We next specialize to admit only boundaries which correspond in a certain way to boundary curves in the sense of classical analysis, and characterize the set of all L with boundary in this class. Finally, we specialize further by imposing a condition of indecomposibility with respect to the boundary. As an application, some conditions are established for equivalence of indefinite variation problems which have fixed boundary conditions with positive definite problems. 1. Introduction. The main definitions and results of the paper areoutlined in this section. Details and auxiliary results are left to later sections. Let Em (mr> 3) denote the space of all continuous functions f(x, J) of an m-dimensional vector x= [xi, . . ., xm] and an mXm skew-symmetric matrix J = [jr8] of rank 2 which satisfy the homogeneity condition f(x, kJ) = kf(x, J) for k _ O. A mapping x(u, v) from the square U: [O < u < 1; 0 < v <1] to m-space Rm is termed a Dirichlet (parametric) representation if: (1) x(u, v) is absolutely continuous in Tonelli's sense; and

Equivalence of a problem in measure theory to a problem in the theory of Vector Lattices

In connection with some other work of the author a question arises concerning the form of the general bounded linear functional on certain vector lattices. It is the purpose of this note to show that this question is completely equivalent to a question in measure theory which has been discussed and partially answered by Ulam [2]. Let 5 be an abstract set and let % be the vector lattice of all realvalued functions defined on S. For each So in S the function F on gf such that F(f) =/(so) for all ƒ in g? is clearly a linear functional. We shall call it the point functional belonging to s0 or simply a point functional. Obviously every point functional and hence every finite linear combination of point functionals is bounded in the sense that it carries every bounded subset of g into a bounded set of real numbers. Our question is as to whether every bounded linear functional on § is a finite linear combination of point functionals. We shall show that this is the case if and only if there exists no countably additive measure a which is defined for all subsets of 5, which is zero at points, which takes on only the values zero and one, and which does not vanish identically. It is well known that every bounded linear functional on a vector lattice is a difference of non-negative linear functionals and it is obvious that a non-negative linear functional is bounded. It follows that we need only consider non-negative linear functionals. Passages from a measure to a non-negative linear functional defined on a class of functions and vice versa are of frequent occurrence in mathematical literature. The proof of our theorem rests basically on the fact that when the methods used in effecting these passages are applied to the case at hand one obtains a natural one-to-one correspondence between the non-negative linear functionals on g and the countably ad-