Showing papers on "Concave function published in 2015"

Mixed f-divergence and inequalities for log-concave functions

Abstract: Mixed $f$-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log concave functions. Special cases of $f$-divergences are mixed $L_\lambda$-affine surface areas for log concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santalo type inequality.

An efficient algorithm for contextual bandits with knapsacks, and an extension to concave objectives

Abstract: We consider a contextual version of multi-armed bandit problem with global knapsack constraints. In each round, the outcome of pulling an arm is a scalar reward and a resource consumption vector, both dependent on the context, and the global knapsack constraints require the total consumption for each resource to be below some pre-fixed budget. The learning agent competes with an arbitrary set of context-dependent policies. This problem was introduced by Badanidiyuru et al. (2014), who gave a computationally inefficient algorithm with near-optimal regret bounds for it. We give a computationally efficient algorithm for this problem with slightly better regret bounds, by generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. The computational time of our algorithm scales logarithmically in the size of the policy space. This answers the main open question of Badanidiyuru et al. (2014). We also extend our results to a variant where there are no knapsack constraints but the objective is an arbitrary Lipschitz concave function of the sum of outcome vectors.

Polyhedral aspects of Submodularity, Convexity and Concavity.

Abstract: Seminal work by Edmonds and Lovasz shows the strong connection between submodularity and convexity. Submodular functions have tight modular lower bounds, and subdifferentials in a manner akin to convex functions. They also admit poly-time algorithms for minimization and satisfy the Fenchel duality theorem and the Discrete Seperation Theorem, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular maximization, though NP hard, admits constant factor approximation guarantees. Concave functions composed with modular functions are submodular, and they also satisfy diminishing returns property. This manuscript provides a more complete picture on the relationship between submodularity with convexity and concavity, by extending many of the results connecting submodularity with convexity to the concave aspects of submodularity. We first show the existence of superdifferentials, and efficiently computable tight modular upper bounds of a submodular function. While we show that it is hard to characterize this polyhedron, we obtain inner and outer bounds on the superdifferential along with certain specific and useful supergradients. We then investigate forms of concave extensions of submodular functions and show interesting relationships to submodular maximization. We next show connections between optimality conditions over the superdifferentials and submodular maximization, and show how forms of approximate optimality conditions translate into approximation factors for maximization. We end this paper by studying versions of the discrete seperation theorem and the Fenchel duality theorem when seen from the concave point of view. In every case, we relate our results to the existing results from the convex point of view, thereby improving the analysis of the relationship between submodularity, convexity, and concavity.

Information-theoretic lower bounds for convex optimization with erroneous oracles

Abstract: We consider the problem of optimizing convex and concave functions with access to an erroneous zeroth-order oracle. In particular, for a given function x → f (x) we consider optimization when one is given access to absolute error oracles that return values in [f (x) - ∊, f (x) + ∊] or relative error oracles that return value in [(1 - ∊)f (x), (1 + ∊)f (x)], for some ∊ > 0. We show stark information theoretic impossibility results for minimizing convex functions and maximizing concave functions over polytopes in this model.

Contextual Bandits with Global Constraints and Objective.

Abstract: We consider the contextual version of a multi-armed bandit problem with global convex constraints and concave objective function. In each round, the outcome of pulling an arm is a context-dependent vector, and the global constraints require the average of these vectors to lie in a certain convex set. The objective is a concave function of this average vector. The learning agent competes with an arbitrary set of context-dependent policies. This problem is a common generalization of problems considered by Badanidiyuru et al. (2014) and Agrawal and Devanur (2014), with important applications. We give computationally efficient algorithms with near-optimal regret, generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. For the special case of budget constraints our regret bounds match those of Badanidiyuru et al. (2014), answering their main open question of obtaining a computationally efficient algorithm.

Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions

Abstract: In this note, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete-Szegő problem for normalized concave functions with a prescribed pole in the unit disk.

Stability and competitive equilibria in multi-unit trading networks with discrete concave utility functions

Abstract: Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp showed the existence of stable outcomes and competitive equilibria in a model of trading networks under the assumption that all agents’ preferences satisfy a condition called the full substitutes condition. In this paper, we extend their model by using discrete concave utility functions called twisted $$\hbox {M}^{ atural }$$ -concave functions. We show that a valuation function of an agent is twisted $$\hbox {M}^{ atural }$$ -concave if and only if the agent’s preference satisfies the generalized variant of the full substitutes condition. We also show that under the generalized full substitutes condition, there exist stable outcomes and competitive equilibria in the extended model and the set of competitive equilibrium price vectors forms a lattice. In addition, we discuss the connection among competitive equilibria, stability, and efficiency. Finally, we investigate the relationship among stability, strong group stability, and chain stability and verify these three stability concepts are equivalent as long as valuation functions of all agents are twisted $$\hbox {M}^{ atural }$$ -concave.

Linear Contextual Bandits with Global Constraints and Objective

Abstract: We consider the linear contextual bandit problem with global convex constraints and a concave objective function. In each round, the outcome of pulling an arm is a vector, that depends linearly on the context of that arm. The global constraints require the average of these vectors to lie in a certain convex set. The objective is a concave function of this average vector. This problem turns out to be a common generalization of classic linear contextual bandits (linContextual) [Auer 2003], bandits with concave rewards and convex knapsacks (BwCR) [Agrawal, Devanur 2014], and the online stochastic convex programming (OSCP) problem [Agrawal, Devanur 2015]. We present algorithms with near-optimal regret bounds for this problem. Our bounds compare favorably to results on the unstructured version of the problem [Agrawal et al. 2015, Badanidiyuru et al. 2014] where the relation between the contexts and the outcomes could be arbitrary, but the algorithm only competes against a fixed set of policies.

A further extension of Rotfel’d theorem

Abstract: We prove an inequality for concave functions and partitioned matrices whose numerical ranges lie in a sector. This complements a theorem by E.Y. Lee concerning the positive semi-definite case.

Twisted trees and inconsistency of tree estimation when gaps are treated as missing data -- the impact of model mis-specification in distance corrections

Abstract: Statistically consistent estimation of phylogenetic trees or gene trees is possible if pairwise sequence dissimilarities can be converted to a set of distances that are proportional to the true evolutionary distances Susko et al (2004) reported some strikingly broad results about the forms of inconsistency in tree estimation that can arise if corrected distances are not proportional to the true distances They showed that if the corrected distance is a concave function of the true distance, then inconsistency due to long branch attraction will occur If these functions are convex, then two "long branch repulsion" trees will be preferred over the true tree -- though these two incorrect trees are expected to be tied as the preferred true Here we extend their results, and demonstrate the existence of a tree shape (which we refer to as a "twisted Farris-zone" tree) for which a single incorrect tree topology will be guaranteed to be preferred if the corrected distance function is convex We also report that the standard practice of treating gaps in sequence alignments as missing data is sufficient to produce non-linear corrected distance functions if the substitution process is not independent of the insertion/deletion process Taken together, these results imply inconsistent tree inference under mild conditions For example, if some positions in a sequence are constrained to be free of substitutions and insertion/deletion events while the remaining sites evolve with independent substitutions and insertion/deletion events, then the distances obtained by treating gaps as missing data can support an incorrect tree topology even given an unlimited amount of data

Quadratic minimization with portfolio and terminal wealth constraints

Abstract: We address a problem of stochastic optimal control drawn from the area of mathematical finance. The goal is to minimize the expected value of a general quadratic loss function of the wealth at close of trade when there is a specified convex constraint on the portfolio over the trading interval, together with a specified almost-sure lower-bound on the wealth at close of trade. We use a variational approach of Rockafellar which leads naturally to an appropriate vector space of dual variables, a dual functional on the space of dual variables such that the dual problem of maximizing the dual functional is guaranteed to have a solution (i.e. a Lagrange multiplier) when a simple and natural Slater condition holds for the terminal wealth constraint, and obtain necessary and sufficient conditions for optimality of a candidate wealth process. The dual variables are pairs, each comprising an Ito process paired with a member of the adjoint of the space of essentially bounded random variables measurable with respect to the event $$\sigma $$ -algebra at close of trade. The necessary and sufficient conditions are used to construct an optimal portfolio in terms of the Lagrange multiplier. The dual problem simplifies to maximization of a concave function over the real line when the portfolio is unconstrained but the terminal wealth constraint is maintained.

Robust randomized matchings

Abstract: The following zero-sum game is played on a weighted graph G: Alice selects a matching M in G and Bob selects a number k. Then, Alice receives a payoff equal to the ratio of the weight of the top k edges of M to optk, which is the maximum weight of a matching of size at most k in G. If M guarantees a payoff of at least α then it is called α-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a [EQUATION]-robust matching, which is best possible for this setting.In this paper, we show that Alice can improve on the guarantee of [EQUATION] when allowing her to play a randomized strategy. For this setting, we devise a simple algorithm that returns a 1/ln(4)-robust randomized matching. The algorithm is based on the following non-trivial observation: If all edge weights are integer powers of 2, then any lexicographically optimum matching is 1-robust. We prove this property not only for matchings but for any independence system in which optk is a concave function of k. This class of systems includes matroid intersection, b-matchings, and strong 2-exchange systems. We also show that our robustness results for randomized matchings translate to an asymptotic robustness guarantee for deterministic matchings: When restricting Bob's choice to cardinalities larger than a given constant, then Alice can find a single deterministic matching with approximately the same guaranteed payoff as in the randomized setting. In addition to the above results, we also give a new simple LP-based proof of Hassin and Rubinstein's original result.

Primal-dual and dual-fitting analysis of online scheduling algorithms for generalized flow-time problems

Abstract: We study online scheduling problems on a single processor that can be viewed as extensions of the well-studied problem of minimizing total weighted flow time. In particular, we provide a framework of analysis that is derived by duality properties, does not rely on potential functions and is applicable to a variety of scheduling problems. A key ingredient in our approach is bypassing the need for "black-box" rounding of fractional solutions, which yields improved competitive ratios. We begin with an interpretation of Highest-Density-First (HDF) as a primal-dual algorithm, and a corresponding proof that HDF is optimal for total fractional weighted flow time (and thus scalable for the integral objective). Building upon the salient ideas of the proof, we show how to apply and extend this analysis to the more general problem of minimizing $\sum_j w_j g(F_j)$, where $w_j$ is the job weight, $F_j$ is the flow time and $g$ is a non-decreasing cost function. Among other results, we present improved competitive ratios for the setting in which $g$ is a concave function, and the setting of same-density jobs but general cost functions. We further apply our framework of analysis to online weighted completion time with general cost functions as well as scheduling under polyhedral constraints.

Positive Solutions for a System of Discrete Boundary Value Problem

Abstract: This paper deals with the existence and multiplicity of positive solutions for a system of second-order discrete boundary value problem. The main results are obtained via Jensen’s inequalities, properties of concave and convex functions, and the Krasnosel’skii-Zabreiko fixed point theorem. Furthermore, concave and convex functions are employed to emphasize the coupling behaviors of nonlinear terms $$f$$ and $$g$$ and we provide two explicit examples to illustrate our main results and the coupling behaviors.

A global optimization approach to fractional optimal control

Abstract: In this paper, we consider a fractional optimal control problem governed by system of linear differential equations, where its cost function is expressed as the ratio of convex and concave functions. The problem is a hard nonconvex optimal control problem and application of Pontriyagin's principle does not always guarantee finding a global optimal control. Even this type of problems in a finite dimensional space is known as NP hard. This optimal control problem can, in principle, be solved by Dinkhelbach algorithm [10]. However, it leads to solving a sequence of hard D.C programming problems in its finite dimensional analogy. To overcome this difficulty, we introduce a reachable set for the linear system. In this way, the problem is reduced to a quasiconvex maximization problem in a finite dimensional space. Based on a global optimality condition, we propose an algorithm for solving this fractional optimal control problem and we show that the algorithm generates a sequence of local optimal controls with improved cost values. The proposed algorithm is then applied to several test problems, where the global optimal cost value is obtained for each case.

Inequalities of Hermite-Hadamard-like Type for the Functions whose Second Derivatives in Absolute Value are Convex and Concave

Abstract: In this article, new estimates on generalization of Hermite-Hadamardlike type inequalities for functions whose second derivatives in absolute value at certain powers are convex and concave are established. Mathematics Subject Classication: 26D15, 26A51

Design optimization based on state problem functionals

Abstract: This paper presents a general mathematical structure for design optimization problems, where state problem functionals are used as design objectives.It extends to design optimization the general model of physical theories pioneered by Tonti (1972, 1976) and Oden and Reddy (1974, 1983). It turns out that the classical structural optimization problem of compliance minimization is a member of the treated general class of problems. Other particular examples, discussed in the paper, are related to Darcy-Stokes flow and pipe flow models. A main novel feature of the paper is the unification of seemingly different design problems, but the general mathematical structure also explains some previously not fully understood phenomena. For instance, the self-penalization property of Stokes flow design optimization receives an explanation in terms of minimization of a concave function over a convex set.

Refinements of hermite-hadamard type inequalities for differentiable co-ordinated convex functions and applications

Abstract: In this paper, we shall establish some inequalities for differentiable co-ordinated convex and concave functions on a rectangle from the plane. They are connected with the left side of extended Hermite-Hadamard inequality in two variables. Also, these inequalities are able to be applied to some special means and cubature formulae.

Combined effects of concave-convex nonlinearities and indefinite potential in some elliptic problems

Abstract: We consider a nonlinear Dirichlet problem driven by the p-Laplacian and a reaction which exhibits the combined effects of concave (that is, sublinear) terms and of convex (that is, superlinear) terms. The concave term is indefinite and the convex term need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies.

A General Model of Random Variation

Abstract: A statistical distribution of a random variable is uniquely represented by its normal-based quantile function. For a symmetrical distribution it is S-shaped (for negative kurtosis) and inverted S-shaped (otherwise). As skewness departs from zero, the quantile function gradually transforms into a monotone convex function (positive skewness) or concave function (otherwise). Recently, a new general modeling platform has been introduced, response modeling methodology, which delivers good representation to monotone convex relationships due to its unique “continuous monotone convexity” property. In this article, this property is exploited to model the normal-based quantile function, and explored using a set of 27 distributions.

Stability in supply chain networks: an approach by discrete convex analysis

Abstract: Ostrovsky generalized the stable marriage model of Gale and Shapley to a model on an acyclic directed graph, and showed the existence of a chain stable allocation under the conditions called same- side substitutability and cross-side complementarity. In this paper, we extend Ostrovsky's model and the concepts of same-side substitutability and cross-side complementarity by using value functions which are defined on integral vectors and allow indifference. We give a characterization of chain stability under the extended versions of same-side substitutability and cross-side complementarity, and develop an algorithm which always finds a chain stable allocation. We also verify that twisted M ♮ -concave functions, which are variants of M ♮ -concave functions central to discrete convex analysis, satisfy these extended conditions. For twisted M ♮ -concave value functions of the agents, we analyze the time-complexity of our algorithm.

On Certain Class of Meromorphic Harmonic Concave Functions

Abstract: In this paper, a class of meromorphic harmonic functions concave in the unit disc is introduced. Coefficient bounds, distortion inequalities, extreme points, geometric convolution, integral convolution for the functions belonging to this class are obtained.

Berwald type inequality for extremal universal integrals based on (α,m)-concave function

Abstract: The aim of this work is to show a Berwald type inequality for the extremal universal integrals based on (α ,m) concave function. Some examples are given to illustrate the validity of these inequalities.