A note on ordinally concave functions

The multiple exchange property for matroid bases has recently been generalized for valuated matroids and $$\hbox {M}^{\natural }$$ -concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a cardinality condition on the exchangeable subset. The stronger form immediately implies the defining exchange property of $$\hbox {M}^{\natural }$$ -concave set functions, which was not the case with the recently established multiple exchange property without the cardinality condition.

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.

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 \(\hbox {M}^{\natural }\)-concave functions. We show that a valuation function of an agent is twisted \(\hbox {M}^{\natural }\)-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 \(\hbox {M}^{\natural }\)-concave.

Evolution of optimal Hill coefficients in nonlinear public goods games

Capacity planning in manufacturing and computer networks

We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions.

The equivalence of markets and games concerns the relationship between two sorts of structures that appear fundamentally different -- markets and games. Shapley and Shubik (1969) demonstrates that: (1) games derived from markets with concave utility functions generate totally balanced games where the players in the game are the participants in the economy and (2) every totally balanced game generates a market with concave utility functions. A particular form of such a market is one where the commodities are the participants themselves, a labor market for example. But markets are very special structures, more so when it is required that utility functions be concave. Participants may also get utility from belonging to groups, such as marriages, or clubs, or productive coalitions. It may be that participants in an economy even derive utility (or disutility) from engaging in processes that lead to the eventual exchange of commodities. The question is when are such economic structures equivalent to markets with concave utility functions. This paper summarizes research showing that a broad class of large economies generate balanced market games. The economies include, for example, economies with clubs where individuals may have memberships in multiple clubs, with indivisibile commodities, with nonconvexities and with non-monotonicities. The main assumption are: (1) that an option open to any group of players is to break into smaller groups and realize the sum of the worths of these groups, that is, essential superadditivity is satisfied and :(2) relatively small groups of participants can realize almost all gains to coalition formation. The equivalence of games with many players and markets with many participants indicates that relationships obtained for markets with concave utility functions and many participants will also hold for diverse social and economic situations with many players. These relationships include: (a) equivalence of the core and the set of competitive outcomes; (b) the Shapley value is contained in the core or approximate cores; (c) the equal treatment property holds -- that is, both market equilibrium and the core treat similar players similarly. These results can be applied to diverse economic models to obtain the equivalence of cooperative outcomes and competitive, price taking outcomes in economies with many participants and indicate that such results hold in yet more generality.

This chapter discusses the elements of convex analysis which are very important in the study of optimization problems. In section 2.1 the fundamentals of convex sets are discussed. In section 2.2 the subject of convex and concave functions is presented, while in section 2.3 generalizations of convex and concave functions are outlined. This section introduces the fundamental concept of convex sets, describes their basic properties, and presents theoretical results on the separation and support of convex sets.

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

In this paper, a new identity for n-times differntiable functions is established and by using the obtained identity, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are convex and concave functions. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are convex and concave functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal formula and special means of established results are given.

Recently, many fractional integral operators were introduced by different mathematicians. One of these fractional operators, Atangana-Baleanu fractional integral operator, was defined by Atangana and Baleanu (Atangana and Baleanu, 2016). In this study, firstly, a new identity by using Atangana-Baleanu fractional integral operators is proved. Then, new fractional integral inequalities have been obtained for convex and concave functions with the help of this identity and some certain integral inequalities.

This research considers a broad range of scheduling problems in the parallel machines environment. Schedules are evaluated according to two independent components of the objective function: (1) machine cost consisting of a fixed cost and a variable cost; and (2) a regular measure of performance. This study is only one of a few that take the selection of machines among those available as a decision variable. For machine cost with concave functions, we derive the general characteristics of optimal solutions with respect to decisions on the number of machines to use and the way to load the machines. Our analysis is not restricted to the machine cost criterion, but may be extended to other measures with concave functions. Furthermore, we provide a Pareto efficient perspective in understanding the tradeoff between machine cost and any regular measure of performance.

In this paper, some inequalities for convex and concave functions are demonstrated. A new concept of majorization is applied for two pairs of vectors. Instead of the standard approach based on one doubly stochastic matrix, here two matrices are employed: one is both row stochastic and column substochastic, while the other is both column stochastic and row superstochastic. An inequality of Hardy–Littlewood–Polya–Karamata type is derived for a row stochastic–column substochastic matrix.

We explore alternative functional or transport-entropy formulations of the Blaschke–Santaló inequality. In particular, we obtain new Blaschke–Santaló inequalities for s -concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions.

Regularization and concave loss functions for estimation of chemical kinetic models