Quasiconvex Research Articles

Capturing preferences for inequality aversion in decision support

We investigate the situation where there is interest in ranking distributions (of income, of wealth, of health, of service levels) across a population, in which individuals are considered preferentially indistinguishable and where there is some limited information about social preferences. We use a natural dominance relation, generalised Lorenz dominance, used in welfare comparisons in economic theory. In some settings there may be additional information about preferences (for example, if there is policy statement that one distribution is preferred to another) and any dominance relation should respect such preferences. However, characterising this sort of conditional dominance relation (specifically, dominance with respect to the set of all symmetric increasing quasiconcave functions in line with given preference information) turns out to be computationally challenging. This challenge comes about because, through the assumption of symmetry, any one preference statement (“I prefer giving $100 to Jane and $110 to John over giving $150 to Jane and $90 to John”) implies a large number of other preference statements (“I prefer giving $110 to Jane and $100 to John over giving $150 to Jane and $90 to John”; “I prefer giving $100 to Jane and $110 to John over giving $90 to Jane and $150 to John”). We present theoretical results that help deal with these challenges and present tractable linear programming formulations for testing whether dominance holds between any given pair of distributions. We also propose an interactive decision support procedure for ranking a given set of distributions and demonstrate its performance through computational testing.

Relationships Between Convexificators and Greenberg–Pierskalla Subdifferentials for Quasiconvex Functions

ABSTRACTIn this paper, the relationship between convexificators and Greenberg–Pierskalla-based (GP-based) subdifferentials for quasiconvex functions is proved. The established results lead to a mean value theorem, a chain rule, and the closedness property for GP-based subdifferentials. Furthermore, the connection between Clarke generalized gradient and Mordukhovich subdifferential with GP-based subdifferentials is highlighted.

Properties of the quasilinear clones containing creative functions

We study the problem of characterizing clones on a three-element set by hyperidentities. We prove that there exists a hyperidentity separating any clone of quasilinear functions defined on the set {0, 1, 2} each of them is either a selector or such that all its values belong to {0, 1} from any noncreative clone constituted by such functions incomparable with the initial clone.

Relay Selection in Full-Duplex Energy-Harvesting Two-Way Relay Networks

In this paper, we investigate the relay selection (RS) problem in full-duplex (FD) two-way relay networks, where the relays are wirelessly powered by harvesting a portion of the received signal power from the sources. To the best of the authors’ knowledge, this is the first investigation of FD two-way relays with simultaneous wireless and information transfer capabilities. For each relay, we prove the quasi convexity of the power splitting (PS) factor optimization, and obtain the optimal PS factor in terms of the outage probability by linear search. Both single relay selection (SRS) and general relay selection (GRS) without the limit on the number of cooperating relays are investigated and the corresponding RS methods are proposed. Since the optimal GRS problem is computationally intractable with exponential complexity, we propose several low-complexity heuristic GRS methods based on various relay ordering and greedy selection criteria. Simulations indicate that the proposed GRS methods perform better than the SRS methods and achieve very similar performance compared with the optimal RS method achieved by exhaustive search but with dramatically reduced complexity.

Optimality Conditions of Approximate Solutions for Nonsmooth Semi-infinite Programming Problems

In this paper, we study optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems. Three new classes of functions, namely $$\varepsilon $$ -pseudoconvex functions of type I and type II and $$\varepsilon $$ -quasiconvex functions are introduced, respectively. By utilizing these new concepts, sufficient optimality conditions of approximate solutions for the nonsmooth semi-infinite programming problem are established. Some examples are also presented. The results obtained in this paper improve the corresponding results of Son et al. (J Optim Theory Appl 141:389–409, 2009).

Global minimization of the difference of increasing co-radiant and quasi-concave functions

Functions which are increasing, co-radiant and quasi-concave have found many applications in microeconomic analysis. In production theory it is commonly assumed that the production function is increasing and quasi-concave. Likewise in consumer theory one often assumes that the utility function has these properties. In this paper, we first examine characterizations of the dual problem for the difference of two increasing, co-radiant and quasi-concave functions. Next, we give various characterizations of the minimal elements of the upper support set of co-radiant functions, by applying a type of duality, which is used in microeconomic theory. As an application, we obtain necessary and sufficient conditions for the global minimum of the difference of two increasing, co-radiant and quasi-concave functions defined on a real locally convex topological vector space X.

$$\left( p,q\right) $$ p , q -Hermite–Hadamard inequalities and $$\left( p,q\right) $$ p , q -estimates for midpoint type inequalities via convex and quasi-convex functions

In this paper, we prove the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, some new \(\left( p,q\right) \)-Hermite–Hadamard inequalities, and generalized \(\left( p,q\right) \)-Hermite–Hadamard inequality. By using the left hand part of the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some \(\left( p,q\right) \)-midpoint type integral inequalities through \(\left( p,q\right) \)-differentiable convex and \(\left( p,q\right) \)-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works.

Distortion theorems for normalized biholomorphic quasi-convex mappings

In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball Bn respectively.

Approximating Quasiconvex Functions with Strictly Quasiconvex Ones in Banach Space

In this paper we show how to approximate a quasiconvex function with a sequence of strictly quasiconvex functions in a reflexive Banach space X. An important role in our approximation procedure is played by a real valued convex function defined on X, and parameterized by a pair of closed bounded convex sets, which is a generalization of the classical Minkowski functional on X; for this reason, we investigate some of its properties. In particular, we prove the continuity of this map, seen as a function acting from a specific family of pairs of closed convex subsets of X, to the space of the real valued continuous functions on X. In the domain space we use the (bounded) Hausdorff topology, while the target space is endowed with the topology of the uniform convergence on bounded sets. Our results also need to approximate a closed convex set, in the sense of the bounded Hausdorff topology, with a sequence of strictly convex sets. The result particularizes to Hausdorff topology if the limit set is bounded.

Hybrid Biogeography-based Optimization Algorithm for Solving Nonlinear Bilevel Programming

The article deals with a class of nonlinear bilevel programming problems in which the follower’s objective is a quasi-concave function, and a new hybrid biogeography-based optimization algorithm is proposed to search for the value of the leader’s variables. First, an extreme point searching approach is constructed to obtain the optimal solution of the follower’s programming. An efficient mutation operator is then designed by using Zoutendijk feasible direction method so that it can generate high quality potential offspring. Furthermore, a new fitness function is established that can be easily used to force the individuals moving toward the feasible region and improve the feasible solutions gradually. Under some assumptions, theoretical analysis of the algorithm has been presented. Finally, to verify the effectiveness of the algorithm, numerical experiments on 8 test problems are made and performance analysis verify the proposed algorithm can converge to the global optimal solution of bilevel programming problems.

Redistribution by means of lotteries

A government designs anonymous income transfers between a continuum of citizens whose income valuation is privately known. When transfers are deterministic, the incentive constraints imply equal treatment independently of the government's taste for redistribution. We study whether random transfers may locally improve upon the egalitarian outcome. A suitable Taylor expansion offers an approximation of the utility function by a quasilinear function. The methodology developed by Myerson to deal with incentive constraints then yields a necessary and sufficient condition for the existence of a socially useful randomization. When this condition is met a large set of lotteries are locally improving. A special menu made of two lotteries only is of interest: all the agents with low risk aversion receive the same random transfer, financed by a deterministic tax paid by the high risk aversion agents.

Hermite-Hadamard inequality for geometrically quasiconvex functions on co-ordinates

In this paper we introduce the concept of geometrically quasiconvex functions on the co-ordinates and establish some Hermite-Hadamard type integral inequalities for functions defined on rectangles in the plane. Some inequalities for product of two geometrically quasiconvex functions on the co-ordinates are considered.

On the microeconomic foundations of linear demand for differentiated products

This paper provides a thorough exploration of the microeconomic foundations for the multi-variate linear demand function for differentiated products, which is widely used in industrial organization. The setting is the standard representative consumer with a quasi-linear utility function. A key finding is that strict concavity of the quadratic utility function is critical for the demand system to be well defined. Otherwise, the true demand function may be quite complex: Multi-valued, non-linear and income-dependent. We uncover failures of duality relationships between substitute products and complementary products, as well as the incompatibility between high levels of complementarity and concavity. The two-good case emerges as a special case with strong but non-robust properties. A key implication is that all conclusions derived in applied economic models via the use of linear demand that does not satisfy the Law of Demand ought to be regarded with some suspicion.

Generalized higher-order cone-convex functions and higher-order duality in vector optimization

In this paper, we introduce a new class of higher-order cone-convex, \((K_1, K_2)\)-pseudoconvex and quasiconvex functions which encapsulates several already known functions. Higher-order sufficient optimality conditions have been established for a vector optimization problem over cones by using these functions, under weaker conditions on multipliers as compared to other papers in this domain. Wolfe type and Mond–Weir type higher-order duals are formulated and corresponding duality results are established. A number of previously studied problems appear as special cases of our primal-dual models. In case of nonlinear programming problem, our higher-order duals reduce to the corresponding higher-order duals given by Mangasarian (J Math Anal Appl 51:607–620, 1975) and Mond and Zhang (Generalized convexity, generalized monotonicity: recent results. Kluwer, Dordrecht, pp 357–372, 1998).

Formulas for Asymptotic Functions via Conjugates, Directional Derivatives and Subdifferentials

The q-asymptotic function is a new tool that permits to study nonconvex optimization problems with unbounded data. It is particularly useful when dealing with quasiconvex functions. In this paper, we obtain formulas for the q-asymptotic function via c-conjugates, directional derivatives and subdifferentials. We obtain them under lower semicontinuity or local Lipschitz assumptions. The well-known formulas for the asymptotic function in the convex case are consequences of these ones. We obtain a new formula for the convex case.

Performance share plans: Valuation and empirical tests

Performance share plans are an increasingly important component of executive compensation. They are equity-based, long-term incentive plans where the number of shares to be awarded is a quasi-linear function of a performance result over a fixed time period. We derive closed-form formulas for the value of a performance share plan when the performance measure is: (1) a non-traded measure following an Arithmetic Brownian Motion (e.g., earnings per share), (2) a non-traded measure following a Geometric Brownian Motion (e.g., revenue), or (3) a rank-order tournament of traded asset returns that are following Arithmetic Brownian Motions (e.g., percentile of ranked stock returns). Then we empirically test our valuation formulas. We find that our valuation formulas are more accurate for performance share plans based on earnings per share when forecasting using analyst consensus prior to the grant date. We also find that the efficiency of our valuation model greatly depends on the method used to forecast future firm performance. The policy implication is that FASB should consider requiring that grant date fair value be estimated using valuation formulas such as ours.

On the resolution of certain discrete univariate max–min problems

We analyze a class of discrete, univariate, and strictly quasiconcave max–min problems. A distinctive feature of max–min-type optimization problems is the nonsmoothness of the objective being maximized. Here we exploit strict quasiconcavity of the given set of functions to prove existence and uniqueness of the optimizer, and to provide computable bounds for it. The analysis inspires an efficient algorithm for computing the optimizer without having to resort to any regularization or heuristics. We prove correctness of the proposed algorithm and briefly discuss the effect of tolerances and approximate computation. Our study finds direct application in the context of certain mesh deformation methods, wherein the optimal perturbation for a vertex is computed as the solution of a max–min problem of the type we consider here. We include examples demonstrating improvement of simplicial meshes while adopting the proposed algorithm for resolution of the optimization problems involved, and use these numerical experiments to examine the performance of the algorithm.

Secrecy Throughput Maximization for MISO Multi-Eavesdropper Wiretap Channels

In this paper, the secure transmission strategy for a multi-input-single-output multi-eavesdropper system with coexistence of a secure user (Bob) and a normal user (NU) is investigated. The NU and Bob require normal and secure data transmissions, respectively, and thus, the stream for the NU can be exploited to confuse the eavesdroppers. To guarantee the security of Bob, artificial noise (AN) is also deliberately injected into the null space of Bob and NU. The power allocation among Bob, NU, and AN, as well as the wiretap code rates, is jointly optimized to maximize the effective secrecy throughput (EST), under the average throughput constraint of the NU and statistical channel state information (CSI) of eavesdroppers. Both non-adaptive and adaptive transmission schemes are proposed, based on the statistical and instantaneous CSI of the legitimate channels, respectively. An alternative optimization algorithm is proposed to obtain the optimal parameters. It is proved that the EST is a quasi-concave function of the secrecy rate and the power allocated to Bob, and for fixed wiretap code rates, the optimal power allocation is derived in a closed-form expression. Numerical results show that the EST increases with the increase in transmitting power and the number of transmit antennas, and decreases with the increasing throughput constraint of the NU. Improved EST can be achieved through injecting AN and concurrent transmission of Bob and NU.

Results on Energy- and Spectral-Efficiency Tradeoff in Cellular Networks With Full-Duplex Enabled Base Stations

In this paper, we address the tradeoff between energy efficiency (EE) and spectral efficiency (SE) for cellular networks with full-duplex (FD) communications enabled base stations. To be backward compatible with legacy LTE systems, it is assumed that user devices still work in the conventional half-duplex (HD) mode. There usually exists residual self-interference (RSI) in FD communications after advanced interference suppression techniques are applied. In this paper, we consider two different RSI models: constant RSI model and linear RSI model. First, the necessary conditions for an FD transceiver to achieve better EE-SE tradeoff than an HD one are derived for both the RSI models. Then, for the constant RSI model, a closed-form EE-SE expression is obtained in the scenario of single pair of users. We further extend our result and prove that EE is a quasi-concave function of SE in the scenario of multiple user pairs. Accordingly, an optimal algorithm to achieve the maximum EE based on the Lagrange dual decomposition technique is developed. For the linear RSI model, the EE-SE relation is difficult to deal with and we develop a heuristic algorithm by decoupling the problem into two sub-problems: power control and resource allocation. Our analysis and algorithms are finally verified by comprehensive numerical results.

Quasilinear Utility and Two Market Monopoly

The use of quasilinear utility functions in economic analyses is widespread. This paper presents an overdue clarification on the implications of quasilinear utility for two market monopoly. The paper begins by deriving the demands facing a two market monopoly from a representative consumer with quasilinear utility. Expressions are derived for the profit margins expressed solely in terms of the own and cross-price elasticities of demand. The paper also analyzes the implications of quasilinear utility for other issues in two market monopoly: pricing below marginal cost in a market, third-degree price discrimination when the monopoly products are substitutes and pricing in the inelastic region of demands.