Concave Cost Functions Research Articles

Compensatory traits can explain the concave cost function of purely sexual traits.

The cost of ornamentation is often measured experimentally to study the relative importance of sexual and viability selection for ornamentation, but these experiments can lead to a misleading conclusion when compensatory trait is ignored. For example, a classic experiment on the outermost tail feathers in the barn swallow Hirundo rustica explains that the concave (or U-shaped) aerodynamic performance cost of the outermost tail feathers would be the evolutionary outcome through viability selection for optimal tail length, but this conclusion depends on the assumption that compensatory traits do not cause reduced performance. Using a simple "toy model" experiment, I demonstrated that ornamentation evolved purely though sexual selection can produce a concave cost function under the presence of compensatory traits, which was further reinforced by a simple mathematical model. Therefore, concave cost function (and the low performance of individuals with reduced ornaments) cannot be used to infer the evolutionary force favoring ornamentation, due to a previously overlooked concept, "overcompensation," which can worsen the whole body performance.

Smoother Entropy for Active State Trajectory Estimation and Obfuscation in POMDPs

We study the problem of controlling a partially observed Markov decision process (POMDP) to either aid or hinder the estimation of its state trajectory. We encode the estimation objectives via the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">smoother entropy</i> , which is the conditional entropy of the state trajectory given measurements and controls. Consideration of the smoother entropy contrasts with previous approaches that instead resort to marginal (or instantaneous) state entropies due to tractability concerns. By establishing novel expressions for the smoother entropy in terms of the POMDP belief state, we show that both the problems of minimising and maximising the smoother entropy in POMDPs can surprisingly be reformulated as belief-state Markov decision processes with concave cost and value functions. The significance of these reformulations is that they render the smoother entropy a tractable optimisation objective, with structural properties amenable to the use of standard POMDP solution techniques for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">both</i> active estimation and obfuscation. Simulations illustrate that optimisation of the smoother entropy leads to superior trajectory estimation and obfuscation compared to alternative approaches.

The Generalized c/μ Rule for Queues with Heterogeneous Server Pools

Dynamic Routing of Queues with Heterogeneous Server Pools In “The Generalized c/μ Rule for Queues with Heterogeneous Server Pools,” Long, Zhang, Zhang, and Zhang study the optimal control of queueing systems with heterogeneous server pools and a single customer class. The goal is to balance the holding cost of the queue with the operating costs of the server pools. They introduce the target-allocation policy, the Gc/μ rule, and the fixed priority policy for systems with general, convex, and concave cost functions, respectively. They also consider an extension to minimize operating costs and maintain a service-level target for customers waiting in the queue. Moreover, they show that their asymptotically optimal routing policies coincide with several classic policies in the literature in special cases.

Generic catastrophic poverty when selfish investors exploit a degradable common resource.

The productivity of a common pool of resources may degrade when overly exploited by a number of selfish investors, a situation known as the tragedy of the commons. Without regulations, agents optimize the size of their individual investments into the commons by balancing incurring costs with the returns received. The resulting Nash equilibrium involves a self-consistency loop between individual investment decisions and the state of the commons. As a consequence, several non-trivial properties emerge. For N investing actors we prove rigorously that typical payoffs do not scale as 1/N, the expected result for cooperating agents, but as (1/N)2. Payoffs are hence reduced with regard to the functional dependence on N, a situation denoted catastrophic poverty. We show that catastrophic poverty results from a fine-tuned balance between returns and costs. Additionally, a finite number of oligarchs may be present. Oligarchs are characterized by payoffs that are finite and not decreasing when N increases. Our results hold for generic classes of models, including convex and moderately concave cost functions. For strongly concave cost functions the Nash equilibrium undergoes a collective reorganization, being characterized instead by entry barriers and sudden death forced market exits.

Distribution of decarbonization costs and externality regulation

The transformation of the economy to a low-carbon level is constrained by a high level of costs and the problem of balancing interests in the distribution of these costs among participants. The paper proposes to use the corporate game theory, in particular the Shapley value, for cost allocation. In contrast to the classical division of additional utility for cooperative games, in this case the costs are divided, and the concave cost function is minimized. The Shapley value coordinates determine the center of gravity of the multidimensional figure of possible cost distributions and are associated with a formal representation of fairness without taking into account financial opportunities and additional, historically established conditions. A comparative analysis of two approaches (carbon tax and carbon credits) to managing negative externalities shows that it is preferable to use the Shapley value for the base allocation of carbon credits. Both approaches to emissions regulation are constrained by institutional barriers to the transformation of the economy: the level of development of national institutions, the achievement of international agreements in the face of economic competition and political confrontation, and the lack of objective information. The authors come to the conclusion that the use of the Shapley value can contribute to the objective formation of quotas and reduce barriers to decarbonization.

Computer Programming Language Solution to a Transportation Problem Involving a Concave Cost Function

The work is on computer programming language solution to a transportation problem involving a concave cost function. Two computer programming languages; Wolfram Mathematica and Anaconda Python programming tools were employed in this study to effectively solve four real life examples from published works. The results from the programming languages yielded an optimal value of N253,000 with an optimal solution as z12 = 13, z22 = 5, z23 = 8, z31 = 11 and z33 = 4 in the first example and the remaining three examples were successfully solved with optimal values of N377,000, GH¢ 236,000 and N509,000 respectively, and the results agreed with the results of existing Karush-Kuhn-Tucker (KKT) procedure of Modified Distribution Method.

Measurement-outlier robust Kalman filter for discrete-time dynamic systems

This paper proposes a recursive filter for discrete-time linear dynamic systems subject to output outliers or heavy-tailed noises. First, we introduce a weight matrix in the conventional MAP estimation. It is shown that this matrix plays an influential role in the innovation whitening and asymptotic variance of the modified MAP estimation and, consequently, can be used in outlier detection. Then, we propose two different constrained optimization problems to obtain this weight. These constraints, stemming from environmental noise characteristics, help to obtain the weight matrix more precisely, which increases the filtering performance significantly. In the first approach, we introduce a convex optimization problem to minimize the estimation upper bound of the error covariance matrix. The second approach converts the modified MAP estimation to a min–min optimization problem with a concave cost function. Consequently, to reduce the effect of outliers in estimation, a semidefinite program (SDP) is proposed for outlier detection. At last, simulation results show the effectiveness and verify the performance of the proposed filter for dynamic systems in the presence of measurement outliers.

Formulation of branched transport as geometry optimization

The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different models used in the literature, another advantage of the urban planning formulation for branched transport is that it provides a more transparent interpretation of the overall cost by separation into a transport (Wasserstein-1-distance) and a network maintenance term, and it splits the problem into the actual transportation task and a geometry optimization.

Solving the non-linear multi-index transportation problems with genetic algorithms

In this paper we study the non-linear multi-index transporta\-tion problem with concave cost functions. We solved the non-linear transportation problem on a network with 5 indices (NTPN5I) described by sources, destinations, intermediate nodes, types of products, and types of transport, that is formulated as a non-linear transportation problem on a network with 3 indices (NTPN3I) described by arcs, types of products, and types of transport. We propose a genetic algorithm for solving the large-scale problems in reasonable amount of time, which was proven by the various tests shown in this paper. The convergence theorem of the algorithm is formulated and proved. The algorithm was implemented in Wolfram Language and tested in Wolfram Mathematica.

Estimating Monotone Concave Stochastic Production Frontiers

Recent research shows that the search for Bayesian estimation of concave production functions is a fruitful area of investigation. In this article, we use a flexible cost function that satisfies globally the monotonicity and curvature properties to estimate features of the production function. Specification of a globally monotone concave production function is a difficult task which is avoided here by using the first-order conditions for cost minimization from a globally monotone concave cost function. The problem of unavailable factor prices is bypassed by assuming structure for relative prices in the first-order conditions. The new technique is shown to perform well in a Monte Carlo experiment as well as in an empirical application to rice farming in India.

Location and two-echelon inventory network design with economies and diseconomies of scale in facility operating costs

Most existing facility location and network design models assume the facility cost as either a fixed cost term or a linear/concave function of the volume of demand assigned to a facility. In particular, a concave cost function implies decreasing marginal cost and exhibits economies of scale. However, when the assigned demand volume exceeds a certain level, diseconomies of scale may occur due to facility congestions, overuse of resources, increasing operational complexity, and so on, leading to a convex cost function with increasing marginal cost. This paper studies a warehouse–retailer network design problem, which simultaneously optimizes the warehouse locations, the warehouse–retailer assignment, and the multi-echelon inventory replenishment policy to minimize the total cost, including the warehouse location and operating costs, the inventory replenishment cost, and the transportation cost. In this paper, the warehouse operating cost is defined as a general function of the volume of demand assigned to a warehouse, which could be concave first due to economies of scale and then convex due to diseconomies of scale. We formulate this problem as a set-covering model and propose a column generation algorithm to solve its linear relaxation. The pricing problem of the column generation process, although formulated as a nonlinear mixed integer program, has nice structural properties and can be solved efficiently by a branch-and-bound method. To validate the effectiveness and efficiency of the proposed column generation algorithm, we conduct numerical studies based on randomly generated instances. Numerical results also demonstrate the impact of economies and diseconomies of scale on network design decisions and warehouse operating cost.

Stochastic optimal transport revisited

We prove the Duality Theorems for the stochastic optimal transportation problems with a convex cost function without a regularity assumption that is often supposed in the proof of the lower semicontinuity of an action integral. In our new approach, we prove that the stochastic optimal transportation problems with a convex cost function are equivalent to a class of variational problems for the Fokker-Planck equation, which lets us revisit them. It is done by the so-called superposition principle and by an idea from the mather theory. The superposition principle is the construction of a semimartingale from the Fokker-Planck equation and can be considered a class of the so-called marginal problems that construct stochastic processes from given marginal distributions. It was first considered in stochastic mechanics by E. Nelson, called Nelson's problem, and was proved by E. Carlen first. The semimartingale is called the Nelson process, provided it is Markovian. We also consider the Markov property of a minimizer of the stochastic optimal transportation problem with a nonconvex cost in a one-dimensional case. In the proof, the superposition principle and the minimizer of an optimal transportation problem with a concave cost function play crucial roles. Lastly, we prove the semiconcavity and the Lipschitz continuity of Schrodinger's problem that is a typical example of the stochastic optimal transportation problem.

The Properties of Conjectural Variations in the Nonlinear Stackelberg Oligopoly Model

The game-theoretic problem of choosing optimal strategies for oligopoly market agents with linear demand functions and nonlinear cost functions is considered. The conjectural variations of each agent, i.e., the expected responses (changes in actions) of his counteragents that optimize their utility functions, are studied. Formulas for calculating the conjectural variations of each agent and also the sum of the conjectural variations of all agents in the environment of each agent are derived. The signs of conjectural variations under an arbitrary level of Stackelberg leadership are analyzed. The following properties of conjectural variations are established: 1) the variations are negative if the cost functions of all environmental agents are either convex or concave; 2) the variations are positive if the agents with concave cost functions (the ones with the positive scale effect) prevail in the environment over the agents with convex cost functions (the ones with the negative scale effect). The sum of the agent’s conjectural variations is: 1) negative and its magnitude is bounded above by 1 if the environmental agents mainly have convex cost functions; 2) positive and unlimited if the agents with concave cost functions prevail in the environment.

Global Optimization from Concave Minimization to Concave Mixed Variational Inequality

We use techniques from global optimization to develop an algorithm for finding a global solution of nonconvex mixed variational inequality problems involving separable DC cost functions. In contrast to the convex mixed variational inequality, in these problems, a local solution may not be a global one. The proposed algorithm uses the convex envelope of the separable cost function over boxes to approximate a DC cost problem with a convex cost one that can be solved by available methods. To obtain better approximate solutions, the algorithm uses an adaptive rectangular bisection which is performed only in the space of concave variables. The algorithm is applied to solve the Nash-Cournot and Bertrand equilibrium models with logarithm and quadratic concave costs. Computational results on a lot number of randomly generated data show that the proposed algorithm is efficient for these models, when the number of the concave cost functions is moderate, while the size of the model may be much larger.

Genetic algorithm for solving transportation problems on networks with one source and multiple sinks

In this paper we propose a genetic algorithm for solving the nonlinear transportation problem on a network with multiple sinks and concave piecewise cost functions. We prove that the complexity of one iteration of the algorithm is O(n2) and the algorithm converges to a local optimum solution. We show that the algorithm can be used to solve large-scale problems and present the implementation and several testing examples of the algorithm using Wolfram Language.

Cost Sharing Methods for Capacity Restricted Cooperative Purchasing Situations

This paper analyzes cooperative purchasing situations with two suppliers with limited supply capacity. In these capacity restricted cooperative purchasing (CRCP) situations we consider a group of cooperating purchasers for whom it turns out to be optimal as a group to or- der as much as possible at one supplier and the possible remainder at the other. To find suitable cost allocations of the total purchasing costs, a CRCP-situation is modeled as a cost sharing problem, in which the corresponding cost function is piecewise concave and the intervals of concavity are determined by the restricted capacity of the suppliers. It is seen that in the setting with piecewise concave cost functions, standard cost sharing mechanisms lose their axiomatic appeal, e.g. the serial cost sharing mechanism will neither satisfy unit cost monotonicity (UCM) nor monotonic vulnerability for the absence of the smallest player (MOVASP). In fact, it is shown that these two properties are incompatible now. We in- troduce a new context specific class of piecewise serial rules, in which the vector of order quantities is divided into separate vectors for the different intervals of concavity, using a bankruptcy rule and subsequently apply the serial rule to each interval. We show that the proportional rule is the only bankruptcy rule for which the corresponding piecewise serial rule satisfies UCM. Moreover, we show that the piecewise serial rule corresponding to the constrained equal losses rule satisfies MOVASP.

Assigning orders to suppliers with linear piecewise concave costs

Purpose: Once a set of suppliers has been determined, according to criteria of quality, price and reliability, among others, there remains the problem of assigning orders to the selected suppliers, in order to cover the needs at the lowest cost. We consider the case in which the needs of a component for a set of plants should be covered by suppliers with linear piecewise concave cost functions, a lower bound on the order size for the non-zero deliveries and a capacity constraint. The purpose is to design procedures for solving this problem.Design/methodology/approach: With the aim of providing practical tools to solve the problem of assigning orders to suppliers with linear piecewise concave costs, two mixed integer linear programs are proposed.Findings: The two MILP models are compared through an extensive computational experiment. This shows that both models, with a slight advantage for one of them, can be solved within a very short time, even when the dimensions of the instance largely exceed those that can occur in real cases.Originality/value: The paper proposes novel models that can be used to solve the problem to optimality in reasonable times and with standard optimization software.

Design Tradeoffs in Concave Cost-Sharing Games

This note focuses on the design of cost-sharing rules to optimize the efficiency of the resulting equilibria in cost-sharing games with concave cost functions. Our analysis focuses on two well-studied measures of efficiency, termed the price of anarchy and price of stability , which provide worst-case guarantees on the performance of the (worst or best) equilibria. Our first result characterizes the cost-sharing design that optimizes the price of anarchy, followed by the price of stability. This optimal cost-sharing rule is precisely the Shapley value cost-sharing rule. Our second result characterizes the cost-sharing design that optimizes the price of stability, followed by the price of anarchy. This optimal cost-sharing rule is precisely the marginal contribution cost-sharing rule. This analysis highlights a fundamental tradeoff between the price of anarchy and price of stability in the considered class of games. That is, given the optimality of both the Shapley value and marginal cost distribution rules in each of their respective domains, it is impossible to improve either the price of anarchy or price of stability without degrading its counterpart.

Ultrametricity of optimal transport substates for multiple interacting paths over a square lattice network.

We model a set of point-to-point transports on a network as a system of polydisperse interacting self-avoiding walks (SAWs) over a finite square lattice. The ends of each SAW may be located both at random, uniformly distributed, positions or with one end fixed at a lattice corner. The total energy of the system is computed as the sum over all SAWs, which may represent either the time needed to complete the transport over the network, or the resources needed to build the networking infrastructure. We focus especially on the second aspect by assigning a concave cost function to each site to encourage path overlap. A simulated annealing optimization, based on a modified Berg-Foerster-Aragao de Carvalho-Caracciolo-Froehlich (BFACF) algorithm developed for polymers, is used to probe the complex conformational substate structure at zero temperature. We characterize the average cost gains (and path-length variations) for increasing polymer density with respect to a Dijkstra routing and find a nonmonotonic behavior as recently found for random networks. We observe the emergence of ergodicity breaking and of nontrivial overlap distributions among replicas when switching from a convex to a concave cost function (e.g., x^{γ}, where x represents the node overlap). Finally, we show that the space of ground states for γ<1 is compatible with an ultrametric structure, as seen in many complex systems such as some spin glasses.

Differentiated oligopolistic markets with concave cost functions via Ky Fan inequalities

A Nash–Cournot model for oligopolistic markets with concave cost functions and a differentiated commodity is analyzed. Equilibrium states are characterized through Ky Fan inequalities. Relying on the minimization of a suitable merit function, a general algorithmic scheme for solving them is provided. Two concrete algorithms are therefore designed that converge under suitable convexity and monotonicity assumptions. The results of some numerical tests on randomly generated markets are also reported.