Class Of Cost Functions Research Articles

A Sensitivity Analysis of the Price of Anarchy in Nonatomic Congestion Games

The price of anarchy (PoA) is a standard measure to quantify the inefficiency of equilibria in nonatomic congestion games. Most publications have focused on worst-case bounds for the PoA. Only a few have analyzed the sensitivity of the PoA against changes of the demands or cost functions, although that is crucial for empirical computations of the PoA. We analyze the sensitivity of the PoA with respect to (w.r.t.) simultaneous changes of demands and cost functions. The key to this analysis is a metric for the distance between two games that defines a topological metric space consisting of all games with the same combinatorial structure. The PoA is then a locally pointwise Hölder continuous function of the demands and cost functions, and we analyze the Hölder exponent for different classes of cost functions. We also apply our approach to the convergence analysis of the PoA when the total demand tends to zero or infinity. Our results further develop the recent seminal work on the sensitivity of the PoA w.r.t. changes of only the demands under special conditions. Funding: This work was supported by the Science Foundation of Anhui Science and Technology Department [Grant 1908085QF262]; Science Foundation of the Anhui Education Department [Grant KJ2019A0834]; National Science Foundation of China [Grants 61906062, 72271085]; Talent Foundation of Hefei University [Grant 1819RC29].

On barren plateaus and cost function locality in variational quantum algorithms

Variational quantum algorithms rely on gradient based optimization to iteratively minimize a cost function evaluated by measuring output(s) of a quantum processor. A barren plateau is the phenomenon of exponentially vanishing gradients in sufficiently expressive parametrized quantum circuits. It has been established that the onset of a barren plateau regime depends on the cost function, although the particular behavior has been demonstrated only for certain classes of cost functions. Here we derive a lower bound on the variance of the gradient, which depends mainly on the width of the circuit causal cone of each term in the Pauli decomposition of the cost function. Our result further clarifies the conditions under which barren plateaus can occur.

ω♯-Algebras

This article explores a generalization of the algebraic theory of formal languages. Having, as starting point, the work of T. Colcombet on cost functions and stabilization monoids, and of Daviaud et al. on stabilization algebras, this class of algebras is extended to ω♯-algebras and ω♯-automata are also introduced. The equality problem for order ideals (of free ω♯-algebras) recognized by finite ω♯-algebras is answered positively in this context. Various results on formal languages and monoids are generalized to this setting of order ideals and ω♯-algebras. The class of cost functions is proved to be embeddable in the class of recognizable order ideals.

Band gap optimization of one-dimension elastic waveguides using spatial Fourier plane wave expansion coefficients

Periodic elastic waveguides, such as rods, beams, and shafts, exhibit frequency bands where wave reflections at impedance discontinuities cause strong wave attenuation by Bragg scattering. Such frequency bands are known as stop bands or band gaps. This work presents a shape optimization technique for one-dimensional periodic structures. The proposed approach, which aims to maximize the width of the first band gap, uses as tuning parameters the spatial Fourier coefficients that describe the shape of the cell cross-section variation along its length. Since the optimization problem is formulated in terms of Fourier coefficients, it can be directly applied to the Plane Wave Expansion (PWE) method, commonly used to obtain the dispersion diagrams, which indicate the presence of band gaps. The proposed technique is used to optimize the shape of a straight bar with both solid and hollow circular cross-sections. First, the optimization is performed using the elementary rod, the Euler-Bernoulli and Timoshenko beam, and the shaft theoretical models in an independent way. Then, the optimization is conducted to obtain a complete band gap in the dispersion diagrams, which includes the three wave types, i.e., longitudinal, bending, and torsional. All numerical results provided feasible shapes that generate wide stop bands in the dispersion diagrams. The proposed technique can be extended to two- and three-dimensional periodic frame structures, and can also be adapted for different classes of cost functions.

Minimizing maximum cost for a single machine under uncertainty of processing times

One of the most known results in the machine scheduling is Lawler’s algorithm to minimize the maximum cost of jobs processed by a single machine subject to precedence constraints. We consider an uncertain version of the same min-max cost scheduling problem. The cost function of each job depends on the job completion time and on an additional generalized numerical parameter, which may be a tuple of parameters. For each job, both, its processing time and the additional parameter are uncertain, only intervals of possible values of these parameters are known.We analyse certain classes of cost functions and develop polynomial algorithms which construct min-max regret solutions. The considered problems cover the most general range of studied cases of interval uncertainty.In the only two papers that present algorithms for minimizing the maximum regret for the problem with uncertain job processing times, the algorithms are based on extremal scenarios, where some uncertain parameters take their maximum values, while all others take their minimum possible values.We show that it is impossible to always limit the search to extremal scenarios. Our approach is based on new ideas different from those underlying previous work.Finally, we show that our approach outperforms all known results for constructing min-max regret solutions for the min-max cost scheduling problem under uncertainty of job processing times.

Optimal AoI-Aware Scheduling and Cycles in Graphs

We study the task of scheduling in a multi-source system where sources report their time-varying information to a central monitoring station via multiple orthogonal channels. The Age-of-Information of a source is defined as the amount of time elapsed since the latest update from that source was received at the monitoring station. At each time instant, the system pays a cost that is a function of the current Ages-of-Information of the sources. Our goal is to design scheduling policies to minimize this cost. We draw a novel parallel between our scheduling problem and the minimum mean cost cycle problem in weighted graphs and use this insight to design optimal scheduling policies for a very general class of cost functions. In addition, we compare the performance of our policy with naive greedy policies. We show that while greedy policies can be optimal if the cost function is symmetric with respect to the sources, our policy strictly outperforms greedy policies when the cost function is asymmetric across sources.

A theoretical investigation of Brockett’s ensemble optimal control problems

This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework. Following Brockett’s formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key to get (under suitable assumptions) also uniqueness of optimal controls.

Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces

In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context of SCE Density Functional Theory introduced in Strong-interaction limit of density-functional theory by Seidl [Phys. Rev. A 60 (1999) 4387].

Identification problems with given material interfaces

This paper is devoted to the identification of coefficients in scalar elliptic partial differential equations using an optimal control approach. The coefficients usually can be interpreted as material characteristics and play the role of the control variable. The paper focuses on processes in domains which can be split into a finite number of materially homogeneous subdomains, i.e. the coefficients to be identified are piecewise constant on them. In addition, we assume that the material interfaces are a priori known. We prove the existence of at least one solution of the optimal control problem for a large class of cost functionals and show that solutions can be obtained as limits of solutions to the problems which are governed by finite element discretizations of the state equations. Further, the unified algebraic sensitivity analysis of the first and the second order for several least squares type cost functions is investigated. Finally, a model problem of the identification of coefficients characterizing hydraulic conductivity by pumping tests in groundwater flow modeling is numerically solved using the sequential approach and the scalarization technique.

Tractability-preserving transformations of global cost functions

Graphical model processing is a central problem in artificial intelligence. The optimization of the combined cost of a network of local cost functions federates a variety of famous problems including CSP, SAT and Max-SAT but also optimization in stochastic variants such as Markov Random Fields and Bayesian networks. Exact solving methods for these problems typically include branch and bound and local inference-based bounds.In this paper we are interested in understanding when and how dynamic programming based optimization can be used to efficiently enforce soft local consistencies on Global Cost Functions, defined as parameterized families of cost functions of unbounded arity. Enforcing local consistencies in cost function networks is performed by applying so-called Equivalence Preserving Transformations (EPTs) to the cost functions. These EPTs may transform global cost functions and make them intractable to optimize.We identify as tractable projection-safe those global cost functions whose optimization is and remains tractable after applying the EPTs used for enforcing arc consistency. We also provide new classes of cost functions that are tractable projection-safe thanks to dynamic programming.We show that dynamic programming can either be directly used inside filtering algorithms, defining polynomially DAG-filterable cost functions, or emulated by arc consistency filtering on a Berge-acyclic network of bounded-arity cost functions, defining Berge-acyclic network-decomposable cost functions. We give examples of such cost functions and we provide a systematic way to define decompositions from existing decomposable global constraints.These two approaches to enforcing consistency in global cost functions are then embedded in a solver for extensive experiments that confirm the feasibility and efficiency of our proposal.

Playing the Wrong Game

In many situations a player may act so as to maximize a perceived utility that is not exactly her utility function, but rather some other, biased, utility. Examples of such biased utility functions are common in behavioral economics, and include risk attitudes, altruism, present-bias and so on. When analyzing a game, one may ask how inefficiency, measured by the Price of Anarchy (PoA) is a?ected by the perceived utilities. The smoothness method [16, 15] naturally extends to games with such perceived utilities or costs, regardless of the game or the behavioral bias. We show that such biasedsmoothness is broadly applicable in the context of nonatomic congestion games. First, we show that on series-parallel networks we can use smoothness to yield PoA bounds even for diverse populations with di?erent biases. Second, we identify various classes of cost functions and biases that are smooth, thereby substantially improving some recent results from the literature.

Price of anarchy for non-atomic congestion games with stochastic demands

We generalize the notions of user equilibrium, system optimum and price of anarchy to non-atomic congestion games with stochastic demands. In this generalized model, we extend the two bounding methods from Roughgarden and Tardos (2004) and Correa et al. (2008) to bound the price of anarchy, and compare the upper bounds we have obtained. Our results show that the price of anarchy depends not only on the class of cost functions but also demand distributions and, to some extent, the network topology. The upper bounds are tight in some special cases, including the case of deterministic demands.

Convergence Rates of Distributed Nesterov-Like Gradient Methods on Random Networks

We consider distributed optimization in random networks where N nodes cooperatively minimize the sum Σi=1N fi(x) of their individual convex costs. Existing literature proposes distributed gradient-like methods that are computationally cheap and resilient to link failures, but have slow convergence rates. In this paper, we propose accelerated distributed gradient methods that 1) are resilient to link failures; 2) computationally cheap; and 3) improve convergence rates over other gradient methods. We model the network by a sequence of independent, identically distributed random matrices {W(k)} drawn from the set of symmetric, stochastic matrices with positive diagonals. The network is connected on average and the cost functions are convex, differentiable, with Lipschitz continuous and bounded gradients. We design two distributed Nesterov-like gradient methods that modify the D-NG and D-NC methods that we proposed for static networks. We prove their convergence rates in terms of the expected optimality gap at the cost function. Let k and K be the number of per-node gradient evaluations and per-node communications, respectively. Then the modified D-NG achieves rates O(logk/k) and O(logK/ K), and the modified D-NC rates O(1/k2) and O(1/ K2-ξ), where ξ > 0 is arbitrarily small. For comparison, the standard distributed gradient method cannot do better than Ω(1/k2/3) and Ω(1/ K2/3), on the same class of cost functions (even for static networks). Simulation examples illustrate our analytical findings.

021 Cost functional dependence of optimal CPA equilibration trajectories

Rapid and safe equilibration of cells with and from high concentrations of permeating solutes is of great interest to cryobiologists. This equilibration is limited by at least two competing factors: cells and tissues have volume or concentration constraints (known as osmotic tolerance limits) and the toxicity of permeating solutes is considered to be a monotonically increasing function of exposure time. Because of this, there has been considerable effort to establish equilibration protocols that minimize time of exposure while keeping cells within known bounds. Our approach has been to define the concept of an equilibration damage cost function that depends on the cellular state during the entire equilibration protocol, namely its CPA equilibration state space trajectory. We recently cast and solved this problem formally in terms of the mathematical theory of optimal control, determining extracellular CPA and nonpermeating solute concentrations that minimize a cost function of total protocol time. While it has been hypothesized that the equilibration damage cost function should also depend on intracellular concentration, the precise dependence has not yet been determined. Here we extend our recent work by mathematically determining the dependence of optimal CPA equilibration state space trajectories for classes of cost functions, defining optimal extracellular CPA and nonpermeating solute concentrations. We are able to show that there are fundamental classes of toxicity cost functions that correspond to a limited number of optimal equilibration trajectories, indicating that the precise determination of the toxicity cost function (e.g. cellular damage as a function of concentration and time) is not as important as its class membership (e.g. does the damage depend super or sublinearly as a function of concentration). In other words, we are able to mathematically determine optimal CPA equilibration protocols given relatively few cell specific parameters. Source of funding: None declared. Conflict of interest: None declared. benson@math.niu.edu

Including the Size of Regions in Image Segmentation by Region-Based Graph

Applying a fast over-segmentation algorithm to image and working on a region-based graph (instead of the pixel-based graph) is an efficient approach to reduce the computational complexity of graph-based image segmentation methods. Nevertheless, some undesirable effects may arise if the conventional cost functions, such as Ncut, AverageCut, and MinCut, are employed for partitioning the region-based graph. This is because these cost functions are generally tailored to pixel-based graphs. In order to resolve this problem, we first introduce a new class of cost functions (containing Ncut and AverageCut) for graph partitioning whose corresponding suboptimal solution can be efficiently computed by solving a generalized eigenvalue problem. Then, among these cost functions, we propose one that considers the size of regions in the partitioning procedure. By simulation, the performance of the proposed cost function is quantitatively compared with that of the Ncut and AverageCut.

Group strategyproof cost sharing: The role of indifferences

Every agent reports his willingness to pay for one unit of a good. A mechanism allocates goods and cost shares to some agents. We characterize the group strategyproof (GSP) mechanisms under two alternative continuity conditions interpreted as tie-breaking rules. With the maximalist rule (MAX) an indifferent agent is always served. With the minimalist rule (MIN) an indifferent agent does not get a unit of the good.GSP and MAX characterize the cross-monotonic mechanisms. These mechanisms are appropriate whenever symmetry is required. On the other hand, GSP and MIN characterize the sequential mechanisms. These mechanisms are appropriate whenever there is scarcity of the good.Our results are independent of an underlying cost function; they unify and strengthen earlier results for particular classes of cost functions.

Automatically generating personalized user interfaces with Supple

Today's computer–human interfaces are typically designed with the assumption that they are going to be used by an able-bodied person, who is using a typical set of input and output devices, who has typical perceptual and cognitive abilities, and who is sitting in a stable, warm environment. Any deviation from these assumptions may drastically hamper the person's effectiveness—not because of any inherent barrier to interaction, but because of a mismatch between the person's effective abilities and the assumptions underlying the interface design. We argue that automatic personalized interface generation is a feasible and scalable solution to this challenge. We present our Supple system, which can automatically generate interfaces adapted to a person's devices, tasks, preferences, and abilities. In this paper we formally define interface generation as an optimization problem and demonstrate that, despite a large solution space (of up to 10 17 possible interfaces), the problem is computationally feasible. In fact, for a particular class of cost functions, Supple produces exact solutions in under a second for most cases, and in a little over a minute in the worst case encountered, thus enabling run-time generation of user interfaces. We further show how several different design criteria can be expressed in the cost function, enabling different kinds of personalization. We also demonstrate how this approach enables extensive user- and system-initiated run-time adaptations to the interfaces after they have been generated. Supple is not intended to replace human user interface designers—instead, it offers alternative user interfaces for those people whose devices, tasks, preferences, and abilities are not sufficiently addressed by the hand-crafted designs. Indeed, the results of our study show that, compared to manufacturers' defaults, interfaces automatically generated by Supple significantly improve speed, accuracy and satisfaction of people with motor impairments.

A Linear Programming Approach to Parameter Fitting for the Master Equation

This technical note proposes a new framework for the design of continuous time, finite state space Markov processes. In particular, we propose a paradigm for selecting an optimal matrix within a pre-specified pencil of transition rate matrices. Given any transition rate matrix specifying the time-evolution of the Markov process, we propose a class of figures of merit that upper-bounds the long-term evolution of any statistical moment. We show that optimization with respect to the aforementioned class of cost functions is tractable via dualization and linear programming methods. In addition, we suggest how this approach can be used as a tool for the sub-optimal design of the master equation, with performance guarantees. Our results are applied to illustrative examples.

Sorting by transpositions: dealing with length-weighted models

For the first time, we study the sorting of permutations by length-weighted transpositions under a wide class of cost functions, namely f(l)=l(alpha), where l is the length of the transposition. For different alpha, we give corresponding upper and lower bounds of the cost of sorting any binary sequences or any permutations. Furthermore, an O(log n)-approximation algorithm and an exact algorithm are given to determine the optimal transposition series of sorting a permutation of length n when 1< alpha < 2 and alpha > or =2 respectively. Our work poses some interesting questions to both biologists and computer scientists and suggests some new bioinformatic insights that are currently being studied.

OPTIMAL ORGANIZATIONAL HIERARCHIES IN FIRMS

One of the main problems in modern economy is to construct an efficient organizational hierarchy allowing to control the firm with minimal cost. This paper describes the mathematical model of optimal hierarchies in firms. Optimal hierarchies for several classes of cost functions are obtained. Particularly, sufficient conditions for tree optimality, 2hier‐archy (any manager has two immediate subordinates) optimality and two‐tier hierarchy optimality are defined.