Showing papers on "Concave function published in 2021"

Joint Resource, Trajectory, and Artificial Noise Optimization in Secure Driven 3-D UAVs With NOMA and Imperfect CSI

Abstract: Driven by the practicality of unmanned aerial vehicle (UAV), we consider a dual-UAV based non-orthogonal multiple access (NOMA) scenario, which consists of one communication UAV for services and one jamming UAV against eavesdropping. The goal is to maximize the secrecy energy efficiency through the successive convex approximation based communication resource, UAV trajectory, and artificial noise optimization. Considering the probabilistic constraint of outage probability from imperfect channel state information, we transform it to a non-probabilistic problem by Markov inequality and Marcum $Q$ -function, then the problem is decomposed into three subproblems. We apply matching-swapping method to assign subchannel in non-orthogonal multiple access (NOMA) UAV networks before the joint process, then convert the power optimization problem to a standard convex optimization form by upper bound of the concave function. The communication UAV trajectory is studied under the constraints of flying energy consumption, maximum speed, and flying altitude. To track this NP-hard problem, Taylor expansion and various slack variables sets are introduced to transform the non-convex problem to convex one. For the artificial noise optimization problem, we use the lower bound to replace the convex term turning it into an easy-to-solve convex optimization problem. In the end, simulations results reveal that: 1) The reasonable jamming scheme can improve the secrecy energy efficiency of the NOMA UAV networks, even if it can cause interference for legitimate users; 2) UAV will fly to a place where the performance gain from users is high when flying energy consumption permits.

Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions.

Abstract: 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 in [2]. In this study, firstly, a new identity by using Atangana-Baleanu fractional integral operators are 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

Fractional Integral Inequalities via Atangana-Baleanu Operators for Convex and Concave Functions

Abstract: 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.

The Loewner function of a log-concave function

Abstract: We introduce the notion of Lowner (ellipsoid) function for a log-concave function and show that it is an extension of the Lowner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid) function (Alonso-Gutierrez et al. in J Geom Anal 28:1182–1201, 2018). For convex bodies, John and Lowner ellipsoids are dual to each other. Interestingly, this need not be the case for the John function and the Lowner function.

A stochastic Prékopa–Leindler inequality for log-concave functions

Abstract: The Brunn–Minkowski and Prekopa–Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functio...

On sum sets of convex functions.

Abstract: In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$ This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to also prove the three-variable expansion result \[ |AB+A|\gtrsim |A|^{\frac32 +\frac3{170}}\,. \] Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, that enables us to find positive proportion subsets with certain desirable properties.

Concave elliptic equations and generalized Khovanskii–Teissier inequalities

Abstract: We explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii-Teissier inequalities.

Smart Charging of Electric Vehicles Considering SOC-Dependent Maximum Charging Powers

Abstract: The aim of this work is to schedule the charging of electric vehicles (EVs) at a single charging station such that the temporal availability of each EV as well as the maximum available power at the station are considered. The total costs for charging the vehicles should be minimized w.r.t. time-dependent electricity costs. A particular challenge investigated in this work is that the maximum power at which a vehicle can be charged is dependent on the current state of charge (SOC) of the vehicle. Such a consideration is particularly relevant in the case of fast charging. Considering this aspect for a discretized time horizon is not trivial, as the maximum charging power of an EV may also change in between time steps. To deal with this issue, we instead consider the energy by which an EV can be charged within a time step. For this purpose, we show how to derive the maximum charging energy in an exact as well as an approximate way. Moreover, we propose two methods for solving the scheduling problem. The first is a cutting plane method utilizing a convex hull of the, in general, nonconcave SOC–power curves. The second method is based on a piecewise linearization of the SOC–energy curve and is effectively solved by branch-and-cut. The proposed approaches are evaluated on benchmark instances, which are partly based on real-world data. To deal with EVs arriving at different times as well as charging costs changing over time, a model-based predictive control strategy is usually applied in such cases. Hence, we also experimentally evaluate the performance of our approaches for such a strategy. The results show that optimally solving problems with general piecewise linear maximum power functions requires high computation times. However, problems with concave, piecewise linear maximum charging power functions can efficiently be dealt with by means of linear programming. Approximating an EV’s maximum charging power with a concave function may result in practically infeasible solutions, due to vehicles potentially not reaching their specified target SOC. However, our results show that this error is negligible in practice.

Resource Allocation for Energy-Efficient NOMA System in Coordinated Multi-Point Networks

Abstract: This paper studies user scheduling and power allocation problem to maximize the energy efficiency (EE) for non-orthogonal multiple access (NOMA) in downlink Coordinated Multi-Point networks In the proposed framework, a more practical scenario the imperfect channel state information, imperfect successive interference cancellation and data outage are investigated To address the considered problem, the optimization problem is formulated constrained by the total power and the outage probability requirements However, the EE objective function is with a non-convex structure Accordingly, we first convert the optimization problem to make the objective function concave and analytically tractable Furthermore, we split the joint optimization problem to find a suboptimal solutions to the original problem As a result, we first propose a suboptimal user-scheduling algorithm to improve the system's EE Due to the non-convex function of the transmit power, we invoke a sequential successive convex approach to address the non-convex problem by its lower bound concave function Besides, the fractional objective function is converted to its equivalent subtractive form Finally, we derive a power control scheme to address the proposed framework Simulation results endorse the effectiveness of the proposed algorithm and their performance gains in terms of EE compared to both NOMA and OFDMA variants

Nonlinear Unbalanced Urn Models via Stochastic Approximation

Abstract: This paper presents a link between unbalanced non-linear urn model (a two-colored urn model) and stochastic approximation theory. Findings of our study reveal a successful establishment of limit laws for the urn composition, obtained under a drawing rule reinforced by an $\mathbb {R}_{+}$ -valued concave function and a non-balanced replacement matrix.

Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions

Abstract: Dragomir introduced the Jensen-type inequality for harmonic convex functions (HCF) and Baloch et al. studied its different variants, such as Jensen-type inequality for harmonic - convex functions. In this paper, we aim to establish the functional form of inequalities presented by Baloch et al. and prove the superadditivity and monotonicity properties of these functionals. Furthermore, we derive the bound for these functionals under certain conditions. Furthermore, we define more generalized functionals involving monotonic nondecreasing concave function as well as evince superadditivity and monotonicity properties of these generalized functionals.

Accelerated proximal stochastic variance reduction for DC optimization

Abstract: In this article, we study an important class of stochastic difference-of-convex (SDC) programming whose objective is given in the form of the sum of a continuously differentiable convex function, a simple convex function and a continuous concave function. Recently, a proximal stochastic variance reduction difference-of-convex algorithm (Prox-SVRDCA) (Xu et al., 2019) is developed for this problem. And, Prox-SVRDCA reduces to the proximal stochastic variance reduction gradient (Prox-SVRG) (Xiao and Zhang, 2014) as the continuous concave function is disappeared, and hence Prox-SVRDCA is potentially slow in practice. Inspired by recently proposed acceleration techniques, an accelerated proximal stochastic variance reduction difference-of-convex algorithm (AProx-SVRDCA) is proposed. Different from Prox-SVRDCA, an extrapolation acceleration step that involves the latest two iteration points is incorporated in AProx-SVRDCA. The experimental results show that, for a fairly general choice of the extrapolation parameter, the acceleration will be achieved for AProx-SVRDCA. Then, a rigorous theoretical analysis is presented. We first show that any accumulation point of the generated iteration sequences is a stationary point of the objective function. Furthermore, different from the traditional convergence analysis in the existing nonconvex stochastic optimizations, a global convergence property with respect to the generated sequences is established under the assumption: the Kurdyka-Łojasiewicz property together with the continuity and differentiability of the concave part in objective function. To the best of our knowledge, this is the first time that the acceleration trick is incorporated into nonconvex nonsmooth SDC programming. Finally, extended experimental results show the superiority of our proposed algorithm.

Geodesic length and shifted weights in first-passage percolation

Abstract: We study first-passage percolation through related optimization problems over paths of restricted length The path length variable is in duality with a shift of the weights This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift For points far enough away from the origin, the ratio of the geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded away from one The shape function is a strictly concave function of the weight shift Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case When the weight distribution has two or more atoms, a dense set of shifts produce singularities The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments

Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows

Abstract: A recent article (Li and Lv, J Geom Anal 30:417–447, 2020) considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.

Contraction of convex hypersurfaces by nonhomogeneous functions of curvature

Abstract: A recent article Li and Lv (J. Geom. Anal. 30: 417–447, 2020) considered contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in certain cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article, we extend the result to various other cases that are analogous to those considered in other earlier work, and we show that in all cases, where sufficient pinching conditions are assumed on the initial hypersurface, then under suitable rescaling the final point is asymptotically round and convergence is exponential in the $$C^\infty $$ -topology.

Transport-majorization to analytic and geometric inequalities.

Abstract: We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the R\'enyi and Tsallis entropies of all orders among all strongly log-concave densities.

Functional portfolio optimization in stochastic portfolio theory

Abstract: In this paper we develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets, and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, is flexible as various regularizations and constraints can be imposed on the generating function. We prove that the optimization problem is well-posed and provide a stability estimate in terms of a Wasserstein metric of the input measure. We then give a careful treatment of its discretization and the optimization algorithm. Finally, we present empirical examples using CRSP data from the US stock market.

A Riesz representation theorem for log-concave functions

Abstract: The classic Riesz representation theorem characterizes all linear and increasing functionals on the space $C_{c}(X)$ of continuous compactly supported functions. A geometric version of this result, which characterizes all linear increasing functionals on the set of convex bodies in $\mathbb{R}^{n}$, was essentially known to Alexandrov. This was used by Alexandrov to prove the existence of mixed area measures in convex geometry. In this paper we characterize linear and increasing functionals on the class of log-concave functions on $\mathbb{R}^{n}$. Here "linear" means linear with respect to the natural addition on log-concave functions which is the sup-convolution. Equivalently, we characterize pointwise-linear and increasing functionals on the class of convex functions. For some choices of the exact class of functions we prove that there are no non-trivial such functionals. For another choice we obtain the expected analogue of the result for convex bodies. And most interestingly, for yet another choice we find a new unexpected family of such functionals. Finally, we explain the connection between our results and recent work done in convex geometry regarding the surface area measure of a log-concave functions. An application of our results in this direction is also given.

Concentration Inequalities for Ultra Log-Concave Distributions

Abstract: We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).

On mutually inverse transforms of functions on a half-line

Abstract: Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is the identity transform on the class of nonnegative concave functions. Applications of this result to some problems of mathematical physics are indicated. Several open questions are formulated.

Joint Optimization of Multi-Objective Reinforcement Learning with Policy Gradient Based Algorithm.

Abstract: Many engineering problems have multiple objectives, and the overall aim is to optimize a non-linear function of these objectives. In this paper, we formulate the problem of maximizing a non-linear concave function of multiple long-term objectives. A policy-gradient based model-free algorithm is proposed for the problem. To compute an estimate of the gradient, a biased estimator is proposed. The proposed algorithm is shown to achieve convergence to within an $\epsilon$ of the global optima after sampling $\mathcal{O}(\frac{M^4\sigma^2}{(1-\gamma)^8\epsilon^4})$ trajectories where $\gamma$ is the discount factor and $M$ is the number of the agents, thus achieving the same dependence on $\epsilon$ as the policy gradient algorithm for the standard reinforcement learning.

Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints

Abstract: We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of $f(a^\top x)$, where $f$ is a univariate concave function, $a$ is a non-negative vector, and $x$ is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when $a$ has two distinct values. We show how to use these inequalities to obtain valid inequalities for general $a$ that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when $a$ contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework.

On Min-Max Affine Approximants of Convex or Concave Real-Valued Functions from \(\mathbb R^k\), Chebyshev Equioscillation and Graphics

Abstract: We study min-max affine approximants of a continuous convex or concave function \(f:\Delta \subseteq \mathbb R^k\xrightarrow {} \mathbb R\), where Δ is a convex compact subset of \(\mathbb R^k\). In the case when Δ is a simplex, we prove that there is a vertical translate of the supporting hyperplane in \(\mathbb R^{k+1}\) of the graph of f at the vertices which is the unique best affine approximant to f on Δ. For k = 1, this result provides an extension of the Chebyshev equioscillation theorem for linear approximants. Our result has interesting connections to the computer graphics problem of rapid rendering of projective transformations.

The Min-Cost Matching with Concave Delays Problem

Abstract: We consider the problem of online min-cost perfect matching with concave delays. We begin with the single location variant. Specifically, requests arrive in an online fashion at a single location. The algorithm must then choose between matching a pair of requests or delaying them to be matched later on. The cost is defined by a concave function on the delay. Given linear or even convex delay functions, matching any two available requests is trivially optimal. However, this does not extend to concave delays. We solve this by providing an $O(1)$-competitive algorithm that is defined through a series of delay counters. Thereafter we consider the problem given an underlying $n$-points metric. The cost of a matching is then defined as the connection cost (as defined by the metric) plus the delay cost. Given linear delays, this problem was introduced by Emek et al. and dubbed the Min-cost perfect matching with linear delays (MPMD) problem. Liu et al. considered convex delays and subsequently asked whether there exists a solution with small competitive ratio given concave delays. We show this to be true by extending our single location algorithm and proving $O(\log n)$ competitiveness. Finally, we turn our focus to the bichromatic case, wherein requests have polarities and only opposite polarities may be matched. We show how to alter our former algorithms to again achieve $O(1)$ and $O(\log n)$ competitiveness for the single location and for the metric case.