Concave-convex Function Research Articles

IRS-Based Energy Efficiency and Admission Control Maximization for IoT Users With Short Packet Lengths

In this paper, we study a multiuser multiple-input single-output (MISO) machine type communication (MTC)-enabled intelligent reflecting surface (IRS) system, where a multi-antenna access point (AP) transmits information symbols to a set of Internet of Things (IoT) users with short packet transmission. In particular, the total energy efficiency (EE) and the number of IoT users that could be served fairly are maximized by jointly optimizing active and passive beamformers. An efficient algorithm based on alternating optimization (AO) is proposed to solve the main optimization problem iteratively. To this end, we adopt the difference of convex functions (DC) and successive convex approximation (SCA) to make a concave-convex function. Then, we employ the fractional programming based on the quadratic form to obtain a sub-optimal solution for the active beamformers at the AP and the number of admitted users. In the passive beamforming case, a penalty-based approach is utilized together with the SCA technique to handle the unit-modulus constraints at the IRS. Simulation results unveil an interesting tradeoff between EE and user admissibility performance. Besides, the results show the effectiveness of the IRS deployment in improving EE and successfully admitted users.

Splitting games over finite sets

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales \(\{p_t,q_t\}_t\), in order to control a terminal payoff \(u(p_\infty ,q_\infty )\). A first part introduces the notion of “Mertens–Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens–Zamir system for continuous functions on the square \([0,1]^2\). A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237–1257, 2020), we show that the value exists by constructing non Markovian \(\varepsilon \)-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.

Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems

<p style='text-indent:20px;'>We prove the existence of a bounded positive solution for the following stationary Schrödinger equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a vanishing potential and <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> has a sublinear growth at the origin (for example if <inline-formula><tex-math id="M3">\begin{document}$ f(x,u) $\end{document}</tex-math></inline-formula> is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [<xref ref-type="bibr" rid="b6">6</xref>]. In addition, if <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type <inline-formula><tex-math id="M5">\begin{document}$ \rho(x)f(u) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a concave-convex function and <inline-formula><tex-math id="M7">\begin{document}$ \rho $\end{document}</tex-math></inline-formula> satisfies the <inline-formula><tex-math id="M8">\begin{document}$ \mathrm{(H)} $\end{document}</tex-math></inline-formula> property introduced in [<xref ref-type="bibr" rid="b6">6</xref>]. We also note that we do not impose any integrability assumptions on the function <inline-formula><tex-math id="M9">\begin{document}$ \rho $\end{document}</tex-math></inline-formula>, which is imposed in most works.

Stability and Instability in Saddle Point Dynamics—Part I

We consider the problem of convergence to a saddle point of a concave-convex function via gradient dynamics. Since first introduced by Arrow, Hurwicz, and Uzawa, such dynamics have been extensively used in diverse areas; there are, however, features that render their analysis nontrivial. These include the lack of convergence guarantees when the concave-convex function considered does not satisfy additional strictness properties and also the nonsmoothness of subgradient dynamics. Our aim in this two-part article is to provide an explicit characterization to the asymptotic behavior of general gradient and subgradient dynamics applied to a general concave-convex function in C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . We show that despite the nonlinearity and nonsmoothness of these dynamics, their ω-limit set is comprised of trajectories that solve only explicit linear ODEs characterized within this article. More precisely, in part I, an exact characterization is provided to the asymptotic behavior of unconstrained gradient dynamics. We also show that when convergence to a saddle point is not guaranteed, then the system behavior can be problematic, with arbitrarily small noise leading to an unbounded second moment for the magnitude of the state vector. In part II, we consider a general class of subgradient dynamics that restrict trajectories in an arbitrary convex domain, and show that when an equilibrium point exists, the limiting trajectories belong to a class of dynamics characterized in part I as linear ODEs. These results are used to formulate corresponding convergence criteria and are demonstrated with examples.

Stability and Instability in Saddle Point Dynamics Part II: The Subgradient Method

In part I we considered the problem of convergence to a saddle point of a concave-convex function in C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> via gradient dynamics and an exact characterization was given to their asymptotic behavior. In part II we consider a general class of subgradient dynamics that provide a restriction in a convex domain. We show that despite the nonlinear and nonsmooth character of these dynamics their ω-limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.

Spectral Efficiency Optimization of Spatially-Correlated Multi-Pair Full-Duplex Massive MIMO Relaying

We consider a multi-pair two-way full-duplex (FD) amplify-and-forward relaying, where multiple FD multi-input multi-output (MIMO) users exchange information via a shared FD massive MIMO relay. We derive a closed-form spectral efficiency (SE) lower-bound for maximal-ratio combining/maximal-ratio transmission relay processing considering spatially correlated relay and user antennas, which most of the existing works have ignored. The derived SE lower-bound is applicable for arbitrary number of relay antennas, and simplifies to the following results available in the literature i) the asymptotic SE for number of relay antennas tending to infinity; and ii) the SE lower-bound with independent relay antennas and single-antenna users. We use this SE lower-bound to maximize the non-convex SE, which is of matrix fractional form. We optimize SE by approximating it as concave-convex function, and then by applying matrix quadratic transformation. We numerically validate the SE lower-bound for various system configurations and also characterize their performance for different spatial correlation and interference values. We investigate the gains achieved by the optimal SE over equal power allocation.

Critical quasilinear elliptic problems using concave–convex nonlinearities

It is established existence, multiplicity, and asymptotic behavior of nonnegative solutions for a quasilinear elliptic problem driven by the $$\Phi $$ -Laplacian operator. One of these solutions is obtained as ground-state solution by applying the well-known Nehari method. The nonlinear term is a concave–convex function which presents a critical behavior at infinity. The concentration-compactness principle is used in order to recover the compactness required in variational methods.

Stability Analysis and Control Design of Discrete-Time Switched Affine Systems

This technical note focuses on stability analysis and control design of switched affine systems in discrete-time domain. The stability conditions are obtained by taking into account that the system trajectories, governed by a certain switching function, converge to a set of attraction ${\boldsymbol{\mathcal V}}$ containing a desired equilibrium point. These conditions follow from the adoption of a general quadratic Lyapunov function whose time variation is bounded above by a concave-convex function with center determined by minimax theory. Our main contribution is to provide a stabilizing state feedback switching function and an invariant set of attraction ${\boldsymbol{\mathcal V}}_*$ with minimum volume as far as this class of quadratic Lyapunov functions is adopted. The results are applied to sampled-data control of continuous-time switched affine systems with chattering avoidance. The speed control of a DC motor is presented.

An Excel based approach to optimise trade-offs between freight expediting and safety stock inventory costs

In this paper, we present an Excel based methodology for the computation of two optimal policy quantities, order size and reorder point of a (Q, r) inventory system under uncertainty when backorder costs are independent of the amount of the backorder and dependent on whether there is a shortage. It is shown that the total cost curve for the backorder model is a concave-convex function. Solving the problem using a spreadsheet solver would not necessarily yield the optimal solution. It is first necessary to identify the convex and concave domains prior to the application of Excel Solver to determine the optimal solution of the model. This approach is flexible and capable of handling different lead time demand distributions as well as the more realistic scenario when both the demand and lead times are randomly distributed. The methodology is illustrated with a simple problem.

Optimal licensing policy for an incumbent firm with fixed cost advantage

This paper primarily focuses on two problems which aim at improving the results of the recent work of Yi S. S. (Yi S. S., Entry, licensing and research joint ventures, International Journal of Industrial Organization, 1999,17:1–24). Firstyly, two improper viewpoints are pointed out and rectified. Under his model, Yi argued that the potential entrants can be better off or worse off under the incumbent firm’s strategic use of ex-post licensing policy, but it is shown in this paper that the strategic licensing undoubtedly reduces the entrants’ expected payoffs. Yi also noted wrongly that industry profit is a convex function of the number of firms in the product market by the Cournot oligopoly with a linear demand and constant marginal costs. However, in this paper it is demonstrated that industry profit in such a market is a concave-convex function of the number of firms. Secondly, a thorough study of the optimal licensing policy for the incumbent is made on the basis of the correction of the second error. Necessary and sufficient conditions under which the incumbent will choose ex-post licensing, ex-ante licensing of one entrant or ex-ante licensing of both extrants are given.

One Optimal Control Method in Certain Class of Problems

Abstract The determination of the optimal control vector of parameters for a many-step-type game problem is considered in the multi-di-mentional case. The method consists of the system of two algorithms, The first one represents the iteration procedure to find a solution for any many-step-type game problem if the criterion, intended for the optimization, is a strictly concave-convex function of the control parameters. At the same time any method may be employed to find the optimal control in the non-game problem. It is proved, that the proposed algorithm is a convergent one. Another algorithm represents the procedure intended for the optimal control of the multi-dimentional non-game problem determination. It is supposed in this case, that the criterion represents a strictly concave or convex control parameters function, assuming the linearization in any point; restrictions, that are imposed upon, must be linear ones. The method is based on the criterion function linearization and the optimal control foundation by the linear programming method with the nonlinearity correction as the following one.

On the sign of the solutions to some semilinear Dirichlet problems

SynopsisWe study the sign of solutions to a class of semilinear Dirichlet problems when the nonlinearity is, for instance, a concave-convex function which interacts with the spectrum of the linear part. We are able to prove, in some cases, exact multiplicity results for positive and negative solutions. For the proofs we employ a device which splits the given problem into two others satisfying a suitable version of the Ambrosetti–Prodi result.

Polyextremal principles and separably-infinite programs

As a direct extension of Charnes' characterization of two-person zero-sum constrained games by linear programming, we show how a general class of saddle value problems can be reduced to a pair of uniextremal dual separably-infinite programs. These programs have an infinite number of variables and an infinite number of constraints, but only a finite number of variables appear in an infinite number of constraints and only a finite number of constraints have an infinite number of variables. The conditions under which the characterization holds are among the more general ones appearing in the literature sufficient to guarantee the existence of a saddle point of a concave-convex function.

Hamiltonian Systems in a Neighborhood of a Saddle Point

The behavior of Hamiltonian differential systems associated with a concave convex function

Hamiltonian systems in a neighborhood of a saddle point

The behavior of Hamiltonian differential systems associated with a concave convex function H in a Hilbert space is studied by variational methods. It is shown that under quite general conditions on the function H the system behaves in a neighborhood of a minimax saddle point of H much like as in the classical theory of ordinary differential systems. The results extend previous work of R. T. Rockafellar.