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The global 6G vision has taken its shape after years of international research and development efforts. This work culminated in ITU-R Recommendation on "IMT-2030 Framework". While the definition phase of technological requirements is currently ongoing, 3GPP standardization process on 6G networks is expected to start in 2025 and worldwide commercialization around 2029-2030. This article serves as a comprehensive guide to 6G by providing an overall vision, a contemporary survey of the main literature, and an informative tutorial-type presentation style. In our vision, 6G will be based on three fundamental elements: wireless, artificial intelligence, and Internet of Everything. Consequently, 6G can ultimately become the Intelligent Network of Everything while serving as an enabling platform for the next major disruption in mobile communication, called mobile intelligence. The potential of mobile intelligence is that anything can be made connected, intelligent, and aware of its environment. This will revolutionize the way how devices, systems, and applications are designed; how they operate and interact with humans and each other; and how they can be used for the benefit of people, society, and the world in general. After high-level visioning, the main details of 6G are discussed, including fundamental elements, disruptive applications, key use cases, main performance requirements, potential technologies, and defining features. A special focus is given to a comprehensive set of potential 6G technologies, each of which is introduced in a tutorial manner. Finally, we speculate on what comes after 6G and sketch the first high-level vision of 7G. All in all, the objective of this article is to provide a thorough guide to 6G in order to serve as a source of knowledge and inspiration for further research and development work in academia, industry, and standardization bodies., Comment: Accepted for publication in IEEE Access, 101 pages, 50 figures, 16 tables

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This paper develops a general framework for stochastic modeling of goals and other events in football (soccer) matches. The events are modelled as Cox processes (doubly stochastic Poisson processes) where the event intensities may depend on all the modeled events as well as external factors. The model has a strictly concave log-likelihood function which facilitates its fitting to observed data. Besides event times, the model describes the random lengths of stoppage times which can have a strong influence on the final score of a match. The model is illustrated on eight years of data from Campeonato Brasileiro de Futebol S\'erie A. We find that dynamic regressors significantly improve the in-game predictive power of the model. In particular, a) when a team receives a red card, its goal intensity decreases more than 30%; b) the goal rate of a team increases by 10% if it is losing by one goal and by 20% if its losing by two goals; and c) when the goal difference at the end of the second half is less than or equal to one, the stoppage time is on average more than one minute longer than in matches with a difference of two goals.

This paper characterizes the actual science performance of the James Webb Space Telescope (JWST), as determined from the six month commissioning period. We summarize the performance of the spacecraft, telescope, science instruments, and ground system, with an emphasis on differences from pre-launch expectations. Commissioning has made clear that JWST is fully capable of achieving the discoveries for which it was built. Moreover, almost across the board, the science performance of JWST is better than expected; in most cases, JWST will go deeper faster than expected. The telescope and instrument suite have demonstrated the sensitivity, stability, image quality, and spectral range that are necessary to transform our understanding of the cosmos through observations spanning from near-earth asteroids to the most distant galaxies., Comment: 5th version as accepted to PASP; 31 pages, 18 figures; https://iopscience.iop.org/article/10.1088/1538-3873/acb293

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control and financial mathematics. Our analysis builds on the theory of Fr\'echet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models e.g.\ on stochastic optimal control, portfolio optimization and mathematical programming.

This paper studies duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in 1976. We derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new frameworks in stochastic programming and stochastic optimal control.

This paper studies the dynamic programming principle for general convex stochastic optimization problems introduced by Rockafellar and Wets in [30]. We extend the applicability of the theory by relaxing compactness and boundedness assumptions. In the context of financial mathematics, the relaxed assumption are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. Besides financial mathematics, we obtain several new results in linear and nonlinear stochastic programming and stochastic optimal control.

To satisfy the capacity requirements of future mobile systems, under-utilized millimeter wave frequencies can be efficiently exploited by employing massive MIMO technology with highly directive beamforming. Hybrid analog-digital beamforming has been recognised as a promising approach for large-scale MIMO implementations with a reduced number of costly and power-hungry RF chains. In comparison to fully connected architecture, hybrid beamforming (HBF) with partially connected RF architecture is particularly appealing for the practical implementation due to less complex RF power division and combining networks. In this paper, we first formulate single- and multi-user rate maximization problems as weighted minimum mean square error (WMMSE) and derive solutions for hybrid beamformers using alternating optimization. The algorithms are designed for the full-array- and sub-array-based processing strategies of partially connected HBF architecture. In addition to the rate maximizing WMMSE solutions, we propose lower complexity sub-array-based zero-forcing algorithms. The performance of the proposed algorithms is evaluated in two different channel models, i.e., a simple geometric model and a realistic statistical millimeter wave model known as NYUSIM. The performance results of the WMMSE HBF algorithms are meant to reveal the potential of partially connected HBF and serve as upper bounds for lower complexity methods. Numerical results imply that properly designed partially connected HBF has the potential to provide an good compromise between hardware complexity and system performance in comparison to fully digital beamforming., Comment: 13 pages

This paper studies topological duals of Banach function spaces (BFS). We assume a finite measure but our arguments extend to general locally convex function spaces whose topology is generated by seminorms that satisfy the usual BFS axioms. The dual is identified with the direct sum of another space of random variables (K\"othe dual), a space of purely finitely additive measures and the annihilator of $L^\infty$. In the special case of rearrangement invariant spaces, the second component in the dual vanishes and we obtain various classical as well as new duality results e.g. on Lebesgue, Orlicz, Lorentz-Orlicz spaces and spaces of finite moments. Beyond rearrangement invariant spaces, we find the topological duals of Musielak-Orlicz spaces and those associated with general convex risk measures.

We study indifference pricing of exotic derivatives by using hedging strategies that take static positions in quoted derivatives but trade the underlying and cash dynamically over time. We use real quotes that come with bid-ask spreads and finite quantities. Galerkin method and integration quadratures are used to approximate the hedging problem by a finite dimensional convex optimization problem which is solved by an interior point method. The techniques are extended also to situations where the underlying is subject to bid-ask spreads. As an illustration, we compute indifference prices of path-dependent options written on S&P500 index. Semi-static hedging improves considerably on the purely static options strategy as well as dynamic trading without options. The indifference prices make good economic sense even in the presence of arbitrage opportunities that are found when the underlying is assumed perfectly liquid. When transaction costs are introduced, the arbitrage opportunities vanish but the indifference prices remain almost unchanged., Comment: 25 pages, 21 figures

This paper describes a general approach for stochastic modeling of assets returns and liability cash-flows of a typical pensions insurer. On the asset side, we model the investment returns on equities and various classes of fixed-income instruments including short- and long-maturity fixed-rate bonds as well as index-linked and corporate bonds. On the liability side, the risks are driven by future mortality developments as well as price and wage inflation. All the risk factors are modeled as a multivariate stochastic process that captures the dynamics and the dependencies across different risk factors. The model is easy to interpret and to calibrate to both historical data and to forecasts or expert views concerning the future. The simple structure of the model allows for efficient computations. The construction of a million scenarios takes only a few minutes on a personal computer. The approach is illustrated with an asset-liability analysis of a defined benefit pension fund.

This paper proposes a simple descriptive model of discrete-time double auction markets for divisible assets. As in the classical models of exchange economies, we consider a finite set of agents described by their initial endowments and preferences. Instead of the classical Walrasian-type market models, however, we assume that all trades take place in a centralized double auction where the agents communicate through sealed limit orders for buying and selling. We find that, under nonstrategic bidding, the double auction clears with zero trades precisely when the agents' current holdings are on the Pareto frontier. More interestingly, the double auctions implement Adam Smith's "invisible hand" in the sense that, when starting from disequilibrium, repeated double auctions lead to a sequence of allocations that converges to individually rational Pareto allocations.

This paper proves the existence of optimal stopping times via elementary functional analytic arguments. The problem is first relaxed into a convex optimization problem over a closed convex subset of the unit ball of the dual of a Banach space. The existence of optimal solutions then follows from the Banach--Alaoglu compactness theorem and the Krein--Millman theorem on extreme points of convex sets. This approach seems to give the most general existence results known to date. Applying convex duality to the relaxed problem gives a dual problem and optimality conditions in terms of martingales that dominate the reward process., Comment: 12 pages

This paper studies the energy efficiency and sum rate trade-off for coordinated beamforming in multi-cell multi-user multigroup multicast multiple-input single-output systems. We first consider a conventional network energy efficiency maximization (EEmax) problem by jointly optimizing the transmit beamformers and antennas selected to be used in transmission. We also account for per-antenna maximum power constraints to avoid non-linear distortion in power amplifiers and user-specific minimum rate constraints to guarantee certain service levels and fairness. To be energy-efficient, transmit antenna selection is employed. It eventually leads to a mixed-Boolean fractional program. We then propose two different approaches to solve this difficult problem. The first solution is based on a novel modeling technique that produces a tight continuous relaxation. The second approach is based on sparsity-inducing method, which does not require the introduction of any Boolean variable. We also investigate the trade-off between the energy efficiency and sum rate by proposing two different formulations. In the first formulation, we propose a new metric that is the ratio of the sum rate and the so-called weighted power. Specifically, this metric reduces to EEmax when the weight is 1, and to sum rate maximization when the weight is 0. In the other method, we treat the trade-off problem as a multi-objective optimization for which a scalarization approach is adopted. Numerical results illustrate significant achievable energy efficiency gains over the method where the antenna selection is not employed. The effect of antenna selection on the energy efficiency and sum rate trade-off is also demonstrated., Comment: Accepted for publication in the IEEE Transactions on Signal Processing (17 pages, 14 figures)

We develop a model for indifference pricing in derivatives markets where price quotes have bid-ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investor's beliefs, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimisation allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between super- and subhedging prices., Comment: 19 pages, 12 figures

This article characterizes topological duals of spaces of cadlag processes. We obtain extensions of functional analytic results of Dellacherie and Meyer that underlie many fundamental results in stochastic analysis. In particular, we obtain a characterization of the dual of cadlag processes of class $(D)$ in terms of optional measures of essentially bounded variation. When specialized to regular processes, we find extensions of the main result of Bismut \cite{bis78} on projections of continuous processes. The dual characterizations yield existence results and optimality conditions for many fundamental problems from optimal stopping to singular stochastic control well beyond classical formulations.

This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalizations., Comment: 31 pages

This paper considers coordinated multicast beamforming in a multi-cell multigroup multiple-input single-output system. Each base station (BS) serves multiple groups of users by forming a single beam with common information per group. We propose centralized and distributed beamforming algorithms for two different optimization targets. The first objective is to minimize the total transmission power of all the BSs while guaranteeing the user-specific minimum quality-of-service targets. The semidefinite relaxation (SDR) method is used to approximate the non-convex multicast problem as a semidefinite program (SDP), which is solvable via centralized processing. Subsequently, two alternative distributed methods are proposed. The first approach turns the SDP into a two-level optimization via primal decomposition. At the higher level, inter-cell interference powers are optimized for fixed beamformers while the lower level locally optimizes the beamformers by minimizing BS-specific transmit powers for the given inter-cell interference constraints. The second distributed solution is enabled via an alternating direction method of multipliers, where the inter-cell interference optimization is divided into a local and a global optimization by forcing the equality via consistency constraints. We further propose a centralized and a simple distributed beamforming design for the signal-to-interference-plus-noise ratio (SINR) balancing problem in which the minimum SINR among the users is maximized with given per-BS power constraints. This problem is solved via the bisection method as a series of SDP feasibility problems. The simulation results show the superiority of the proposed coordinated beamforming algorithms over traditional non-coordinated transmission schemes, and illustrate the fast convergence of the distributed methods., Comment: Accepted for publication in the IEEE Transactions on Signal Processing, 14 pages, 10 figures

This paper studies energy-efficient coordinated beamforming in multi-cell multi-user multigroup multicast multiple-input single-output systems. We aim at maximizing the network energy efficiency by taking into account the fact that some of the radio frequency chains can be switched off in order to save power. We consider the antenna specific maximum power constraints to avoid non-linear distortion in power amplifiers and user-specific quality of service (QoS) constraints to guarantee a certain QoS levels. We first introduce binary antenna selection variables and use the perspective formulation to model the relation between them and the beamformers. Subsequently, we propose a new formulation which reduces the feasible set of the continuous relaxation, resulting in better performance compared to the original perspective formulation based problem. However, the resulting optimization problem is a mixed-Boolean non-convex fractional program, which is difficult to solve. We follow the standard continuous relaxation of the binary antenna selection variables, and then reformulate the problem such that it is amendable to successive convex approximation. Thereby, solving the continuous relaxation mostly results in near-binary solution. To recover the binary variables from the continuous relaxation, we switch off all the antennas for which the continuous values are smaller than a small threshold. Numerical results illustrate the superior convergence result and significant achievable gains in terms of energy efficiency with the proposed algorithm., Comment: 6 pages, 5 figures, accepted to IEEE ICC 2017 - International Workshop on 5G RAN Design

This article characterizes conjugates and subdifferentials of convex integral functionals over the linear space $\mathcal N^\infty$ of stochastic processes of essentially bounded variation (BV) when $\mathcal N^\infty$ is identified with the Banach dual of the space of regular processes. Our proofs are based on new results on the interchange of integration and minimization of integral functionals over BV processes. Under mild conditions, the domain of the conjugate is shown to be contained in the space of semimartingales which leads to several applications in the duality theory in stochastic control and mathematical finance., Comment: 19 pages

This paper studies convex duality in optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into tree terms, corresponding to the agent's risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid when the market model satisfies an appropriate generalization of the no-arbitrage condition and the agent's utility function satisfies an appropriate generalization of asymptotic elasticity conditions. When applied to classical liquid market models or models with bid-ask spreads, we recover well-known pricing formulas in terms of martingale measures and consistent price systems. Building on the general theory of convex stochastic optimization, we also derive optimality conditions in terms of an extended notion of a "shadow price"., Comment: 37 pages

The shadow price of information has played a central role in stochastic optimization ever since its introduction by Rockafellar and Wets in the mid-seventies. This article studies the concept in an extended formulation of the problem and gives relaxed sufficient conditions for its existence. We allow for general adapted decision strategies, which enables one to establish the existence of solutions and the absence of a duality gap e.g. in various problems of financial mathematics where the usual boundedness assumptions fail. As applications, we calculate conjugates and subdifferentials of integral functionals and conditional expectations of normal integrands. We also give a dual form of the general dynamic programming recursion that characterizes shadow prices of information., Comment: 17 pages

This paper considers coordinated multicast beamforming in a multi-cell wireless network. Each multiantenna base station (BS) serves multiple groups of single antenna users by generating a single beam with common data per group. The aim is to minimize the sum power of BSs while satisfying user-specific SINR targets. We propose centralized and distributed multicast beamforming algorithms for multi-cell multigroup systems. The NP-hard multicast problem is tackled by approximating it as a convex problem using the standard semidefinite relaxation method. The resulting semidefinite program (SDP) can be solved via centralized processing if global channel knowledge is available. To allow a distributed implementation, the primal decomposition method is used to turn the SDP into two optimization levels. The higher level is in charge of optimizing inter-cell interference while the lower level optimizes beamformers for given inter-cell interference constraints. The distributed algorithm requires local channel knowledge at each BS and scalar information exchange between BSs. If the solution has unit rank, it is optimal for the original problem. Otherwise, the Gaussian randomization method is used to find a feasible solution. The superiority of the proposed algorithms over conventional schemes is demonstrated via numerical evaluation., Comment: Submitted to IEEE ICC 2016

This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be identified with the space of optional Radon measures with essentially bounded variation. Combined with classical Banach space techniques, our results allow for a systematic treatment of stochastic optimization problems over BV processes and, in particular, yields a maximum principle for a general class of singular stochastic control problems., Comment: 31 pages

This article studies convex duality in stochastic optimization over finite discrete-time. The first part of the paper gives general conditions that yield explicit expressions for the dual objective in many applications in operations research and mathematical finance. The second part derives optimality conditions by combining general saddle-point conditions from convex duality with the dual representations obtained in the first part of the paper. Several applications to stochastic optimization and mathematical finance are given., Comment: 17 pages

This paper introduces a kinetic code that simulates gamma-ray burst (GRB) afterglow emission from the external forward shock and presents examples of some of its applications. One interesting research topic discussed in the paper is the high-energy radiation produced by Compton scattering of the prompt GRB photons against the shock-accelerated electrons. The difference between the forward shock emission in a wind-type and a constant-density medium is also studied, and the emission due to Maxwellian electron injection is compared to the case with pure power-law electrons. The code calculates the time-evolving photon and electron distributions in the emission region by solving the relativistic kinetic equations for each particle species. For the first time, the full relativistic equations for synchrotron emission/absorption, Compton scattering, and pair production/annihilation were applied to model the forward shock emission. The synchrotron self-absorption thermalization mechanism, which shapes the low-energy end of the electron distribution, was also included in the electron equation. The simulation results indicate that inverse Compton scattering of the prompt GRB photons can produce a luminous TeV emission component, even when pair production in the emission region is taken into account. This very high-energy radiation may be observable in low-redshift GRBs. The test simulations also show that the low-energy end of a pure power-law distribution of electrons can thermalize owing to synchrotron self-absorption in a wind-type environment, but without an observable impact on the radiation spectrum. Moreover, a flattening in the forward shock X-ray light curve may be expected when the electron injection function is assumed to be purely Maxwellian instead of a power law., Comment: 16 pages, 11 figures, accepted for publication in A&A

This paper studies convex problems of Bolza in the conjugate duality framework of Rockafellar. We parameterize the problem by a general Borel measure which has direct economic interpretation in problems of financial economics. We derive a dual representation for the optimal value function in terms of continuous dual arcs and we give conditions for the existence of solutions. Combined with well-known results on problems of Bolza over absolutely continuous arcs, we obtain optimality conditions in terms of extended Hamiltonian conditions.

This paper studies optimal investment from the point of view of an investor with longevity-linked liabilities. The relevant optimization problems rarely are analytically tractable, but we are able to show numerically that liability driven investment can significantly outperform common strategies that do not take the liabilities into account. In problems without liabilities the advantage disappears, which suggests that the superiority of the proposed strategies is indeed based on connections between liabilities and asset returns.