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This paper develops the geometry of rational functions on non-singular real algebraic varieties that are locally bounded. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of {\L}ojasiewicz's inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions., Comment: 24 pages

Centralized training for decentralized execution paradigm emerged as the state-of-the-art approach to epsilon-optimally solving decentralized partially observable Markov decision processes. However, scalability remains a significant issue. This paper presents a novel and more scalable alternative, namely sequential-move centralized training for decentralized execution. This paradigm further pushes the applicability of Bellman's principle of optimality, raising three new properties. First, it allows a central planner to reason upon sufficient sequential-move statistics instead of prior simultaneous-move ones. Next, it proves that epsilon-optimal value functions are piecewise linear and convex in sufficient sequential-move statistics. Finally, it drops the complexity of the backup operators from double exponential to polynomial at the expense of longer planning horizons. Besides, it makes it easy to use single-agent methods, e.g., SARSA algorithm enhanced with these findings applies while still preserving convergence guarantees. Experiments on two- as well as many-agent domains from the literature against epsilon-optimal simultaneous-move solvers confirm the superiority of the novel approach. This paradigm opens the door for efficient planning and reinforcement learning methods for multi-agent systems.

Soil properties, such as wetting collapse behavior and permeability, are strongly correlated to the soil microstructure. To date, several techniques including mercury intrusion porosimetry (MIP), can be used to characterize the microstructure of soil, but all techniques have their own limitations. In this study, the features of mercury that penetrated and has been entrapped in the pore network of the specimens through MIP testing were investigated by X-Ray microtomography (X-$\mu$CT), in order to give an insight into the geometry of macropores and possible ink-bottle geometry. Two conditions of water content and density were selected for the compacted Maryland clay. The distribution and geometry features of mercury entrapped in the microstructure after MIP were characterized and pore size distributions were also reconstructed. The results suggest that, for the two conditions studied in this paper, macropores were evenly distributed within the specimens, and most of them with a non-spherical shape, and with aspect ratio (ratio between the maximum and minimum thickness along a given segment) smaller than three. Different dominant entrance pore size of macropore was obtained from MIP and X-$\mu$CT, due to the specific experimental protocol used in tests and the effect of ink-bottle geometry. Only the large pore bodies with high aspect ratio were imaged in X-$\mu$CT, due to the extrusion of mercury during the process of depressurization and subsequent sample preparation for X- $\mu$CT. But entire pore space was accessible in MIP. The difference in dominant entrance pore size was more significant for specimens with lower void ratio due to a more pronounced aspect ratio.

The field of Contextual Optimization (CO) integrates machine learning and optimization to solve decision making problems under uncertainty. Recently, a risk sensitive variant of CO, known as Conditional Robust Optimization (CRO), combines uncertainty quantification with robust optimization in order to promote safety and reliability in high stake applications. Exploiting modern differentiable optimization methods, we propose a novel end-to-end approach to train a CRO model in a way that accounts for both the empirical risk of the prescribed decisions and the quality of conditional coverage of the contextual uncertainty set that supports them. While guarantees of success for the latter objective are impossible to obtain from the point of view of conformal prediction theory, high quality conditional coverage is achieved empirically by ingeniously employing a logistic regression differentiable layer within the calculation of coverage quality in our training loss. We show that the proposed training algorithms produce decisions that outperform the traditional estimate then optimize approaches.

A recent theory shows that a multi-player decentralized partially observable Markov decision process can be transformed into an equivalent single-player game, enabling the application of \citeauthor{bellman}'s principle of optimality to solve the single-player game by breaking it down into single-stage subgames. However, this approach entangles the decision variables of all players at each single-stage subgame, resulting in backups with a double-exponential complexity. This paper demonstrates how to disentangle these decision variables while maintaining optimality under hierarchical information sharing, a prominent management style in our society. To achieve this, we apply the principle of optimality to solve any single-stage subgame by breaking it down further into smaller subgames, enabling us to make single-player decisions at a time. Our approach reveals that extensive-form games always exist with solutions to a single-stage subgame, significantly reducing time complexity. Our experimental results show that the algorithms leveraging these findings can scale up to much larger multi-player games without compromising optimality.

Inverse optimization has been increasingly used to estimate unknown parameters in an optimization model based on decision data. We show that such a point estimation is insufficient in a prescriptive setting where the estimated parameters are used to prescribe new decisions. The prescribed decisions may be low-quality and misaligned with human intuition and thus are unlikely to be adopted. To tackle this challenge, we propose conformal inverse optimization, which seeks to learn an uncertainty set for the unknown parameters and then solve a robust optimization model to prescribe new decisions. Under mild assumptions, we show that our method enjoys provable guarantees on solution quality, as evaluated using both the ground-truth parameters and the decision maker's perception of the unknown parameters. Our method demonstrates strong empirical performance compared to classic inverse optimization.

Recently there has been a surge of interest in operations research (OR) and the machine learning (ML) community in combining prediction algorithms and optimization techniques to solve decision-making problems in the face of uncertainty. This gave rise to the field of contextual optimization, under which data-driven procedures are developed to prescribe actions to the decision-maker that make the best use of the most recently updated information. A large variety of models and methods have been presented in both OR and ML literature under a variety of names, including data-driven optimization, prescriptive optimization, predictive stochastic programming, policy optimization, (smart) predict/estimate-then-optimize, decision-focused learning, (task-based) end-to-end learning/forecasting/optimization, etc. Focusing on single and two-stage stochastic programming problems, this review article identifies three main frameworks for learning policies from data and discusses their strengths and limitations. We present the existing models and methods under a uniform notation and terminology and classify them according to the three main frameworks identified. Our objective with this survey is to both strengthen the general understanding of this active field of research and stimulate further theoretical and algorithmic advancements in integrating ML and stochastic programming.

The abundance of data has led to the emergence of a variety of optimization techniques that attempt to leverage available side information to provide more anticipative decisions. The wide range of methods and contexts of application have motivated the design of a universal unitless measure of performance known as the coefficient of prescriptiveness. This coefficient was designed to quantify both the quality of contextual decisions compared to a reference one and the prescriptive power of side information. To identify policies that maximize the former in a data-driven context, this paper introduces a distributionally robust contextual optimization model where the coefficient of prescriptiveness substitutes for the classical empirical risk minimization objective. We present a bisection algorithm to solve this model, which relies on solving a series of linear programs when the distributional ambiguity set has an appropriate nested form and polyhedral structure. Studying a contextual shortest path problem, we evaluate the robustness of the resulting policies against alternative methods when the out-of-sample dataset is subject to varying amounts of distribution shift.

Plasma ignition is critical in various scientific and industrial applications, demanding an efficient and robust execution mechanism. In this work, we present an innovative approach to plasma ignition by incorporating the analysis of fundamental aspects of light scattering in the complex frequency plane. For the first time, we demonstrate the high-power virtual perfect absorption (VPA) regime, a groundbreaking method for perfectly capturing light within a resonator. By carefully designing the temporal profile of the incident wave, we effectively minimize reflections during the ignition stages, thereby significantly enhancing the efficiency and resilience of the process. Through comprehensive experimental investigations, we validate the viability of this approach, establishing VPA as a powerful tool for reflectionless excitation and optimal control of plasma discharge. By addressing the limitations of conventional plasma ignition methods, this research represents a pivotal step towards transformative advancements in plasma technology, with promising implications for improving the performance and sustainability of numerous applications.

Optimizing static risk-averse objectives in Markov decision processes is difficult because they do not admit standard dynamic programming equations common in Reinforcement Learning (RL) algorithms. Dynamic programming decompositions that augment the state space with discrete risk levels have recently gained popularity in the RL community. Prior work has shown that these decompositions are optimal when the risk level is discretized sufficiently. However, we show that these popular decompositions for Conditional-Value-at-Risk (CVaR) and Entropic-Value-at-Risk (EVaR) are inherently suboptimal regardless of the discretization level. In particular, we show that a saddle point property assumed to hold in prior literature may be violated. However, a decomposition does hold for Value-at-Risk and our proof demonstrates how this risk measure differs from CVaR and EVaR. Our findings are significant because risk-averse algorithms are used in high-stake environments, making their correctness much more critical.

[Abridged] Aims. We provide an important step toward a better understanding of the magnetorotational instability (MRI)-dust coevolution in protoplanetary disks by presenting a proof of concept that dust evolution ultimately plays a crucial role in the MRI activity. Methods. First, we study how a fixed power-law dust size distribution with varying parameters impacts the MRI activity, especially the steady-state MRI-driven accretion, by employing and improving our previous 1+1D MRI-driven turbulence model. Second, we relax the steady-state accretion assumption in this disk accretion model, and partially couple it to a dust evolution model in order to investigate how the evolution of dust (dynamics and grain growth processes combined) and MRI-driven accretion are intertwined on million-year timescales. Results. Dust coagulation and settling lead to a higher gas ionization degree in the protoplanetary disk, resulting in stronger MRI-driven turbulence as well as a more compact dead zone. On the other hand, fragmentation has an opposite effect because it replenishes the disk in small dust particles. Since the dust content of the disk decreases over million years of evolution due to radial drift, the MRI-driven turbulence overall becomes stronger and the dead zone more compact until the disk dust-gas mixture eventually behaves as a grain-free plasma. Furthermore, our results show that dust evolution alone does not lead to a complete reactivation of the dead zone. Conclusions. The MRI activity evolution (hence the temporal evolution of the MRI-induced $\alpha$-parameter) is controlled by dust evolution and occurs on a timescale of local dust growth, as long as there is enough dust particles in the disk to dominate the recombination process for the ionization chemistry. Once it is no longer the case, it is expected to be controlled by gas evolution and occurs on a viscous evolution timescale., Comment: 23 pages, 13 figures, Accepted for publication in A&A

Earth-sized exoplanets that transit nearby, late spectral type red dwarfs will be prime targets for atmospheric characterization in the coming decade. Such systems, however, are difficult to find via wide-field transit surveys like Kepler or TESS. Consequently, the presence of such transiting planets is unexplored and the occurrence rates of short-period Earth-sized planets around late M dwarfs remain poorly constrained. Here, we present the deepest photometric monitoring campaign of 22 nearby late M dwarf stars, using data from over 500 nights on seven 1-2 meter class telescopes. Our survey includes all known single quiescent northern late M dwarfs within 15 pc. We use transit-injection-and-recovery tests to quantify the completeness of our survey, successfully identify most ($>80\%$) transiting short-period (0.5-1 d) super-Earths ($R > 1.9 R_\oplus$), and are sensitive ($\sim50\%$) to transiting Earth-sized planets ($1.0-1.2 R_\oplus$). Our high sensitivity to transits with a near-zero false positive rate demonstrates an efficient survey strategy. Our survey does not yield a transiting planet detection, yet it provides the most sensitive upper limits on transiting planets orbiting our target stars. Finally, we explore multiple hypotheses about the occurrence rates of short-period planets (from Earth-sized planets to giant planets) around late M dwarfs. We show, for example, that giant planets at short periods ($<1$ day) are uncommon around our target stars. Our dataset provides some insight into occurrence rates of short-period planets around TRAPPIST-1-like stars, and our results can help test planetary formation and system evolution models, as well as guide future observations of nearby late M dwarfs., Comment: 27 pages, 11 figures

We compute the deformation of the spinor fields of the super-Polyakov action leading to the so-called tensionless super-string theory. This allows us to obtain a satisfying Majorana condition in the tensionless limit, which is the main result of this paper. Our tensionless limit is shown to possess previously computed features of tensionless super-string theory., Comment: 18 pages

This research focuses on the bid optimization problem in the real-time bidding setting for online display advertisements, where an advertiser, or the advertiser's agent, has access to the features of the website visitor and the type of ad slots, to decide the optimal bid prices given a predetermined total advertisement budget. We propose a risk-aware data-driven bid optimization model that maximizes the expected profit for the advertiser by exploiting historical data to design upfront a bidding policy, mapping the type of advertisement opportunity to a bid price, and accounting for the risk of violating the budget constraint during a given period of time. After employing a Lagrangian relaxation, we derive a parametrized closed-form expression for the optimal bidding strategy. Using a real-world dataset, we demonstrate that our risk-averse method can effectively control the risk of overspending the budget while achieving a competitive level of profit compared with the risk-neutral model and a state-of-the-art data-driven risk-aware bidding approach., Comment: Accepted for CIKM '22

State-of-the-art methods for solving 2-player zero-sum imperfect information games rely on linear programming or regret minimization, though not on dynamic programming (DP) or heuristic search (HS), while the latter are often at the core of state-of-the-art solvers for other sequential decision-making problems. In partially observable or collaborative settings (e.g., POMDPs and Dec- POMDPs), DP and HS require introducing an appropriate statistic that induces a fully observable problem as well as bounding (convex) approximators of the optimal value function. This approach has succeeded in some subclasses of 2-player zero-sum partially observable stochastic games (zs- POSGs) as well, but how to apply it in the general case still remains an open question. We answer it by (i) rigorously defining an equivalent game to work with, (ii) proving mathematical properties of the optimal value function that allow deriving bounds that come with solution strategies, (iii) proposing for the first time an HSVI-like solver that provably converges to an $\epsilon$-optimal solution in finite time, and (iv) empirically analyzing it. This opens the door to a novel family of promising approaches complementing those relying on linear programming or iterative methods., Comment: 42 pages, 2 algorithms. arXiv admin note: substantial text overlap with arXiv:2110.14529

We propose a novel approach to solve K-adaptability problems with convex objective and constraints and integer first-stage decisions. A logic-based Benders decomposition is applied to handle the first-stage decisions in a master problem, thus the sub-problem becomes a min-max-min robust combinatorial optimization problem that is solved via a double-oracle algorithm that iteratively generates adverse scenarios and recourse decisions and assigns scenarios to K subsets of the decisions by solving p-center problems. Extensions of the proposed approach to handle parameter uncertainty in both the first-stage objective and the second-stage constraints are also provided. We show that the proposed algorithm converges to an optimal solution and terminates in finite number of iterations. Numerical results obtained from experiments on benchmark instances of the adaptive shortest path problem, the regular knapsack problem, and a generic K-adaptability problem demonstrate the performance advantage of the proposed approach when compared to state-of-the-art methods in the literature., Comment: 29 pages, 7 figures

Complex coupled thermo-hydromechanical (THM) loading paths are expected to occur in clay rocks which serve as host formations for geological radioactive waste repositories. Exothermic waste packages heat the rock, causing thermal strains and temperature induced pore pressure build-up. The drifts are designed in such a way as to limit these effects. One has to anticipate failure and fracturing of the material, should pore pressures exceed the tensile resistance of the rock. To characterise the behaviour of the Callovo-Oxfordian claystone (COx) under effective tension and to quantify the tensile failure criterion, a laboratory program is carried out in this work. THM loading paths which correspond to the expected in situ conditions are recreated in the laboratory. To this end, a special triaxial system was developed, which allows the independent control of radial and axial stresses, as well as of pore pressure and temperature of rock specimens. More importantly, the device allows one to maintain axial effective tension on a specimen. Saturated cylindrical claystone specimens were tested in undrained conditions under constrained lateral deformation and under nearly constant axial stress. The specimens were heated until the induced pore pressures created effective tensile stresses and ultimately fractured the material. The failure happened at average axial effective tensile stresses around 3.0 MPa. Fracturing under different lateral total stresses allows one to describe the failure with a Hoek-Brown or Fairhurst's generalized Griffith criterion. Measured axial extension strains are analysed based on a transversely isotropic thermo-poroelastic constitutive model, which is able to satisfactorily reproduce the observed behaviour.

The NASA InSight Lander on Mars includes the Heat Flow and Physical Properties Package HP$^3$ to measure the surface heat flow of the planet. The package uses temperature sensors that would have been brought to the target depth of 3--5 m by a small penetrator, nicknamed the mole. The mole requiring friction on its hull to balance remaining recoil from its hammer mechanism did not penetrate to the targeted depth. Instead, by precessing about a point midway along its hull, it carved a 7 cm deep and 5-6 cm wide pit and reached a depth of initially 31 cm. The root cause of the failure - as was determined through an extensive, almost two years long campaign - was a lack of friction in an unexpectedly thick cohesive duricrust. During the campaign -- described in detail in this paper -- the mole penetrated further aided by friction applied using the scoop at the end of the robotic Instrument Deployment Arm and by direct support by the latter. The mole finally reached a depth of 40 cm, bringing the mole body 1--2 cm below the surface. The penetration record of the mole and its thermal sensors were used to measure thermal and mechanical soil parameters such as the thermal conductivity and the penetration resistance of the duricrust and its cohesion. The hammerings of the mole were recorded by the seismometer SEIS and the signals could be used to derive a P-wave velocity and a S-wave velocity and elastic moduli representative of the topmost tens of cm of the regolith. The combined data were used to derive a model of the regolith that has an about 20 cm thick duricrust underneath a 1 cm thick unconsolidated layer of sand mixed with dust and above another 10 cm of unconsolidated sand. Underneath the latter, a layer more resistant to penetration and possibly consisting of debris from a small impact crater is inferred., Comment: 78 pages 22 figures, , submitted to Space Science Reviews

Dynamic programming and heuristic search are at the core of state-of-the-art solvers for sequential decision-making problems. In partially observable or collaborative settings (\eg, POMDPs and Dec-POMDPs), this requires introducing an appropriate statistic that induces a fully observable problem as well as bounding (convex) approximators of the optimal value function. This approach has succeeded in some subclasses of 2-player zero-sum partially observable stochastic games (zs-POSGs) as well, but failed in the general case despite known concavity and convexity properties, which only led to heuristic algorithms with poor convergence guarantees. We overcome this issue, leveraging on these properties to derive bounding approximators and efficient update and selection operators, before deriving a prototypical solver inspired by HSVI that provably converges to an $\epsilon$-optimal solution in finite time, and which we empirically evaluate. This opens the door to a novel family of promising approaches complementing those relying on linear programming or iterative methods., Comment: 37 pages, 4 figures, 4 tables, 3 algorithms

Observations of young stars hosting transition disks show that several of them have high accretion rates, despite their disks presenting extended cavities in their dust component. This represents a challenge for theoretical models, which struggle to reproduce both features. We explore if a disk evolution model, including a dead zone and disk dispersal by X-ray photoevaporation, can explain the high accretion rates and large gaps (or cavities) measured in transition disks. We implement a dead zone turbulence profile and a photoevaporative mass loss profile into numerical simulations of gas and dust. We perform a population synthesis study of the gas component, and obtain synthetic images and SED of the dust component through radiative transfer calculations. This model results in long lived inner disks and fast dispersing outer disks, that can reproduce both the accretion rates and gap sizes observed in transition disks. For a dead zone of turbulence $\alpha_{dz} = 10^{-4}$ and extent $r_{dz}$ = 10 AU, our population synthesis study shows that $63\%$ of our transition disks are accreting with $\dot{M}_g > 10^{-11} M_\odot/yr$ after opening a gap. Among those accreting transition disks, half display accretion rates higher than $5\times10^{-10} M_\odot/yr$ . The dust component in these disks is distributed in two regions: in a compact inner disk inside the dead zone, and in a ring at the outer edge of the photoevaporative gap, which can be located between 20 AU and 100 AU. Our radiative transfer calculations show that the disk displays an inner disk and an outer ring in the millimeter continuum, a feature observed in some transition disks. A disk model considering X-ray photoevaporative dispersal in combination with dead zones can explain several of the observed properties in transition disks including: the high accretion rates, the large gaps, and long-lived inner disks at mm-emission., Comment: 16 pages, 15 figures. Accepted in A&A

[abridged] We present a 1+1D global magnetically-driven disk accretion model that captures the essence of the MRI-driven accretion, without resorting to 3D global non-ideal MHD simulations. The gas dynamics is assumed to be solely controlled by the MRI and hydrodynamic instabilities. For given stellar and disk parameters, the Shakura-Sunyaev viscosity parameter $\alpha$ is determined self-consistently under the framework of viscously-driven accretion from detailed considerations of the MRI with non-ideal MHD effects (Ohmic resistivity and ambipolar diffusion), accounting for disk heating by stellar irradiation, non-thermal sources of ionization, and dust effects on the ionization chemistry. Additionally, the magnetic field strength is constrained and adopted to maximize the MRI activity. We demonstrate the use of our framework by investigating steady-state MRI-driven accretion in a fiducial protoplanetary disk model around a solar-type star. We find that the equilibrium solution displays no pressure maximum at the dead zone outer edge, except if a sufficient amount of dust particles have accumulated there before the disk reaches a steady-state accretion regime. Furthermore, the steady-state accretion solution describes a disk that displays a spatially extended long-lived inner disk gas reservoir (the dead zone) accreting a few $10^{-9}\, M_{\odot}.\rm{yr}^{-1}$. By conducting a detailed parameter study, we find that the extend to which the MRI can drive efficient accretion is primarily determined by the total disk gas mass, the representative grain size, the vertically-integrated dust-to-gas mass ratio, and the stellar X-ray luminosity. A self-consistent time-dependent coupling between gas, dust, stellar evolution models and our general framework on million-year timescales is required to fully understand the formation of dead zones and their potential to trap dust particles., Comment: Accepted for publication in A&A

Context. The protoplanetary disk around the star HD 100546 displays prominent substructures in the form of two concentric rings. Recent observations with the Atacama Large Millimeter/sub-millimeter Array (ALMA) have revealed these features with high angular resolution and have resolved the faint outer ring well. This allows us to study the nature of the system further. Aims. Our aim is to constrain some of the properties of potential planets embedded in the disk, assuming that they induce the observed rings and gaps. Methods. We present the self-calibrated $0.9\,$mm ALMA observations of the dust continuum emission from the circumstellar disk around HD 100546. These observations reveal substructures in the disk that are consistent with two rings, the outer ring being much fainter than the inner one. We reproduced this appearance closely with a numerical model that assumes two embedded planets. We varied planet and disk parameters in the framework of the planet-disk interaction code FARGO3D and used the outputs for the gas and dust distribution to generate synthetic observations with the code RADMC-3D. Results. From this comparison, we find that an inner planet located at $r_1 = 13\,$au with a mass $M_1 = 8 M_{\rm{Jup}}$ and an outer planet located at $r_2 = 143\,$au with a mass $M_2 = 3 M_{\rm{Jup}}$ leads to the best agreement between synthetic and ALMA observations (deviation less than $3\sigma$ for the normalized radial profiles). To match the very low brightness of the outer structure relative to the inner ring, the initial disk gas surface density profile needs to follow an exponentially tapered power law (self-similar solution), rather than a simple power-law profile., Comment: Accepted for publication in A&A

The problem of portfolio management represents an important and challenging class of dynamic decision making problems, where rebalancing decisions need to be made over time with the consideration of many factors such as investors preferences, trading environments, and market conditions. In this paper, we present a new portfolio policy network architecture for deep reinforcement learning (DRL)that can exploit more effectively cross-asset dependency information and achieve better performance than state-of-the-art architectures. In particular, we introduce a new property, referred to as \textit{asset permutation invariance}, for portfolio policy networks that exploit multi-asset time series data, and design the first portfolio policy network, named WaveCorr, that preserves this invariance property when treating asset correlation information. At the core of our design is an innovative permutation invariant correlation processing layer. An extensive set of experiments are conducted using data from both Canadian (TSX) and American stock markets (S&P 500), and WaveCorr consistently outperforms other architectures with an impressive 3%-25% absolute improvement in terms of average annual return, and up to more than 200% relative improvement in average Sharpe ratio. We also measured an improvement of a factor of up to 5 in the stability of performance under random choices of initial asset ordering and weights. The stability of the network has been found as particularly valuable by our industrial partner.

Recently equal risk pricing, a framework for fair derivative pricing, was extended to consider dynamic risk measures. However, all current implementations either employ a static risk measure that violates time consistency, or are based on traditional dynamic programming solution schemes that are impracticable in problems with a large number of underlying assets (due to the curse of dimensionality) or with incomplete asset dynamics information. In this paper, we extend for the first time a famous off-policy deterministic actor-critic deep reinforcement learning (ACRL) algorithm to the problem of solving a risk averse Markov decision process that models risk using a time consistent recursive expectile risk measure. This new ACRL algorithm allows us to identify high quality time consistent hedging policies (and equal risk prices) for options, such as basket options, that cannot be handled using traditional methods, or in context where only historical trajectories of the underlying assets are available. Our numerical experiments, which involve both a simple vanilla option and a more exotic basket option, confirm that the new ACRL algorithm can produce 1) in simple environments, nearly optimal hedging policies, and highly accurate prices, simultaneously for a range of maturities 2) in complex environments, good quality policies and prices using reasonable amount of computing resources; and 3) overall, hedging strategies that actually outperform the strategies produced using static risk measures when the risk is evaluated at later points of time.