Stochastic Optimization Framework Research Articles

A Framework for Stochastic Optimization of Parameters for Integrative Modeling of Macromolecular Assemblies

Integrative modeling of macromolecular assemblies requires stochastic sampling, for example, via MCMC (Markov Chain Monte Carlo), since exhaustively enumerating all structural degrees of freedom is infeasible. MCMC-based methods usually require tuning several parameters, such as the move sizes for coarse-grained beads and rigid bodies, for sampling to be efficient and accurate. Currently, these parameters are tuned manually. To automate this process, we developed a general heuristic for derivative-free, global, stochastic, parallel, multiobjective optimization, termed StOP (Stochastic Optimization of Parameters) and applied it to optimize sampling-related parameters for the Integrative Modeling Platform (IMP). Given an integrative modeling setup, list of parameters to optimize, their domains, metrics that they influence, and the target ranges of these metrics, StOP produces the optimal values of these parameters. StOP is adaptable to the available computing capacity and converges quickly, allowing for the simultaneous optimization of a large number of parameters. However, it is not efficient at high dimensions and not guaranteed to find optima in complex landscapes. We demonstrate its performance on several examples of random functions, as well as on two integrative modeling examples, showing that StOP enhances the efficiency of sampling the posterior distribution, resulting in more good-scoring models and better sampling precision.

A Flexible Distributed Stochastic Optimization Framework for Concurrent Tasks in Processing Networks

Distributed stochastic optimization has important applications in the practical implementation of machine learning and signal processing setup by providing means to allow interconnected network of processors to work towards the optimization of a global objective with intermittent communication. Existing works on distributed stochastic optimization predominantly assume all the processors storing related data to perform updates for the optimization task in each iteration. However, such optimization processes are typically executed at shared computing/data centers along with other concurrent tasks. Therefore, it is necessary to develop efficient optimization methods that possess the flexibility to share the computing resources with other ongoing tasks. In this work, we propose a new first-order framework that allows this flexibility through a probabilistic computing resource allocation strategy while guaranteeing the satisfactory performance of distributed stochastic optimization. Our results, both analytical and numerical, show that by controlling a flexibility parameter, our suite of algorithms (designed for various scenarios) can achieve the lower computation and communication costs of distributed stochastic optimization than their inflexible counterparts. This framework also enables the fair sharing of the common resources with other concurrent tasks being processed by the processing network.

Two-stage framework for optimizing the operation of remote off-grid power systems under uncertainties

With the precipitated integration of intermittent renewable sources, operation optimization employed with uncertainty management strategies is imperative to ensure a reliable, cost-effective, and secure energy supply for the remote off-grid power systems. This paper presents a two-stage operation optimization process embedded in a model predictive control framework. The first stage decision variables are derived in a deterministic optimization framework to achieve minimum operational costs and emissions. The second stage uses a stochastic optimization framework to refine the first stage decision variables to achieve a feasible operation considering several candidate scenarios in a computationally inexpensive manner. Several measures are integrated to the proposed framework in order to address the operational requirements pertaining to remote off-grid power systems. The effectiveness of the proposed framework is demonstrated through numerical experiments for an isolated remote power system in Northern Canada for both summer and winter seasons. Quality of the obtained results as well as the computation efficiency of the overall framework has been verified compared to the existing energy management techniques. The overall result confirms the applicability of the proposed method in achieving a cost-effective and environmentally friendly operational trajectory while effectively accounting for the underlying uncertainties with a reduced computational burden.

Distributionally robust optimization under endogenous uncertainty with an application in retrofitting planning

Endogenous uncertainty concerns uncertainty which is dependent of decisions such as link failure in the retrofitting planning application. We propose a marginal-based distributionally robust optimization framework for integer stochastic optimization with decision-dependent discrete distributions that can be applied for the retrofitting planning application. We show that the resulting model can be formulated as a mixed-integer linear optimization problem. In order to solve the problem, we develop a constraint generation algorithm given the exponentially large number of constraints. Numerical results for the retrofitting planning application show that the proposed algorithm once tailored can solve the problem efficiently.

Triply stochastic gradient method for large-scale nonlinear similar unlabeled classification

Similar unlabeled (SU) classification is pervasive in many real-world applications, where only similar data pairs (two data points have the same label) and unlabeled data points are available to train a classifier. Recent work has identified a practical SU formulation and has derived the corresponding estimation error bound. It evaluated SU learning with linear classifiers on medium-sized datasets. However, in practice, we often need to learn nonlinear classifiers on large-scale datasets for superior predictive performance. How this could be done in an efficient manner is still an open problem for SU classification. In this paper, we propose a scalable kernel learning algorithm for SU classification using a triply stochastic optimization framework, called TSGSU. Specifically, in each iteration, our method randomly samples an instance from the similar pairs set, an instance from the unlabeled set, and their random features to calculate the stochastic functional gradient for the model update. Theoretically, we prove that our method can converge to a stationary point at the rate of \(O(1/\sqrt{T})\) after T iterations. Experiments on various benchmark datasets and high-dimensional datasets not only demonstrate the scalability of TSGSU but also show the efficiency of TSGSU compared with existing SU learning algorithms while retaining similar generalization performance.

Optimal dynamic longevity hedge with basis risk

This paper proposes an optimal dynamic strategy for hedging longevity risk in a discrete-time setting. Our proposed hedging strategy relies on standardized mortality-linked securities and minimizes the variance of the hedging error as induced by the population basis risk. While the formulation of our proposed hedging strategy is quite general, we use a stylized pension plan, together with a specified “yearly rollingǥ trading strategy involving q-forwards and a specified stochastic mortality model, to illustrate our proposed strategy. Under these specifications, we show that the resulting hedging problem can be formulated as a stochastic optimal control framework and that a semi-analytic solution can be derived through an extended Bellman equation. Extensive Monte Carlo studies are conducted to highlight the effectiveness of our proposed hedging strategy. We also consider a scheme to approximate the semi-analytic solution in order to reduce the computational time significantly while still retaining its hedge effectiveness. We benchmark our strategy against the “delta” hedging strategy as well as its robustness to q-forwards maturity, reference age, interest rate, and stochastic mortality models. The proposed strategy has many appealing features, including its discrete-time setting which is consistent with market practice and hence conducive to practical implementation, and its generality in that the underlying hedging principle can be applied to other standardized mortality-linked securities and other stochastic models.

A stochastic multi-objective optimization framework for distribution feeder reconfiguration in the presence of renewable energy sources and energy storages

In this paper, a multi-objective optimization framework is proposed to solve the distribution feeder reconfiguration (DFR) problem and its operation considering the demand response (DR) program, renewable energy sources (RES's), and electrical energy storages (EES's). The proposed model is implemented on 33-bus and 118-bus radial distribution systems while the uncertainties of RES's output power, load demand and electricity price are taken into account. The Monte Carlo simulation approach is used to generate scenarios while the backward scenario reduction approach is used to reduce the number of scenarios. The studied problem is modeled using the Epsilon-constrained method as a two objective problem and it is solved in the form of five case studies using the GUROBI solver in GAMS software. Our analysis of the results shows that reducing losses and increasing system reliability increases the production of local generation units, thereby increasing the operating costs. In addition, simulation results demonstrate that considering the dynamic topology reduced losses by 9.73% and increased reliability by 4.7%. The results also show that using the DR program reduces LMP by about 20% during peak hour.

Learning models with uniform performance via distributionally robust optimization

A common goal in statistics and machine learning is to learn models that can perform well against distributional shifts, such as latent heterogeneous subpopulations, unknown covariate shifts or unmodeled temporal effects. We develop and analyze a distributionally robust stochastic optimization (DRO) framework that learns a model providing good performance against perturbations to the data-generating distribution. We give a convex formulation for the problem, providing several convergence guarantees. We prove finite-sample minimax upper and lower bounds, showing that distributional robustness sometimes comes at a cost in convergence rates. We give limit theorems for the learned parameters, where we fully specify the limiting distribution so that confidence intervals can be computed. On real tasks including generalizing to unknown subpopulations, fine-grained recognition and providing good tail performance, the distributionally robust approach often exhibits improved performance.

Mixed-Integer Convex Optimization for DC Microgrid Droop Control

Droop control is a viable method for the operation of island DC microgrids in a decentralized architecture. This paper presents a mixed-integer conic optimization formulation for the design of generator droop control, comprising the parameters of a piecewise linear droop curve. The mixed-integer formulation originates from a stochastic optimization framework that considers several operating scenarios for finding the optimal design. The convexity of the mixed-integer problem continuous relaxation gives global optimality guarantees for the design problem. The paper presents computational results using a tight polyhedral approximation of the conic program, leading to a mixed-integer linear programming (MILP) problem that is solved using a state-of-the-art commercial solver. The results from the proposed approach are contrasted with both a classic linear droop control design and a recent piecewise linear formulation. The Monte-Carlo simulation results quantify the extent to which the MILP solution is superior in reducing voltage violations and power loss, and the degree to which the loss is close to that from a conic optimal power flow solution.

Two-Timescale Uplink Channel Estimation for Dual-Hop MIMO Relay Multi-User Systems

Multiple-input multiple-output (MIMO) relay systems have received great attention for the development of future wireless networks, since they can be invoked for improving system capacity and extending the coverage area. Considering the fixed base station (BS) and relay station (RS) and the highly mobile users (UEs), the dual-hop MIMO relay channel possesses the two-timescale characteristics, i.e., the UEs-RS channel varies in a short-timescale manner whereas the RS-BS channel varies in a long-timescale manner. In order to acquire the knowledge of all individual channels at the BS directly, we propose two-timescale channel estimation algorithms with one-stage training (OST) scheme based on the above time-varying channel characteristics. Specifically, the two-timescale channel estimation problem is firstly formulated as an approximate maximum likelihood (AML) problem. Then, we design batch algorithm to solve the resultant optimization problem. Moreover, to overcome the drawbacks of batch algorithm, a low-complexity online algorithm is also proposed based on the two-stage stochastic optimization framework. Finally, simulation results are provided to verify the effectiveness of the proposed two-timescale channel estimation algorithms.

Downlink Connection Density Maximization for NB-IoT Networks Using NOMA With Perfect and Partial CSI

We address the issue of maximizing the number of connected devices in a Narrowband Internet-of-Things (NB-IoT) network using nonorthogonal multiple access (NOMA) in the downlink. We first propose an optimal joint subcarrier and power allocation strategy assuming perfect channel state information (CSI) called stratified device allocation (SDA), which maximizes the connectivity under data rate, power, and bandwidth constraints. Then, we generalize the connectivity maximization problem to the case of partial CSI, where only the distance-dependent path-loss component of the channel gain is available at the base station (BS). We introduce a novel framework called the stochastic connectivity optimization (SCO) framework. In this framework, we propose a heuristic improvement to SDA, namely, SDA with excess power (SDA-EP) algorithm for operation under partial CSI. Furthermore, we derive a concave approximation (SCO-CA) algorithm of near-optimal performance to SCO given the same amount of CSI. Through computer simulations, we show that SDA-EP and SCO-CA outperform conventional NOMA and OMA schemes in the presence of partial CSI over a wide range of service scenarios.

Chance-constrained optimization-based solar microgrid design and dispatch for radial distribution networks

We consider a solar microgrid design and dispatch problem using an adaptive stochastic optimization framework. First, we propose a two-stage mixed-integer model for optimal placement and planning of distributed generation (DG) units and energy storage system units. We incorporate time series modeling into stochastic optimization approach to characterize the solar irradiance uncertainty. In the first stage, design decisions (e.g., location and sizing of DGs) are made and in the second stage, dispatch decisions (e.g., how much to generate, how much to store) are made such that electricity demand is met in a reliable and cost effective way. Chance constraints are employed to control the real load shedding within a predefined probability level. Then, we propose a combined sample average approximation (SAA) and linearization technique to solve this problem more efficiently. The advantage of this approach is that no additional binary variables are introduced while reformulating the chance constraints. Computational time, quality of solution, and load shedding percentage are compared with the traditional SAA. Moreover, we carry out a comprehensive out-of-sample simulation on a real-world case study in the state of Arizona assessing the effectiveness of our approach. Numerical experiments demonstrate that the chance constraints are effective tools for control of load shedding in distributed generation and the proposed approach outperforms traditional methods both in terms of true load shedding percentage and computational time.

РЕШЕНИЕ ПРЯМОЙ ЗАДАЧИ ГРАВИРАЗВЕДКИ МЕТОДОМ ИМИТАЦИИ ОТЖИГА ДЛЯ ИНТЕРПРЕТАЦИИ ДАННЫХ ГРАВИМЕТРИЧЕСКОГО МОНИТОРИНГА СОСТОЯНИЯ НЕДР

The long-term development of hydrocarbon reservoirs (HCR) in the geological environment, complex deformation processes occur. Gravity monitoring is carried out to evaluate the possible geodynamic risk and negative consequences from HCR. As a result, the interrelationships of the continuously changing field-geological situation (changes in production volumes, changes in reservoir pressure, processes of fluid injection into productive formations) are investigated. The main tool for solving the gravity inversion when determining areas of increased industrial hazard is the solution of the gravity direct problem. In these studies, proceeding from a given initial approximation of the environment, the problem is realized through successive approximations. To assess such distributions, the authors of the article recommend using the simulated annealing technique within the framework of stochastic optimization. It is aimed at fitting the optimal parameters of the medium provided that a minimal residual of the gravity field values occurs. The approach is implemented using three simple mathematical models of the geological medium such as horizontal prism, homogeneous sphere, and vertical ledge. This technique allows fitting the media values simultaneously by a pair of its parameters. The operation of the algorithm is described and the simulation results are provided. The results showed acceptable accuracy of the algorithm for solving the direct gravity problem by the proposed method. The simulated annealing technique made it possible to increase the reliability of the HCR model while reduce the time for the analysis of the gravity field.

Retail supply chain network design with concurrent resilience capabilities

In this paper, we address the problem of designing/redesigning a resilient retail supply chain network under uncertainty. Considering pre- and post-disruption scenarios, this study proposes a two-stage stochastic optimization framework with multiple resilience strategies for designing a resilient retail supply chain. We consider facility fortification, inventory prepositioning, direct-to-store delivery, reserved capacity, inventory sharing and multiple set covering as our resilience strategies, covering all the three dimensions of resilience capabilities, namely, proactive, reactive, and network design quality. Taking into account the prominent role of retail outlets during Covid-19 epidemic outbreaks, through a real case study, we evaluate the impact of random and targeted disruptions on the performance of the retail supply chain network and provide managerial insights regarding the right balance between cost efficiency and resilience. Furthermore, we evaluate the effectiveness of these resilience capabilities in a number of simulated scenarios to show their applicability and importance under different disruptive situations. We show that when implemented concurrently, these resilience capabilities have a synergistic effect where the whole becomes larger than the sum of its parts. Subject to extensive stress testing, our results show that using any mixture of these resilience capabilities increases the retail network's resilience while considerably decreasing post-disruption costs.

Scenario-based stochastic framework for harmonic power markets using plug-in electric vehicles

With respect to growing the nonlinear loads, it is appealing to employ Plug-in electric vehicles (PEVs) as harmonic compensators to distributedly mitigate harmonic pollution. Hence, this study is intended to propose a comprehensive stochastic optimization framework for PEVs participation considering their uncertainties to accurately settling harmonic power market by distribution system operators. The optimization framework consists of the total expected distortion payment function (DTPF) of the accepted harmonic compensators, such as PEVs and active power line conditioners (APLCs) as an objective function while fulfilling the constraints concerned with the grid, vehicle, and market price. The DTPF comprises availability, loss, and loss of opportunity cost terms based on the harmonic capability curve of the harmonic compensators. Furthermore, the Monte-Carlo simulation based on the contingency concept is utilized to generate scenarios for covering the uncertainties of the PEVs, and then the scenarios solve by adaptive particular tunable fuzzy particle swarm method. This study demonstrates the efficiency of the proposed framework and the outperformance of the PEVs compared with APLCs investigated on a 14-node microgrid.

A Stochastic Multiagent Optimization Framework for Interdependent Transportation and Power System Analyses

We study the interdependence between transportation and power systems considering decentralized renewable generators and electric vehicles (EVs). We formulate the problem in a stochastic multiagent optimization framework considering the complex interactions between EV/conventional vehicle drivers, renewable/conventional generators, and independent system operators, with locational electricity and charging prices endogenously determined by markets. We show that the multiagent optimization problems can be reformulated as a single convex optimization problem and prove the existence and uniqueness of the equilibrium. To cope with the curse of dimensionality, we propose the alternating direction method of multipliers (ADMM)-based decomposition algorithm to facilitate parallel computing. Numerical insights are generated using standard test systems in the transportation and power system literature.

Optimal BS Deployment and User Association for 5G Millimeter Wave Communication Networks

Although millimeter wave (mmWave) communications can well support high-data-rate transmissions, the inherent shortcomings, e.g., high path loss and sensitivity to blockage, may cause severe outage problems if the network is not configured properly. This paper aims to minimize the long-term outage probability of an mmWave communication network by optimizing the base station (BS) deployment and user association. For the BS deployment problem, existing works usually assumed that the positions of users are fixed and formulated it as a deterministic optimization problem. With the time-varying nature of positions of user equipments (UEs) taken into account, we establish a stochastic optimization framework for BS deployment optimization. The objective is to maximize the average number of physically accessible BSs of each UE under an inaccessible probability constraint, and a cooperative stochastic approximation (CSA)-based algorithm is developed to effectively search the optimal positions of BSs. For user association, our focus is to properly associate UEs with BSs to minimize the outage probability with balanced workloads among BSs. Combined with the proposed user association scheme, the proposed BS deployment scheme can significantly improve the network outage probability in the long term, especially when the aggregation degree of UEs is large.

Dynamic Transmission Rate Control for Multi‐Interface IoT Devices: A Stochastic Optimization Framework

Recent advances in the Internet of Things (IoT) technologies have enabled ubiquitous smart devices to sense and process various kinds of data. However, these innovations also raise the concern of efficient data transmission. Tackling the above issue is nontrivial since the resource constraints and environmental randomness in IoT require a lightweight transmission scheme while guaranteeing system stability. In this paper, we formulate the transmission scheduling problem of multi‐interface IoT devices as a concave optimization, aimed at accommodating the randomness of the IoT environment within the network capacity. By applying the Lyapunov optimization technique, we divide the stochastic problem into a series of low‐complex subproblems, which can be individually solved per time slot, and develop a dynamical control algorithm that does not require a priori knowledge such as link states. Theoretical analysis shows that our algorithms nicely bound the average queue length and are asymptotically optimal. Finally, extensive simulation results verify the theoretical conclusions and validate the effectiveness of the proposed algorithm.

Unified Algorithm Framework for Nonconvex Stochastic Optimization in Deep Neural Networks

This paper presents a unified algorithmic framework for nonconvex stochastic optimization, which is needed to train deep neural networks. The unified algorithm includes the existing adaptive-learning-rate optimization algorithms, such as Adaptive Moment Estimation (Adam), Adaptive Mean Square Gradient (AMSGrad), Adam with weighted gradient and dynamic bound of learning rate (GWDC), AMSGrad with weighted gradient and dynamic bound of learning rate (AMSGWDC), and Adapting stepsizes by the belief in observed gradients (AdaBelief). The paper also gives convergence analyses of the unified algorithm for constant and diminishing learning rates. When using a constant learning rate, the algorithm can approximate a stationary point of a nonconvex stochastic optimization problem. When using a diminishing rate, it converges to a stationary point of the problem. Hence, the analyses lead to the finding that the existing adaptive-learning-rate optimization algorithms can be applied to nonconvex stochastic optimization in deep neural networks in theory. Additionally, this paper provides numerical results showing that the unified algorithm can train deep neural networks in practice. Moreover, it provides numerical comparisons for unconstrained minimization using benchmark functions of the unified algorithm with certain heuristic intelligent optimization algorithms. The numerical comparisons show that a teaching-learning-based optimization algorithm and the unified algorithm perform well.

A Decision Model for an Electricity Retailer With Energy Storage and Virtual Bidding Under Daily and Hourly CVaR Assessment

This paper presents a short-term decision-making model for an electricity retailer with battery energy storage system (BESS) and virtual bidding through a two-stage stochastic optimization framework. In the first stage, the retailer determines the amount of power to be purchased in the day-ahead wholesale market and the optimal incremental and decremental virtual bidding strategies. In the second stage, the optimal energy storage decisions and the retailer's involvement in the real-time market are determined. The proposed model minimizes the retailer's expected procurement cost and generates the optimal power and virtual bidding curves in the day-ahead market. Two types of Conditional Value at Risk (CVaR) are integrated in the proposed model to manage the retailer's hourly and daily risks, respectively. Case studies with real-world data are performed to verify the retailer's cost reduction obtained with the integration of BESS and virtual bidding and to study how the hourly and daily risk-management strategies affect the retailer's procurement cost distribution for different risk-aversion levels.