Distributed Stochastic Optimization with Compression for Non-Strongly Convex Objectives

We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization parameters through the wireless network,large-scale training models can create communication bottlenecks,resulting in slower training times.To address this issue,CHOCO-SGD was proposed,which allows compressing information with arbitrary precision without reducing the convergence rate for strongly convex objective functions.Nevertheless,most convex functions are not strongly convex(such as logistic regression or Lasso),which raises the question of whether this algorithm can be applied to non-strongly convex functions.In this paper,we provide the first theoretical analysis of the convergence rate of CHOCO-SGD on non-strongly convex objectives.We derive a sufficient condition,which limits the fidelity of compression,to guarantee convergence.Moreover,our analysis demonstrates that within the fidelity threshold,this algorithm can significantly reduce transmission burden while maintaining the same convergence rate order as its no-compression equivalent.Numerical experiments further validate the theoretical findings by demonstrating that CHOCO-SGD improves communication efficiency and keeps the same convergence rate order simultaneously.And experiments also show that the algorithm fails to converge with low compression fidelity and in time-varying topologies.Overall,our study offers valuable insights into the potential applicability of CHOCO-SGD for non-strongly convex objectives.Additionally,we provide practical guidelines for researchers seeking to utilize this algorithm in real-world scenarios.

Distributed Momentum-Based Frank-Wolfe Algorithm for Stochastic Optimization

This paper considers distributed stochastic optimization,in which a number of agents cooperate to optimize a global objective function through local computations and information exchanges with neighbors over a network.Stochastic optimization problems are usually tackled by variants of projected stochastic gradient descent.However,projecting a point onto a feasible set is often expensive.The Frank-Wolfe(FW)method has well-documented merits in handling convex constraints,but existing stochastic FW algorithms are basically developed for centralized settings.In this context,the present work puts forth a distributed stochastic Frank-Wolfe solver,by judiciously combining Nesterov's momentum and gradient tracking techniques for stochastic convex and nonconvex optimization over networks.It is shown that the convergence rate of the proposed algorithm is O(k^(-1/2))for convex optimization,and O(1/log_(2)(k))for nonconvex optimization.The efficacy of the algorithm is demonstrated by numerical simulations against a number of competing alternatives.

Stability Analysis for Stochastic Optimization Problems

Stochastic optimization offers a means of considering the objectives and constrains with stochastic parameters. However, it is generally difficult to solve the stochastic optimization problem by employing conventional methods for nonlinear programming when the number of random variables involved is very large. Neural network models and algorithms were applied to solve the stochastic optimization problem on the basis of the stability theory. Stability for stochastic programs was discussed. If random vector sequence converges to the random vector in the original problem in distribution, the optimal value of the corresponding approximation problems converges to the optimal value of the original stochastic optimization problem.

Application of Stochastic Optimization to Optimal Preventive Maintenance Problem

The contribution deals with the optimization of a sequential preventive maintenance schedule of a technical device. We are given an initial time-to-failure probability distribution, model of changes of this distribution after maintenance actions, as well as the costs of maintenance, of a device acquisition, and of the impact of failure. The maintenance timing and, eventually, its extent, are the matter of optimization. The objective of the contribution is two-fold: first, to formulate a proper (random) objective function evaluating the lifetime of the maintained device relatively to maintenance costs;second, to propose a numerical method searching for a maintenance policy optimizing selected characteristics of this objective function. The method is based on the MCMC random search combined with simulated annealing. It is also shown that such a method is rather universal for different problem specifications. The approach will be illustrated on an artificial example dealing with accelerated lifetime after each maintenance action.

Matrix-valued distributed stochastic optimization with constraints

In this paper,we address matrix-valued distributed stochastic optimization with inequality and equality constraints,where the objective function is a sum of multiple matrix-valued functions with stochastic variables and the considered problems are solved in a distributed manner.A penalty method is derived to deal with the constraints,and a selection principle is proposed for choosing feasible penalty functions and penalty gains.A distributed optimization algorithm based on the gossip model is developed for solving the stochastic optimization problem,and its convergence to the optimal solution is analyzed rigorously.Two numerical examples are given to demonstrate the viability of the main results.

Almost Sure Convergence of Proximal Stochastic Accelerated Gradient Methods

Proximal gradient descent and its accelerated version are resultful methods for solving the sum of smooth and non-smooth problems. When the smooth function can be represented as a sum of multiple functions, the stochastic proximal gradient method performs well. However, research on its accelerated version remains unclear. This paper proposes a proximal stochastic accelerated gradient (PSAG) method to address problems involving a combination of smooth and non-smooth components, where the smooth part corresponds to the average of multiple block sums. Simultaneously, most of convergence analyses hold in expectation. To this end, under some mind conditions, we present an almost sure convergence of unbiased gradient estimation in the non-smooth setting. Moreover, we establish that the minimum of the squared gradient mapping norm arbitrarily converges to zero with probability one.

Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces

In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls U can be chosen to be non-convex but now the space of controls U can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.

TWO-DIMENSIONAL STOCHASTIC AIRFOIL OPTIMIZATION DESIGN METHOD BASED ON NEURAL NETWORKS

To avoid the aerodynamic performance loss of airfoil at non-design state which often appears in single point design optimization, and to improve the adaptability to the uncertain factors in actual flight environment, a two-dimensional stochastic airfoil optimization design method based on neural networks is presented. To provide highly efficient and credible analysis, four BP neural networks are built as surrogate models to predict the airfoil aerodynamic coefficients and geometry parameter. These networks are combined with the probability density function obeying normal distribution and the genetic algorithm, thus forming an optimization design method. Using the method, for GA（W）-2 airfoil, a stochastic optimization is implemented in a two-dimensional flight area about Mach number and angle of attack. Compared with original airfoil and single point optimization design airfoil, results show that the two-dimensional stochastic method can improve the performance in a specific flight area, and increase the airfoil adaptability to the stochastic changes of multiple flight parameters.

Benefits of Stochastic Optimization for Scheduling Energy Storage in Wholesale Electricity Markets

We propose a two-stage stochastic model for optimizing the operation of energy storage. The model captures two important features: uncertain real-time prices when day-ahead operational commitments are made;and the price impact of charging and discharging energy storage. We demonstrate that if energy storage has full flexibility to make real-time adjustments to its day-ahead commitment and market prices do not respond to charging and discharging decisions, there is no value in using a stochastic modeling framework, i.e., the value of stochastic solution is always zero. This is because in such a case the energy storage behaves purely as a financial arbitrageur day ahead, which can be captured using a deterministic model.We show also that prices responding to its operation can make it profitable for energy storage to "waste" energy, for instance by charging and discharging simultaneously, which is normally sub-optimal. We demonstrate our model and how to calibrate the price-response functions from historical data with a practical case study.

Review of stochastic optimization methods for smart grid

This paper presents various approaches used by researchers for handling the uncertainties involved in renewable energy sources, load demands, etc. It gives an idea about stochastic programming (SP) and discusses the formulations given by different researchers for objective functions such as cost, loss, generation expansion, and voltage/V control with various conventional and advanced methods. Besides, it gives a brief idea about SP and its applications and discusses different variants of SP such as recourse model, chance constrained programming, sample average approximation, and risk aversion. Moreover, it includes the application of these variants in various power systems. Furthermore, it also includes the general mathematical form of expression for these variants and discusses the mathematical description of the problem and modeling of the system. This review of different optimization techniques will be helpful for smart grid development including renewable energy resources (RERs).

Multi-source coordinated stochastic restoration for SOP in distribution networks with a two-stage algorithm

After suffering from a grid blackout, distributed energy resources(DERs), such as local renewable energy and controllable distributed generators and energy storage can be used to restore loads enhancing the system’s resilience. In this study, a multi-source coordinated load restoration strategy was investigated for a distribution network with soft open points(SOPs). Here, the flexible regulation ability of the SOPs is fully utilized to improve the load restoration level while mitigating voltage deviations. Owing to the uncertainty, a scenario-based stochastic optimization approach was employed,and the load restoration problem was formulated as a mixed-integer nonlinear programming model. A computationally efficient solution algorithm was developed for the model using convex relaxation and linearization methods. The algorithm is organized into a two-stage structure, in which the energy storage system is dispatched in the first stage by solving a relaxed convex problem. In the second stage, an integer programming problem is calculated to acquire the outputs of both SOPs and power resources. A numerical test was conducted on both IEEE 33-bus and IEEE 123-bus systems to validate the effectiveness of the proposed strategy.

A new look at the Lagrange method for continuous-time stochastic optimization

We formulate a Lagrange method for continuous-time stochastic optimization in an appropriate normed space by using a proper stochastic process as the Lagrange multiplier.The obtained optimality conditions are applied to different types of problems.Some examples selected from control theory and economic theory are studied to test and illustrate the potential applications of the method.

Stochastic Optimization in Cooperative Relay Networks for Revenue Maximization

In cellular networks, cooperative relaying is an economic and promising way to enlarge the network capacity and coverage. In the case that multiple users and multiple relays are taken into account, efficient resource allocation is important in such networks. In this paper, we consider the joint relay power control with amplify-and-forward(AF) strategy and dynamic pricing for uplink cellular networks in order to maximize the network administrator's system revenue. The system revenue is associated with pricing strategies and mobile users' random data request, which is supported by the relay assisted transmission. To deal with the problem of the coupling in pricing and relay resource allocation, we utilize Lyapunov optimization techniques to design online pricing and relay power control without any statistic information of random events in networks. Theoretical analysis shows that the proposed algorithm can achieve a near-optimal performance and simulation results also validate its effectiveness and robustness.

Search for Compromise Solution of the MultistageAxial Compressor's Stochastic Optimization Problem

The aim of this paper is to discuss a method of the compromise region determination for the multistage axial flow compressor stochastic optimization problems. This method is based on the 2-D axisynunetrical mathematical model of the compressor and on the new multicriteria optimization procedure.A specific feature of the multicriteria optimization procedure is a possibility to obtain a set of the Edgeworth-Pareto optimal solutions within the frame of single optimization task. The paper presents some examples of the compressor’s geometrical parameters multicriteria optimization.

Integrating geometallurgical ball mill throughput predictions into short-term stochastic production scheduling in mining complexes

This article presents a novel approach to integrate a throughput prediction model for the ball mill into short-term stochastic production scheduling in mining complexes.The datasets for the throughput prediction model include penetration rates from blast hole drilling(measurement while drilling),geological domains,material types,rock density,and throughput rates of the operating mill,offering an accessible and cost-effective method compared to other geometallurgical programs.First,the comminution behavior of the orebody was geostatistically simulated by building additive hardness proportions from penetration rates.A regression model was constructed to predict throughput rates as a function of blended rock properties,which are informed by a material tracking approach in the mining complex.Finally,the throughput prediction model was integrated into a stochastic optimization model for short-term production scheduling.This way,common shortfalls of existing geometallurgical throughput prediction models,that typically ignore the non-additive nature of hardness and are not designed to interact with mine production scheduling,are overcome.A case study at the Tropicana Mining Complex shows that throughput can be predicted with an error less than 30 t/h and a correlation coefficient of up to 0.8.By integrating the prediction model and new stochastic components into optimization,the production schedule achieves weekly planned production reliably because scheduled materials match with the predicted performance of the mill.Comparisons to optimization using conventional mill tonnage constraints reveal that expected production shortfalls of up to 7%per period can be mitigated this way.

ANew Theoretical Framework forAnalyzing Stochastic Global Optimization Algorithms

In this paper, we develop a new theoretical framework by means of the absorbing Markov process theory for analyzing some stochastic global optimization algorithms. Applying the framework to the pure random search, we prove that the pure random search converges to the global minimum in probability and its time has geometry distribution. We also analyze the pure adaptive search by this framework and turn out that the pure adaptive search converges to the global minimum in probability and its time has Poisson distribution.

A New Stochastic Algorithm of Global Optimization ——Region's Walk and Contraction

This paper presents a new stochastic algorithm for box constrained global optimization problem. Bacause the level set of objective function is always not known, the authors designed a region containing the current minimum point to replace it, and in order to fit the level set well, this region would be walking and contracting in the running process. Thus, the new algorithm is named as region's walk and contraction(RWC). Some numerical experiments for the RWC were conducted, which indicate good property of the algorithm.

Distributed Heterogeneous Multi-Agent Optimization with Stochastic Sub-Gradient

This paper studies the optimization problem of heterogeneous networks under a timevarying topology.Each agent only accesses to one local objective function,which is nonsmooth.An improved algorithm with noisy measurement of local objective functions' sub-gradients and additive noises among information exchanging between each pair of agents is designed to minimize the sum of objective functions of all agents.To weaken the effect of these noises,two step sizes are introduced in the control protocol.By graph theory,stochastic analysis and martingale convergence theory,it is proved that if the sub-gradients are uniformly bounded,the sequence of digraphs is balanced and the union graph of all digraphs is joint strongly connected,then the designed control protocol can force all agents to find the global optimal point almost surely.At last,the authors give some numerical examples to verify the effectiveness of the stochastic sub-gradient algorithms.

Optimizing continuous cover management of boreal forest when timber prices and tree growth are stochastic

Background： Decisions on forest management are made under risk and uncertainty because the stand development cannot be predicted exactly and future timber prices are unknown. Deterministic calculations may lead to biased advice on optimal forest management. The study optimized continuous cover management of boreal forest in a situation where tree growth, regeneration, and timber prices include uncertainty. Methods： Both anticipatory and adaptive optimization approaches were used. The adaptive approach optimized the reservation price function instead of fixed cutting years. The future prices of different timber assortments were described by cross-correlated auto-regressive models. The high variation around ingrowth model was simulated using a model that describes the cross- and autocorrelations of the regeneration results of different species and years. Tree growth was predicted with individual tree models, the predictions of which were adjusted on the basis of a climate-induced growth trend, which was stochastic. Residuals of the deterministic diameter growth model were also simulated. They consisted of random tree factors and cross- and autocorrelated temporal terms. Results： Of the analyzed factors, timber price caused most uncertainty in the calculation of the net present value of a certain management schedule. Ingrowth and climate trend were less significant sources of risk and uncertainty than tree growth. Stochastic anticipatory optimization led to more diverse post-cutting stand structures than obtained in deterministic optimization. Cutting interval was shorter when risk and uncertainty were included in the analyses. Conclusions： Adaptive optimization and management led to 6%-14% higher net present values than obtained in management that was based on anticipatory optimization. Increasing risk aversion of the forest landowner led to earlier cuttings in a mature stand. The effect of risk attitude on optimization results was small.

POLLUTION, GOVERNMENT EXPENDITURE, TAXES AND STOCHASTIC GROWTH

This paper studies a stochastic endogenous growth model with pollution. It introduces government expenditure and exogenous pollution abatement technology to eliminate pollution and proves that under appropriate equilibrium conditions the main economic indexes （including economic growth rate, the optimal government expenditure rate） in the centrally planned economy and decentralized economy can be expressed by the parameters of the model uniquely. The optimal tax policy is analyzed ,and the optimal pollution is derived.