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Iterative method for optimal control problems for Markov chains, discussing degenerate nonlinear elliptic equations solution in optimal stochastic control theory
Linear time dependent stochastic optimal control with nonquadratic performance indices using function space approach
We propose a stochastic gradient descent based optimization algorithm to solve the analytic continuation problem in which we extract real frequency spectra from imaginary time Quantum Monte Carlo data. The procedure of analytic continuation is an ill-posed inverse problem which is usually solved by regularized optimization methods, such like the Maximum Entropy method, or stochastic optimization methods. The main contribution of this work is to improve the performance of stochastic optimization approaches by introducing a supervised stochastic gradient descent algorithm to solve a flipped inverse system which processes the random solutions obtained by a type of Fast and Efficient Stochastic Optimization Method.
Optimal control problems for Markov chains solved by iterative method, using nonlinear finite difference equations to approximate degenerate elliptic functions
Necessary conditions for discrete parameter stochastic optimization problems
The concept of circular economy has been diffused in recent decades to promote economic growth that does not add to the burden on natural resource extraction. Re-X options (e.g., reuse, repair, refurbish, remanufacture, recycle) have been gradually adopted in the product development process and optimized to reduce or eliminate waste and pollution. Although manufacturing incorporating Re-X options can be more environmentally friendly, it involves more sources of uncertainty than traditional manufacturing since the end-of-use products can be collected from multiple origins with various quantities and qualities, and the market demand for both new and remanufactured products cannot be forecasted perfectly. Thus, there is a need to optimize the Re-X policy to alleviate the negative impacts of the higher uncertainty. One option is using the reliability information of new products to estimate the end-of-use conditions and applying multi-stage stochastic optimization to capture multiple demand scenarios. This paper develops a two-stage stochastic optimization model to optimize the quality thresholds for reuse, recycling, and remanufacturing options. Our objective is to minimize the total cost, energy consumption, and environmental impact of producing and providing warranty service for a product family. The model employs reliability information of product components to estimate the warranty service cost and the end-of-use conditions of the returned resources. A case study on a general product family is implemented to illustrate the efficacy of the optimization model. Finally, results show that the two-stage optimization can achieve cost and environmental impact reduction for a hybrid manufacturing and remanufacturing process.
In the construction of a stellarator, the manufacturing and assembling of the coil system is a dominant cost. These coils need to satisfy strict engineering tolerances, and if those are not met the project could be cancelled as in the case of the National Compact Stellarator Experiment (NCSX) project. Therefore, our goal is to find coil configurations that increase construction tolerances without compromising the performance of the magnetic field. In this paper, we develop a gradient-based stochastic optimization model which seeks robust stellarator coil configurations in high dimensions. In particular, we design a two-step method: first, we perform an approximate global search by a sample efficient trust-region Bayesian optimization; second, we refine the minima found in step one with a stochastic local optimizer. To this end, we introduce two stochastic local optimizers: BFGS applied to the sample average approximation; and Adam, equipped with a control variate for variance reduction. Numerical simulations performed on a W7-X-like coil configuration demonstrate that our global optimization approach finds a variety of promising local solutions at less than 0.1% of the cost of previous work, which considered solely local stochastic optimization.
Dynamic programming applications to optimal stochastic orbital transfer strategy, describing computer program
Function space approach to linear stochastic optimal control systems
Dynamic programming applications to optimal stochastic orbital transfer strategy, describing computer program
Linear system optimal stochastic control and observation strategies simultaneous determination with quadratic cost by dynamic programming
As commercial nuclear power plants (NPPs) pursue extended plant operations in the form of Second License Renewals (SLRs), opportunities exist for these plants to provide capital investments to ensure long-term, safe, and economic performance. Several utilities have already announced their intention to pursue extended operations for one or more of their NPPs via SLR2. The goal of this research is to develop a riskinformed approach to evaluate and prioritize plant capital investments made in preparation for, and during the period of, extended plant operations to support decisions in NPP operations. In order to prioritize project selection via a riskinformed approach we developed a single decision-making tool that integrates safety/reliability, cost, and stochastic optimization models to provide users with data analysis capabilities to more cost effectively manage plant assets. Both stochastic analysis methods—such as Monte Carlo-based sampling strategies—and multi-stage stochastic optimization strategies are employed to provide priority lists to decisionmakers in support of risk-informed decisions. We applied the proposed method to a trial application of projected replacement/refurbishment expenditures for plant capital assets (i.e., Structures, Systems, and Components [SSCs]). The objective is to optimize the SSC replacement/refurbishment schedule in terms of economic constraints, data uncertainties, and SSC reliability data, as well to generate a priority list for maximizing returns on investment.
Linear systems simultaneous optimal stochastic control and observation strategies, assuming quadratic observation cost
Abstract variational theory application to continuous parameter stochastic optimization problems to derive maximum principles in linear programming
In this paper, we present a novel derivative-free framework for solving unconstrained stochastic optimization problems. Many problems in fields ranging from simulation optimization to reinforcement learning to quantum computing involve settings where only stochastic function values are obtained via a zeroth-order oracle, which has no available gradient information and necessitates the usage of derivative-free optimization methodologies. Our approach includes estimating gradients using stochastic function evaluations and integrating adaptive sampling techniques to control the accuracy in these stochastic approximations. Our framework encapsulates several gradient estimation techniques, including standard finite-difference, Gaussian smoothing, sphere smoothing, randomized coordinate finite-difference, and randomized subspace finite-difference methods. We provide theoretical convergence guarantees for our framework and analyze the worst-case iteration and sample complexities associated with each gradient estimation method. Finally, we demonstrate the empirical performance of the methods on logistic regression and nonlinear least squares problems.
Rapid integration of variable renewable energy sources (VRES) has made modeling and stochastic optimization of hybrid energy systems crucial for studying their long-term performance and viability. However, most studies have focused on just historical data, which may be unreliable for capturing short-term fluctuations, rare events, and long-term patterns of energy demand, price, and the variability of renewable energy sources. For this study, optimal synthetic time series models were developed using Wasserstein distance. The models were validated by comparing the key statistical measures against those of the historical data. They were then used to optimize the integrated Natrium-style advanced energy systems and their long-term (30 years) economics. The stochastic model performs bi-level optimization to find the optimal sizes for the balance of plant and thermal energy storage, while also optimizing energy dispatch to achieve the maximum net present value. In studies of two deregulated markets (California ISO and the Electric Reliability Council of Texas), the integrated Natrium-style system performed better in CAISO than in ERCOT, given higher and more consistent electricity prices during peak-demand periods. The potentially enlarged cost associated with the variable operation and maintenance of the TES system also plays a significant role in driving the system sizing, thus its impacts on the system are investigated in detail through comparison against a baseline case. The study also finds that the bi-level optimization results based on stochastic gradient descent closely match the grid search results. The uncertainty quantification of the stochastic signals provides further NPV-related insights and probability distributions for the case studies. The normal standard error of the mean of NPV for the case with and without TES VOM for CAISO were found to be 7.73M (plus-minus sign) 1.09M USD and 104.99M (plus-minus sign) 1.25M USD, respectively based on a 95% confidence. Given the relatively small NPV variance based on 150 samples, the analysis affords the most robust possible prediction of the techno-economic performance of the integrated Natrium-style energy systems.
A computer program is described which solves the linear stochastic optimal control and estimation (LSOCE) problem by using a time-domain formulation. The LSOCE problem is defined as that of designing controls for a linear time-invariant system which is disturbed by white noise in such a way as to minimize a performance index which is quadratic in state and control variables. The LSOCE problem and solution are outlined; brief descriptions are given of the solution algorithms, and complete descriptions of each subroutine, including usage information and digital listings, are provided. A test case is included, as well as information on the IBM 7090-7094 DCS time and storage requirements.
A computational approach is taken to solve the optimal nonlinear stochastic control problem. The approach is to systematically solve the stochastic dynamic programming equations forward in time, using a nested stochastic approximation technique. Although computationally intensive, this provides a straightforward numerical solution for this class of problems and provides an alternative to the usual dimensionality problem associated with solving the dynamic programming equations backward in time. It is shown that the cost degrades monotonically as the complexity of the algorithm is reduced. This provides a strategy for suboptimal control with clear performance/computation tradeoffs. A numerical study focusing on a generic optimal stochastic adaptive control example is included to demonstrate the feasibility of the method.