scholarly journals A stability result for linear Markovian stochastic optimization problems

2020 ◽  
Author(s):  
Adriana Kiszka ◽  
David Wozabal
Keyword(s):  
Stochastic Optimization ◽  
Stochastic Processes ◽  
Objective Function ◽  
Markov Processes ◽  
Optimization Problems ◽  
Stability Result ◽  
Extant Literature ◽  
Numerical Example ◽  
Optimal Values ◽  
Objective Value

Abstract In this paper, we propose a semi-metric for Markov processes that allows to bound optimal values of linear Markovian stochastic optimization problems. Similar to existing notions of distance for general stochastic processes, our distance is based on transportation metrics. As opposed to the extant literature, the proposed distance is problem specific, i.e., dependent on the data of the problem whose objective value we want to bound. As a result, we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature. We derive several properties of the proposed semi-metric and demonstrate its use in a stylized numerical example.

scholarly journals Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Piyush Pandita ◽  
Ilias Bilionis ◽  
Jitesh Panchal
Keyword(s):  
Global Optimization ◽  
Stochastic Optimization ◽  
Objective Function ◽  
Information Acquisition ◽  
Optimization Problems ◽  
Epistemic Uncertainty ◽  
Optimization Under Uncertainty ◽  
Expected Improvement ◽  
Oil Well ◽  
Placement Problem

Design optimization under uncertainty is notoriously difficult when the objective function is expensive to evaluate. State-of-the-art techniques, e.g., stochastic optimization or sampling average approximation, fail to learn exploitable patterns from collected data and require a lot of objective function evaluations. There is a need for techniques that alleviate the high cost of information acquisition and select sequential simulations optimally. In the field of deterministic single-objective unconstrained global optimization, the Bayesian global optimization (BGO) approach has been relatively successful in addressing the information acquisition problem. BGO builds a probabilistic surrogate of the expensive objective function and uses it to define an information acquisition function (IAF) that quantifies the merit of making new objective evaluations. In this work, we reformulate the expected improvement (EI) IAF to filter out parametric and measurement uncertainties. We bypass the curse of dimensionality, since the method does not require learning the response surface as a function of the stochastic parameters, and we employ a fully Bayesian interpretation of Gaussian processes (GPs) by constructing a particle approximation of the posterior of its hyperparameters using adaptive Markov chain Monte Carlo (MCMC) to increase the methods robustness. Also, our approach quantifies the epistemic uncertainty on the location of the optimum and the optimal value as induced by the limited number of objective evaluations used in obtaining it. We verify and validate our approach by solving two synthetic optimization problems under uncertainty and demonstrate it by solving the oil-well placement problem (OWPP) with uncertainties in the permeability field and the oil price time series.

scholarly journals STOCHASTIC OPTIMIZATION IN MULTIVARIATE STRATIFIED DOUBLE SAMPLING DESIGN

2020 ◽  
Vol 5 (1) ◽  
pp. 115-122
Author(s):  
Ziaul Bakhshi
Keyword(s):  
Stochastic Optimization ◽  
Objective Function ◽  
Sample Size ◽  
Sampling Design ◽  
Computational Procedure ◽  
Optimum Allocation ◽  
Double Sampling ◽  
Numerical Example

This paper deals with optimum allocation of sample size in stratified double sampling when costs are considered as random in the objective function. When costs function are random, by applying modified E. model, objective function is converted into an equivalent deterministic form. A Numerical example is presented to illustrate the computational procedure and the problem is solved by using LINGO Software.

scholarly journals Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions

2016 ◽  
Author(s):  
Piyush Pandita ◽  
Ilias Bilionis ◽  
Jitesh Panchal
Keyword(s):  
Global Optimization ◽  
Stochastic Optimization ◽  
Objective Function ◽  
Design Optimization ◽  
Information Acquisition ◽  
Optimization Problems ◽  
Epistemic Uncertainty ◽  
Optimization Under Uncertainty ◽  
Expected Improvement ◽  
Convergence Criterion

Design optimization under uncertainty is notoriously difficult when the objective function is expensive to evaluate. State-of-the-art techniques, e.g., stochastic optimization or sampling average approximation, fail to learn exploitable patterns from collected data and, as a result, they tend to require an excessive number of objective function evaluations. There is a need for techniques that alleviate the high cost of information acquisition and select sequential simulations in an optimal way. In the field of deterministic single-objective unconstrained global optimization, the Bayesian global optimization (BGO) approach has been relatively successful in addressing the information acquisition problem. BGO builds a probabilistic surrogate of the expensive objective function and uses it to define an information acquisition function (IAF) whose role is to quantify the merit of making new objective evaluations. Specifically, BGO iterates between making the observations with the largest expected IAF and rebuilding the probabilistic surrogate, until a convergence criterion is met. In this work, we extend the expected improvement (EI) IAF to the case of design optimization under uncertainty. This involves a reformulation of the EI policy that is able to filter out parametric and measurement uncertainties. We by-pass the curse of dimensionality, since the method does not require learning the response surface as a function of the stochastic parameters. To increase the robustness of our approach in the low sample regime, we employ a fully Bayesian interpretation of Gaussian processes by constructing a particle approximation of the posterior of its hyperparameters using adaptive Markov chain Monte Carlo. An addendum of our approach is that it can quantify the epistemic uncertainty on the location of the optimum and the optimal value as induced by the limited number of objective evaluations used in obtaining it. We verify and validate our approach by solving two synthetic optimization problems under uncertainty. We demonstrate our approach by solving a challenging engineering problem: the oil-well-placement problem with uncertainties in the permeability field and the oil price time series.

Chapter 24. Maximum Satisfiabiliy

2021 ◽  
Author(s):  
Fahiem Bacchus ◽  
Matti Järvisalo ◽  
Ruben Martins
Keyword(s):  
Objective Function ◽  
Real World ◽  
Optimization Problems ◽  
Truth Assignment ◽  
Maximum Satisfiability ◽  
Performance Improvements ◽  
Algorithm Portfolios ◽  
Recent Developments ◽  
Significant Performance ◽  
Objective Value

Maximum satisfiability (MaxSAT) is an optimization version of SAT that is solved by finding an optimal truth assignment instead of just a satisfying one. In MaxSAT the objective function to be optimized is specified by a set of weighted soft clauses: the objective value of a truth assignment is the sum of the weights of the soft clauses it satisfies. In addition, the MaxSAT problem can have hard clauses that the truth assignment must satisfy. Many optimization problems can be naturally encoded into MaxSAT and this, along with significant performance improvements in MaxSAT solvers, has led to MaxSAT being used in a number of different application areas. This chapter provides a detailed overview of the approaches to MaxSAT solving that have in recent years been most successful in solving real-world optimization problems. Further recent developments in MaxSAT research are also overviewed, including encodings, applications, preprocessing, incomplete solving, algorithm portfolios, partitioning-based solving, and parallel solving.

Scalable Computing Architecture for Time-Dependent Transportation Optimization Problems Based on High-Performance Computing Techniques

2019 ◽  
Vol 2673 (4) ◽  
pp. 205-216 ◽  
Author(s):  
Pengfei (Taylor) Li ◽  
Peirong (Slade) Wang ◽  
Farzana Chowdhury ◽  
Li Zhang
Keyword(s):  
Objective Function ◽  
High Performance Computing ◽  
High Performance ◽  
Optimization Problems ◽  
Dynamic Network ◽  
Alternative Formulation ◽  
High Fidelity ◽  
Objective Functions ◽  
Performance Computing ◽  
Transportation Optimization

Traditional formulations for transportation optimization problems mostly build complicating attributes into constraints while keeping the succinctness of objective functions. A popular solution is the Lagrangian decomposition by relaxing complicating constraints and then solving iteratively. Although this approach is effective for many problems, it generates intractability in other problems. To address this issue, this paper presents an alternative formulation for transportation optimization problems in which the complicating attributes of target problems are partially or entirely built into the objective function instead of into the constraints. Many mathematical complicating constraints in transportation problems can be efficiently modeled in dynamic network loading (DNL) models based on the demand–supply equilibrium, such as the various road or vehicle capacity constraints or “IF–THEN” type constraints. After “pre-building” complicating constraints into the objective functions, the objective function can be approximated well with customized high-fidelity DNL models. Three types of computing benefits can be achieved in the alternative formulation: ( a) the original problem will be kept the same; ( b) computing complexity of the new formulation may be significantly reduced because of the disappearance of hard constraints; ( c) efficiency loss on the objective function side can be mitigated via multiple high-performance computing techniques. Under this new framework, high-fidelity and problem-specific DNL models will be critical to maintain the attributes of original problems. Therefore, the authors’ recent efforts in enhancing the DNL’s fidelity and computing efficiency are also described in the second part of this paper. Finally, a demonstration case study is conducted to validate the new approach.

scholarly journals Global optimization based on active preference learning with radial basis functions

2020 ◽  
Author(s):  
Alberto Bemporad ◽  
Dario Piga
Keyword(s):  
Global Optimization ◽  
Objective Function ◽  
Decision Maker ◽  
Optimization Problems ◽  
Global Optimum ◽  
Bayesian Optimization ◽  
Preference Learning ◽  
Global Optimizer ◽  
Radial Basis ◽  
Decision Vector

AbstractThis paper proposes a method for solving optimization problems in which the decision-maker cannot evaluate the objective function, but rather can only express a preference such as “this is better than that” between two candidate decision vectors. The algorithm described in this paper aims at reaching the global optimizer by iteratively proposing the decision maker a new comparison to make, based on actively learning a surrogate of the latent (unknown and perhaps unquantifiable) objective function from past sampled decision vectors and pairwise preferences. A radial-basis function surrogate is fit via linear or quadratic programming, satisfying if possible the preferences expressed by the decision maker on existing samples. The surrogate is used to propose a new sample of the decision vector for comparison with the current best candidate based on two possible criteria: minimize a combination of the surrogate and an inverse weighting distance function to balance between exploitation of the surrogate and exploration of the decision space, or maximize a function related to the probability that the new candidate will be preferred. Compared to active preference learning based on Bayesian optimization, we show that our approach is competitive in that, within the same number of comparisons, it usually approaches the global optimum more closely and is computationally lighter. Applications of the proposed algorithm to solve a set of benchmark global optimization problems, for multi-objective optimization, and for optimal tuning of a cost-sensitive neural network classifier for object recognition from images are described in the paper. MATLAB and a Python implementations of the algorithms described in the paper are available at http://cse.lab.imtlucca.it/~bemporad/glis.

On the stochastic optimization problems of plastic metal-working processes

1997 ◽  
Vol 84 (3) ◽  
pp. 1109-1112 ◽  
Author(s):  
M. B. Gitman ◽  
P. V. Trusov ◽  
S. A. Fedoseev
Keyword(s):  
Stochastic Optimization ◽  
Optimization Problems ◽  
Metal Working ◽  
Plastic Metal ◽  
Working Processes

scholarly journals Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.

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