Counting solutions to linear and nonlinear constraints through Ehrhart polynomials

1996 ◽  
Author(s):  
Philippe Clauss
Keyword(s):  
Nonlinear Constraints ◽  
Ehrhart Polynomials ◽  
Linear And Nonlinear Constraints ◽  
Linear And Nonlinear ◽  
Counting Solutions

scholarly journals Incorporating User Input Into Optimal Constraining Procedures for Survey Estimates

2013 ◽  
Vol 29 (3) ◽  
pp. 375-396 ◽  
Author(s):  
Matthew Williams ◽  
Emily Berg
Keyword(s):  
First Principles ◽  
Contingency Table ◽  
Nonlinear Constraints ◽  
Constrained Estimation ◽  
Data Set ◽  
User Input ◽  
Optimal Estimates ◽  
Linear And Nonlinear Constraints ◽  
Linear And Nonlinear ◽  
Survey Estimates

Abstract We examine the incorporation of analyst input into the constrained estimation process. In the calibration literature, there are numerous examples of estimators with “optimal” properties. We show that many of these can be derived from first principles. Furthermore, we provide mechanisms for injecting user input to create user-constrained optimal estimates. We include derivations for common deviance measures with linear and nonlinear constraints and we demonstrate these methods on a contingency table and a simulated survey data set. R code and examples are available at https://github.com/mwilli/Constrained-estimation.git.

Robust Constrained Optimization of Short- and Long-Term Net Present Value for Closed-Loop Reservoir Management

SPE Journal ◽  
2012 ◽  
Vol 17 (03) ◽  
pp. 849-864 ◽  
Author(s):  
C.. Chen ◽  
G.. Li ◽  
A.C.. C. Reynolds
Keyword(s):  
Life Cycle ◽  
Net Present Value ◽  
Production Optimization ◽  
Nonlinear Constraints ◽  
Present Value ◽  
Short Term ◽  
Linear And Nonlinear Constraints ◽  
Linear And Nonlinear ◽  
Reservoir Description

Summary In this paper, we develop an efficient algorithm for production optimization under linear and nonlinear constraints and an uncertain reservoir description. The linear and nonlinear constraints are incorporated into the objective function using the augmented Lagrangian method, and the bound constraints are enforced using a gradient-projection trust-region method. Robust long-term optimization maximizes the expected life-cycle net present value (NPV) over a set of geological models, which represent the uncertainty in reservoir description. Because the life-cycle optimal controls may be in conflict with the operator's objective of maximizing short-time production, the method is adapted to maximize the expectation of short-term NPV over the next 1 or 2 years subject to the constraint that the life-cycle NPV will not be substantially decreased. The technique is applied to synthetic reservoir problems to demonstrate its efficiency and robustness. Experiments show that the field cannot always achieve the optimal NPV using the optimal well controls obtained on the basis of a single but uncertain reservoir model, whereas the application of robust optimization reduces this risk significantly. Experimental results also show that robust sequential optimization on each short-term period is not able to achieve an expected life-cycle NPV as high as that obtained with robust long-term optimization.

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