Stochastic linearization

Atlians, M. (1971) The Role and Use of the Stochastic Linear-Quadratic-Gaussian Problem in Control System Design, IEEE Trans, on Automatic Control AC-16, 6, pp. 529-551. 

The constraints of a two-stage stochastic linear program can be classified into constraints on the first-stage variables only (9.3.2) and constraints on the first and on the second-stage variables (9.3.3). The latter represent the interdependency of the stages. All constraints are represented as linear inequalities with the matrices Aer>x">, Tb6 R 2xn Wm Rm2x 2, and the vectors b Rm2 and K, e R 2. 

Castillo-Garit, J.A., Marrero-Ponce, Y., Torrens, F., Garcia-Domenech, R. Estimation of ADME properties in drug discovery predicting Caco-2 cell permeability using atom-based stochastic and non-stochastic linear indices. J. Pharm. Sci. 2008, 97, 1946-76. 

Consideration of the expected value of profit alone as the objective function, which is characteristic of the classical stochastic linear programs introduced by Dantzig (1955) and Beale (1955), is obviously inappropriate for moderate and high-risk decisions under uncertainty since most decision makers are risk averse in facing important decisions. The expected value objective ignores both the risk attribute of the decision maker and the distribution of the objective values. Hence, variance of each of the random price coefficients can be adopted as a viable risk measure of the objective function, which is the second major component of the MV approach adopted in Risk Model I. 

Wood (1999), Linderoth, Shapiro and Wright (2002) for stochastic linear problems, Kleywegt, Shapiro and Homem-De-Mello (2001), Verweij et al. (2003) for stochastic integer problems, and Wei and Realff (2004), Goyal and Ierapetritou (2007) for MINLP problems. Problem (7.15) can be solved iteratively in order to provide statistical bounds on the optimality gap of the obj ective function value. For details and proofs see N orkin, Pflug and Ruszczysk (1998) and Mark, Morton and Wood (1999). The procedure consists of a number of steps as described in the following section. 

Birge, J.R. (1982) Stochastic linear programs with fixed recourse. 

Madansky, A. (1960) Inequalities for stochastic linear programming problems. Management Science, 6, 197. 

A standard formulation of the two-stage stochastic linear program is ... 

Vladimirou H, Zenios SA (1997) Stochastic linear programs with restricted recourse. European Journal of Operational Research 101 177-192... 

F. Krajewski and M. Parrinello (2005) Stochastic linear scaling for metals and nonmetals. Phys. Rev. B 71, p. 233105 F. Krajewski, M. Parrinello, Linear scaling electronic structure calculations and accurate sampling with noisy forces, cond-mat/05 0842 0... 

Marrero-Ponce, Y, Marrero, R.M., Torrens, F., Martinez, Y, Bernal, M.L., Zaldivar, V.R., Castro, E. A. and Abalo, R.G. (2006) Non-stochastic and stochastic linear indices of the molecular pseudograph s atom-adjacency matrix a novel approach for computational in sllico screening and rational selection of new lead antibacterial agents. J. Mol. Model., 12, 255-271. 

Marrero-Ponce, Y, Montero-Torres, A., Zaldivar, C. R., Veitia, M.I., Mayon Perez, M. and Garda Sanchez, R.N. (2005) Non-stochastic and stochastic linear indices of the molecular pseudographs atom adjacency matrix application to in silico studies for the rational discovery of new antimalarial compounds. Bioorg. Med. Chem., 13, 1293—1304. 

Higle J.L. and Sen S. 1996. Stochastic Decomposition. A Statistical Method for Large Scale Stochastic Linear Programming. Kluwer Academic Publishers, Norwell, MA. 

Infanger G. 1994. Planning under Uncertainty Solving Large-Scale Stochastic Linear Programs. Boyd and Fraser, Danvers, MA. 

A two-stage stochastic linear program with fixed recourse is a two-stage stochastic program of the form... 

A basic assumption of stochastic programming is that the probability distribution of the random variable is known. The target then is to find an optimal solution that makes the expected value of the system to be minimum (or maximum). According to the type of the objective function and constraints, the stochastic programming problem can be divided into stochastic linear programming problems and stochastic nonlinear programming problems. 

The model above which contains many parameters and constraints is a large-scale, stochastic linear mixed integer convex programming model. What follows is the explanation and illustration for components of the model. 

The above model contains many parameters and constraints, which makes it a large-scale and complex stochastic linear mixed integer programming model. The components of the model are described and explained as follows ... 

The main principle of stochastic linearization is the minimization of the expected value of the difference between the original nonlinear Bouc-Wen (Eq. 5) and its linear surrogate... 

Hurtado, J.E. Barbat, A. 1996. Improved stochastic linearization method using mixed distributions. Structural Safety 18, 49-62. 

A very efficient computational procedure for the reliability-based optimization of uncertain stochastic linear dynamical systems... 

In the last contribution, Jensen and Valdebenito (Chapter 35) deal with an efficient computational procedure for the reliability-based optimization of uncertain stochastic linear dynamical systems. The reliability-based optimization problem is formulated as the minimization of an objective function for a specified failure probability. The probability that design conditions are satisfied within a given time interval is used as a measure of the system reliability. Approximation concepts are used to construct... 

Rico-Ramirez, V., 2002, Two-Stage Stochastic Linear Programming A Tutorial, SIAG/OPT Views and News, 13(1), 8-14. 

Atalik, T. S., and Utku, S., "Stochastic Linearization of Multidegree-of-Freedom Nonlinear Systems," Journal of Earthquake Engineering and Structural Dynamics, Vol. 4, 1976. 

T.S.Atallk and S. Utku, Stochastic linearization of multl-degree-of-freedom non-linear systems. Earthquake Engineering andStructurar Dynamics 4,411 -420 (1976). 

P.D. Spanos, Stochastic linearization in structural dynamics, Appl. Mech. Rev.,... 

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