Abstract
Optimization is the art of finding the best among several alternatives in decision making.
Let S be the set of possible decisions. This set is called the feasible set. The decision variable is denoted by x. If x ε S, then x is called feasible, otherwise infeasible. The net costs caused by decision x are measured by a real valued objective function F(x). The goal is to find the best decision, i.e. the decision with minimal costs. We will always assume that there is one single objective function. The case of several competing objective functions, the multi-criteria decision making problem will not be touched here.
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Bibliography
Aaarts E., Korst J. (1989). Simulated annealing and Boltzmann machines: A stochastic approach to combinatorial optimization and neural computing. J. Wiley and Sons, New York.
Andradottir S. (1994). A global search method for discrete stochastic optimization. To appear in: Siam J. of Optimization.
Aarts E., Laarhoven P. (1987). Simulated Annealing: Theory and Applications. Kluwer Academic Publishers.
Bazaraa M.S., Shetty C.M. (1979). Nonlinear Programming-Theory and Algorithms. J. Wiley and Sons, New York.
Clarke F.H. (1983). Optimization and Nonsmooth Analysis. J. Wiley and Sons, New York.
Dantzig G. (1963). Linear Programming and extensions. Princeton University Press.
David H.A. (1970). Order statistics. J. Wiley and Sons, New York.
Ermoliev Yu., Norkin V., Ruszczynski A. (1994). Optimal allocation of indivisibles under uncertainty. IIASA working paper WP-94–21, Laxenburg, Austria.
Fabian V. (1960). Stochastic approximation methods. Czech. Math. Journal 10, 123–159.
Fabian V. (1967). Stochastic approximation of minima with improved asymptotic speed. Ann. Math. Statist. 38, 191–200.
Fletcher R. (1981). Practical Methods of Optimization. J. Wiley and Sons, Chichester.
Frauendorfer K. (1992). Stochastic Two-Stage Programming. Springer Lecture Notes in Economics and Mathematical Systems 392, Springer Verlag.
Gill P.E., Murray W., Wright M.H. (1981). Practical optimization. Academic Press, London.
Glasserman P. (1991). Gradient Estimation via Perturbation Analysis. Kluwer Academic Publishers, Norwell, USA.
Gnedenko B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423–453.
Greenstadt J. (1970). Variations on variable metric methods. Math. Computation 24 (1), 1–22.
Gutjahr W., Pflug G. Ch. (1995). Simulated Annealing for noisy cost functions. J. Global Optimization 8, 1–13.
Hening M.I., Levin M. (1992). Joint planning and product delivery comittments with random yield. Operations Research 40 (2), 404–408.
Hestenes M. R. (1980). Conjugate Direction Methods in Optimization. Applications of Mathematics 12, Springer Verlag, New York.
Hiriart-Urruty J.B. (1977). Algorithms of penalization type and of dual type for the solution of stochastic optimization problems with stochastic constraints. Recent developments in statistics (J.R. Barra et al ed.). North Holland Publishing Company, 183–219.
Higle J., Sen S. (1996). Stochastic Decomposition. Kluwer Academic Publishers, Norwell, USA.
Kesten H. (1958). Accelerated Stochastic Approximation. Ann. Math. Statist. 29, 41–59.
Kiefer J., Wolfowitz J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statist. 23, 462–466.
Kirkpatrick S., Gelatt C.D., Vecchi M.P. (1983). Optimization by simulated annealing. Science 220, 671–680.
Kiwiel K. (1990). Proximy control in bundle methods for convex nondifferentiable optimization. Math. Programming 46, 105–122.
Kushner H. J., Sanvincente E. (1975). Stochastic approxiamation for constrained system with observation noise on the system and constraints. Automatica 11, 375–380.
Lemarechal C. (1989). Nondifferentiable optimization. In: Handbooks of Operations Research and Management Science, Vol. 1, Optimization (Nemhauser G.L., Rinnoy Kan A.H.G., Todd M.J., eds.). North Holland, Amsterdam.
Marti K. (1982). On Accelerations of the Convergence in Random Search Methods. Operations Research Verfahren.
Marquardt D.W. (1963). An algorithm for least squares estimation of nonlinear parameters. SIAM Journal of Ind. and Appl Math. 11, 431–441.
Metropolis N., Rosenbluth A., Rosenbluth M., Teller A., Teller E. (1953). Equation of state calculations by fast computing machines. J. Chemical Physics 21, 1087–1092.
Mihram G.A. (1972). Simulation: Statistical foundations and methodology. Academic Press.
Nelder J.A., Mead R. (1964). A simplex method for function minimization. Computer J. 7, 308–313.
Norkin V., Pflug G. and Ruszczynski A. (1996). Branch and Bound methods for stochastic global optminization. IIAS A working paper, II AS A, Laxenburg, Austria.
Peressini A.L., Sullivan F.E., Uhl J.J. (1988). The Mathematics of nonlinear programming. Undergraduate Texts in Mathematics, Springer Verlag.
Plambeck E.L., Fu B.R., Robinson S.M., Suri R. (1994). Sample Path Optimization of Convex Stochastic Performance Functions. Mathematical Programming, Series B.
Pflug G. (1981). On the convergence of a penalty-type stochastic approximation procedure. J. Information and Optimization Sciences 2, 249–258.
Pflug G. (1990). Non-asymptotic Confidence Bounds for Stochastic Approximation Algorithms with Constant Step Size. Monatshefte für Math-ematik, 110, 297–314.
Polak E. (1971). Computational methods in optimization. Academic Press, New York.
Rockafellar T. (1973). A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Programming 5, 354–373.
Robbins H., Monro S. (1951). A stochastic approximation method. Ann. Math. Statist 22, 400–407.
Ruppert D. (1985). A Newton-Raphson version of the Multivariate Robbins-Monro Procedure. Ann. Statist 13 (1), 236–245.
Rubinstein, R. (1981). Simulation and the Monte Carlo Method. J. Wiley and Sons, New York.
Rubinstein R. Y. (1986). Monte Carlo Optimization, Simulation and Sensitivity of Queuing Networks. J. Wiley and Sons, New York.
Rubinstein R.Y., Shapiro A. (1993). Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. J. Wiley and Sons, New York.
Schramm H., Zowe J. (1992). A restricted step proximal bundle method for nonconvex nondifferentiable optimization. In: Nonsmooth optimization; methods and applications. (Kiwiel K.C. editor). Gordon and Breach, 175–188.
Shapiro A. (1994). Quantitative stability in stochastic programming. Math. Programming 67, 99–108.
Solis F.J., Wets R.J.B. (1981). Minimization by random search techniques. Mathematics of Operations Research 6 (1), 19–30.
Spendley W., Hext G.R., Himsworth F.R. (1962). Sequential Application of Simplex Design in Optimization and Evolutionary Operation. Technometrics 4 (1), 441–461.
Suri R., Dille J.W. (1985). A technique for online sensitivity analysis of flexible manufacturung systems. Annals of Operations Research 3, 381–391.
Yan D., Mukai H. (1992). Stochastic discrete optimization. SIAM J. Control and Optimization 30, 594–612.
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Pflug, G.C. (1996). Optimization. In: Optimization of Stochastic Models. The Kluwer International Series in Engineering and Computer Science, vol 373. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1449-3_1
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DOI: https://doi.org/10.1007/978-1-4613-1449-3_1
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