Stochastic Optimization Methods

Basic methods for treating stochastic optimization problems (SOP), hence, optimization problems with random data are presented: Optimization problems in practice depend mostly on several model parameters, noise factors, uncontrollable parameters, etc., which are not given fixed quantities at the planning stage. Typical examples from engineering and economics/operations research are: Material parameters (e.g., elasticity moduli, yield stresses, allowable stresses, moment capacities, specific gravity), external loadings, friction coefficients, moments of inertia, length of links, mass of links, location of the center of gravity of links, manufacturing errors, tolerances, noise terms, demand parameters, technological coefficients in input-output functions, cost factors, interest rates, exchange rates, etc. Due to several types of stochastic uncertainties (physical uncertainty, economic uncertainty, statistical uncertainty, model uncertainty) these parameters must be modeled by random variables having a certain probability distribution. In most cases at least certain moments of this distribution are known.n order to cope with these uncertainties, a basic procedure in the engineering/economic practice is to replace first the unknown parameters by some chosen nominal values, e.g., estimates, guesses, of the parameters. Then, the resulting and mostly increasing deviation of the performance (output, behavior) of the structure/system from the prescribed performance (output, behavior), i.e., the tracking error, is compensated by (online) input corrections. However, the online correction of a system/structure is often time consuming and causes mostly increasing expenses (correction or recourse costs). Very large recourse costs may arise in case of damages or failures of the plant. This can be omitted to a large extent by taking into account already at the planning stage the possible consequences of the tracking errors and the known prior and sample information about the random data of the problem. Hence, instead of relying on ordinary deterministic parameter optimization methods - based on some nominal parameter values—and applying then just some correction actions, stochastic optimization methods should be applied: Incorporating stochastic parameter variations into the optimization process, expensive and increasing online correction expenses can be omitted or at least reduced to a large extent. Consequently, for the computation of robust optimal decisions/designs, i.e., optimal decisions which are insensitive with respect to random parameter variations, appropriate deterministic substitute problems must be formulated first. Based on decision theoretical principles, these substitute problems depend on probabilities of failure/success and/or on more general expected cost/loss terms. Two basic types of deterministic substitute problems occur mostly in practice: Reliability-Based Optimization Problems: primary cost minimization subject to expected recourse (correction) cost constraints: Minimization of the expected primary costs subject to expected recourse cost constraints (reliability constraints) and remaining deterministic constraints, e.g., box constraints. In case of piecewise constant cost functions, probabilistic objective functions and/or probabilistic constraints occur; Expected Total Cost Minimization Problems: Minimization of the expected total costs (costs of construction, design, recourse/correction, repair costs, etc.) subject to the remaining deterministic constraints. Since probabilities and expectations are defined by multiple integrals in general, the resulting often nonlinear and also non-convex deterministic substitute problems can be solved by approximate methods only.