A basic characteristic of applied optimization problems arising in engineering is the fact that the data (parameters) of these problems, e.g. material parameters (yield stresses, allowable stresses, moment capacities, specific gravity, ...), external loadings, payloads, manufacturing errors, tolerances, cost factors, etc., are not known in advance, at the planning stage, but have to be considered as random variables having a certain probability distribution; in addition, there is always some uncertainty in the mathematical modelling of concrete (technical) problems. Typical problems of this type are e.g. the

• Limit (collapse) load analysis (plastic analysis) and the plastic or elastic design of mechanical structures represented mathematically by means of
• the equilibrium equation,
• (generalized) Hooke's law and
• the compatibility equation.
Here, the basic condition for survival of the structure, for a safe system, are given by certain constraints for the interior member loads (forces, moments) and/or for the joint (node) displacements.
• Optimal trajectory planning and online control of industrial and service robots. Here, the underlying mechanical system is described by the kinematic and dynamic equation of the robot.

Having to deal in practice mostly with unknown random parameters, a standard engineering procedure is to replace first the unknown parameters by some chosen nominal values, e.g. estimates of the parameters. Then, the resulting deviation of the performance (output, behavior) of the structure/system from the prescribed performance (output, behavior) must be compensated by increasing (online) corrections. However, the (online) correction of an optimal decision, i.e., an optimal structural design or an optimal control of a dynamic system, is time-consuming and causes mostly increasing expenses (costs). This can be omitted to a large extent by taking into account at the planning stage also the known prior and - at later (replanning) stages - statistical information about the random data of the problem. Hence, instead of relying on ordinary parameter optimization methods - based on some nominal parameter values - and to apply then just some correction actions, stochastic optimization methods must be applied reducing expensive online correction actions.

Consequently, in order to get robust optimal decisions, i.e., optimal designs, optimal controls, resp., which are insensitive with respect to stochastic parameter variations, the original optimization problem with random data must be replaced - using certain decision theoretical criteria - by an appropriate deterministic substitute problem incorporating random parameter variations.

Fundamental deterministic substitute problems are:

• Reliability-Based Optimization (RBO) Problems
Minimization of the expected costs of construction, design etc., subject to reliability constraints
• Total Cost Minimization
Minimization of the total expected costs (costs of construction, design, penalty costs, etc.) subject to the remaining (simple) deterministic constraints, e.g. box constraints.

Thus, besides the usual optimizational difficulties of engineering-type optimization problems (high nonlinearity, very large scale problems, etc.), a central problem is the analytical and numerical treatment of the occuring expectation (mean) value and probability functions:

• Analytical consideration of the basic mathematical properties (differentiability, convexity, etc.) of functions defined by multiple integrals, i.e., mean value and probability functions; analytical approximations of functions of this type;
• Derivation of differentiation formulas for mean value and probability functions;
• Efficient numerical computation of mean value and probability functions (estimation, regression methods);
• Efficient numerical computation of derivatives of mean value and probability functions.

The purpose of the present Stochastic Optimization homepage is therefore to provide information on

• mathematical results about analytical properties of stochastic optimization problems
• numerical tools (Taylor expansion, simulation methods, as e.g. Regression Techniques, Response Surface Methods (RSM), Design of Experiments (DOE), stochastic approximation methods, especially stochastic gradient methods)
• optimization software
for coping with optimization problems from engineering having random parameters.