Stochastic Programming Methods and Technical Applications

The objective of stochastic linear programs with recourse contains a multivariate integral 2(x) = ∫_Ξ Q(x,ξ)P(dξ), in general. This, although having convenient properties under mild assumptions (like e.g. convexity, smoothness), causes difficulties in computational solution procedures. Therefore we usually replace 2(•) by successively improved lower and upper bounding functions more amenable to optimization procedures, the involved bounding functions being solutions to various (generalized) moment problems.

A power generation system comprising thermal and pumpedstorage hydro plants is considered. Two kinds of models for the cost-optimal generation of electric power under uncertain load are introduced: (i) a dynamic model for the short-term operation and (ii) a power production planning model. In both cases the presence of stochastic data in the optimization model leads to multi-stage and two-stage stochastic programs respectively. Both stochastic programming problems involve a large number of mixed-integer (stochastic) decisions but their constraints are loosely coupled across operating power units. This is used to design Lagrangian relaxation methods for both models which lead to a decomposition into stochastic single unit subproblems. For the dynamic model a Lagrangian decomposition based algorithm is described in more detail. Special emphasis is put on a discussion of the duality gap the efficient solution of the multi-stage single unit subproblems and on solving the dual problem by bundle methods for convex nondifferentiable optimization.

If a robot has to perform a specified manipulation task involving intentional environmental contacts, a certain response behavior is desired to reduce strains and ensure successful completion without damage of the contacting bodies. On the other hand, the dynamic behavior of a manipulator depends strongly on its position and the gains of its joint controllers. Hence, varying these parameters for an optimized performance during manipulation seems to be an obvious task. In order to deal with impacts, oscillations and constrained motion, a model-based optimization approach is suggested, which relies on a detailled dynamic model of the manipulator incorporating finite gear stiffnesses and damping. These models are used to define an optimization problem, which is then solved using numerical programming methods. It is illustrated with an assembly task, namely inserting a rigid peg into a hole with a PUMA 562 manipulator. The expected advantage in industrial applications is a comparatively easy implementation, because performance can be improved by simply adjusting ’external’ parameters as mating position and coefficients of the standard joint controller. Particularly, no modifications of the control architecture and no additional hardware are required. Application of the proposed approach to a rigid peg-in-hole insertion under practical constraints can reduce the measure for impact sensitivity by 17%, that for mating tolerances by 78% and the damping of end-effector oscillations and motor torques by up to 79%. These improvements are shown to be reproducable experimentally.

Particularly in view of the introduction of product liability, reliability-based design procedures, and for that matter optimization (RBO) receive increasing attention. The analysis deals with statistical uncertainties inherent in structural, material, damage parameters, etc., which are modeled by random variables. The mechanical representation of the structure and the component respectively is generally modeled by Finite Elements (FE). In this paper basic mathematical formulations of design objectives and restrictions including reliability measures are discussed. Based on these models structural reliability analyses provide information for design modification and selection of an optimal design solution. As generally minimization of the expected total cost of the structure including initial costs and costs due to failure, minimization of the overall probability of failure and weight minimization with respect to reliability constraints are considered. Design problems commonly denoted as multiobjective or multicriteria optimization problems are treated. For the reliability analysis numerical methods are utilized to estimate the reliability measures. These procedures are already cast in an easy-to-use-software, denoted as COSSAN (Computational Stochastic Structural Analysis). In this context a concept is discussed, which is based on the separation of the tasks of reliability analysis and nonlinear mathematical programming techniques for which pertinent applicable software is already available. The RBO procedure utilizes approximation techniques for estimating the reliability measures. In particular, the reliability analysis makes use of the well known Response Surface Method (RSM) in context with Advanced Monte Carlo Simulation techniques, while the reliability based optimization procedure itself is controlled by the well known NLPQL-algorithm. Finally a number of numerical applications are shown in order to exemplify the approach.

We consider dynamic optimization models the structure of which (functional, equations, constrains) can be changed at time depending on control strategy.

Various aspect of the robust optimization approach [11] are discussed in the context of scenario based stochastic linear programs. The main items are the choice of the model parameter, which can be related to properties of nonlinearly perturbed linear programs [10] or of parametric quadratic programs [1], and an extension of the first results on the robustness of the optimal value with respect to probabilities of the selected scenarios and with respect to out-of-sample scenarios, cf. [5].

In this paper we extend the results of Ermoliev, Norkin and Wets [8] and Ermoliev and Norkin [7] to the case of constrained discontinuous optimization problems. General optimality conditions for problems with nonconvex feasible sets are obtained. Easily implementable random search technique is proposed.

The equivalence between the quantile functional minimization and the probability functional maximization under the assumption that the probability measure may depend on the optimized strategy is discussed. The equivalence ensures an opportunity to obtain a solution of each of these problems by solving another one. The weakened sufficient conditions for the equivalence are presented. These conditions are more general and can be verified easier than the known ones. They are applied to prove the optimality of a satellite orbital correction with respect to a quantile performance index.

Two-stage stochastic programming problems and chance constrained stochastic programming problems belong to deterministic optimization problems depending on a probability measure. Surely, the probability measure can be considered as a parameter of such problems and, moreover, it is reasonable to investigate stability with respect to it. However, to investigate the stability of the above mentioned problems it means, mostly, to investigate simultaneously behaviour of the objective functions and multifunctions corresponding to the constraints sets. The aim of the paper is to investigate stability of the multifunctions and summarize by it (from the constraints point of view) problems with the stable behaviour.

The problem of maximization of the probability of a reliability level under integrable decision rules is approximated by a sequence of finite dimensional problems with discrete distributions. Convergence conditions of the approximation are presented.

In this paper we extend the Stochastic Branch and Bound Method, developed in [7], [8] for stochastic integer and global optimization problems, to optimization problems with stochastic (expectation or chance) constraints. As examples we solve a problem of optimization of probabilities and a chance constrained programming problem with discrete decision variables.

The problem of checking robust stability of interval matrices has been proved to be NP-hard. However a closely related problem can be effectively solved in the framework of the stochastic approach [1]. Moreover the deterministic interval robust stability radius happens to be very conservative for large dimensions from the probabilistic point of view.

Preprocessing in two-stage stochastic programming is considered from the viewpoint of Fourier-Motzkin elimination. Although of exponential complexity in general, Fourier-Motzkin elimination is shown to provide valuable insights into specific topics such as solving integer recourse stochastic programs or verifying stability conditions. Test runs with the computer code PORTA [5] are reported.

In the paper new bounds for the reliability of k-out-of-connected(r,s)-from-(m,n):F lattice systems are given. These bounds are based on the Boole-Bonferroni bounding techniques and one of them on the Hunter-Worsley bound. All of these bounds need the calculation of the first few binomial moments of a random variable introduced according to the special reliability system. The main results of the paper are formulae for calculation of these binomial moments.

Important practical problems of calculus of variations can be transformed into one-parametric optimization ones. Simple engineering examples are presented to show that the two kinds of solution coincide.

The sensitivity to changes in parameters of 12 common deterministic and stochastic search methods has been investigated by measuring the time they need to approach the known minimum within a preset range of function values. These search methods use or do not use derivatives and depend on 1–5 parameters; their choice largely determines the experimental search time. An example demonstrates that an improper choice of parameters reverses the order of efficiency of search methods: i.e. the fastest method becomes the slowest one. To overcome the difficulty of a proper choice of parameters, the optimisation time was measured as a function of these parameters. Then the optimal parameter vector and optimal search time were determined with regard to the parameter space for any search method and any function separately. The functions of optimisation time versus parameter are presented. Their trend, their structure and position of the optimum are discussed.

The optimal parameter, run times, number of function values, number of decreasing function values, number of iterations required to pass half the distance from the value of the test function at start to it’s minimum, etc. at the position of the optimum of optimisation times are discussed being parts of a characterising criterion vector with some 40 components which is typical for each search method and test function.

The practical convergence speed of each search method has been determined by analysing the distribution of function values during iteration of the minimum and their iteration times. The sensitivity of the optimal parameters to changes in the feasibility parameter, the start vector, the distribution of random number vectors in R.S. methods, has been investigated experimentally. In separate programs the function error was investigated for each search method; appropriate termination criteria were selected. By increasing some sample size parameter in many parameter R.S. methods the feasible region could be reached again.

Many problems in mechanics may be described by ordinary differential equations (ODE) and can be solved numerically by a variety of reliable numerical algorithms. For optimization or sensitivity analysis often the derivatives of final values of an initial value problem with respect to certain system parameters have to be computed. This paper discusses some subtle issues in the application of Algorithmic (or Automatic) Differentiation (AD) techniques to the differentiation of numerical integration algorithms. Since AD tools are not aware of the overall algorithm underlying a particular program, and apply the chain rule of differential calculus at the elementary operation level, we investigate how the derivatives computed by AD tools relate to the mathematically desired derivatives in the presence of numerical artifacts such as stepsize control in the integrator. As it turns out, the computation of the final time step is of critical importance. This work illustrates that AD tools compute the derivatives of the program employed to arrive at a solution, not just the derivatives of the solution that one would have arrived at with strictly mathematical means, and that, while the two may be different, highlevel algorithmic insight allows for the reconciliation of these discrepancies.

Linear stochastic multistage programs are considered with uncertain data evolving as a multidimensional discrete-time stochastic process. The associated conditional probability measures are supposed to depend linearly on the past. This ensures convexity of the problem and allows application of barycentric scenario trees. These approximate the discrete-time stochastic process, and provide inner and outer approximation of the value functions.

The main issue is to refine the discretization of the stochastic process efficiently, using the nested optimization and integration of the dynamic, implicitely given value functions. We analyze and illustrate how errors evolve across nodes of the scenario trees.

Solving a stochastic linear programming (SLP) problem involves selecting an SLP solver, transmitting the model data to the solver and retrieving and interpreting the results. After shortly introducing the SLP model classes in the first part of the paper we give a general discussion of these various facets of solving SLP problems. The second part consists of an overview on the model—solver connection as implemented in SLP—IOR, our model management system for SLP. Finally we summarize the main features and capabilities of the solvers in the collection of solvers presently connected to SLP—IOR.

In this paper we study the ON/OFF model of telecommunication traffic source. The time duration of the state ON of this model has heavy-tailed probability distribution with infinity variance. We prove, that Index of Dispersion for Counts of traffic generated by such source is unbounded for t increasing to infinity. It means, that this traffic possesses long-range dependency.

In this paper a method for constrained minimization of functions is outlined. This method is similar to the method developed by Hasofer/Lind and Rackwitz/Fiessler; but firstly it can be generalized to problems with several constraints and secondly under slight regularity conditions its convergence can be demonstrated.

Further it is shown that the sequential quadratic programming schemes, which produce an approximate Hessian of the Lagrangian, can be used easily for calculating SORM approximations, since the determinant of this Hessian divided by the squared length of the gradient of the limit state function is the inverse of the square of the SORM correction factor.

We consider the initial—boundary value problem for the hyperbolic partial differential equations of thermoelasticity theory for non-simple materials. The new approach is based on the fact that we consider the initial—boundary value problem for these equations with control for temperature. We formulate the control for termperature in the terms of maximal monotone set. Existence, uniqueness and regularity of the solution to this initial—boundary value problems are proved in Sobolev space. In our proof, we use the semigroup theory and the method of Hilbert space.

Optimal trajectory planning for robots is a basic tool for improving manufacturing processes.

Response surface methods are used to construct explicit approximations of objective and constraint functions in optimization problems. Serving as interface between analysis code and optimization algorithm, the function approximations provide a fast but approximate evaluation of response quantities in the optimization process. Application to the quiet design of an engine air intake system is discussed.

NVH (Noise, Vibration, Harshness) is one of the main attributes in a passenger car. One of the key systems responsible for the overall NVH behaviour is the powertrain mounting system. The optimal layout of this system leads to the definition of a stochastic optimization problem. In this paper the background of the complexity of the powertrain mounting system is highlighted. In a first approach the optimization problem is solved with Design of Experiments (DOE) methods.