Differentiation Formula for Integrals over Sets Given by Inequalities

The investigation of a problem connected to probabilistic risk assessments for industrial plants led us to the need to optimize integrals calculated over sets that depend upon parameters. The problem was developed for two applications of tested and inspected components as an optimal control problem involving nonsmooth state transitions. In solving the optimization problem it is necessary to calculate the derivatives of an integral over a domain depending upon the parameters to be optimized. Up to date the theory of the differentiation these integrals is not fully developed. In the working paper a new general formula for differentiation of such integrals is proposed. These results were used for calculation of sensitivities for risk functions. This approach can have a wide application for the stochastic programming problems

A Stochastic Quasigradient Algorithm with Variable Metric

This paper deals with a new variable metric algorithm for stochastic optimization problems. The essence of this is as follows: there exist two stochastic quasigradient algorithms working simultaneously -- the first in the main space, the second with respect to the matrices that modify the space variables. Almost sure convergence of the algorithm is proved for the case of the convex (possibly nonsmooth) objective function

Adaptive Variable Metric Algorithms for Generalized Equations

Many of the problems in mathematical economics and game theory may be reduced to the investigation of a generalized equation with a multivalued right-hand side. This paper deals with methods for solving generalized equations. The author has developed a new approach to the construction of variable metric algorithms for these equations. The convergence of the suggested algorithm is proved for X*-antimonotone multivalued maps

Adaptive Variable Metric Algorithms for Nonsmooth Optimization Problems

This paper deals with new variable metric algorithms for nonsmooth optimization problems. The author develops so-called adaptive algorithms. The essence of such algorithms is as follows: there are two simultaneously working gradient algorithms, the first is in the main space and the second with respect to the matrix for modification of the space. The author proves convergence of such algorithms for different cases

On the Anti-Monotonicity of Differential Mappings Connected with General Equilibrium Problem

This paper is concerned with the anti-monotonicity of differential mappings connected with general equilibrium problems. These results can be used for the investigation of different game theory problems, for example Nash equilibria for noncooperative n-person games. Such an approach gives possibility to construct recurrent algorithms for finding the equilibria point

Stochastic Quasi-Gradient Algorithms with Adaptively Controlled Parameters

The paper deals with choosing stepsize and other parameters in stochastic quasi-gradient methods for solving convex problems of stochastic optimization. The principal idea of methods consists in using random estimates of gradients of the objective function to search for the point of extremum. To control algorithm parameters the iterative adaptive procedures are suggested which are quasi-gradient algorithms with respect to parameters. The convergence is proved and the estimates of the rate of convergence of such algorithms are given. The results of computations for several stochastic optimization problems are considered. The paper is part of the research on numerical techniques for stochastic optimization conducted in the Adaptation and Optimization project of the System and Decision Sciences program

Differentiation Formula for Integrals Over Sets Given by Inclusion

Formulae for differentiation with respect to the parameter of an integral over the set given by an inclusion are proposed. Such formulae are useful for solving chance constrained optimization problems. Using these formulae one can compute the gradient (or stochastic quasi-gradient) of the chance constraint and consequently apply gradient (or stochastic quasi-gradient) algorithm for the optimization

On the Optimization Model for Acid Loads on Forest Soils

The emphasis of the model is on the transboundary aspect of air pollution in Europe with the aim to find cost effective environment policies for Europe. The model will be embedded in the IIASA Regional Acidification Information and Simulation (RAINS) model. The spatial coverage of RAINS is all of Europe, and the time horizon begins in 1960 to permit checking of historical calculations, and extends to 2030 to allow examination of long-term consequences of control strategies. In this work we concentrate on soil acidification, which is an important link between air pollution and damage to the terrestrial and aquatic environment. The ability of soil to buffer acid deposition is a key factor in regulating the long-term surface and groundwater acidification. Soil acidification has also been related to forest die-back via its effect in the tree root zone. This work is concentrating on the finding of cost effective pollution control satisfying environment constraints, such as pH-value in forest soils

Optimal Operational Strategies for an Inspected Component - Statement of the Problem

This is the second report on work done on time dependent probabilities initiated in cooperation between the International Atomic Energy Agency (IAEA) and IIASA in 1990. The treatment of the underlying mathematical model is rather theoretical, but the intent is to cover a broad range of applications. The advantage with the problem formulation is that it enables the inclusion also of monetary considerations connected to risks and the actions for decreasing them. The intent in formulating the model is that it will be used for a computerized optimization of selected decision variables. Originally, the formulation was initiated by the problem of optimization of test intervals at nuclear power plants. In this paper the non-destructive testing of major components has been approached. The main result of the paper is the formulation of an optimal rule for decision if continued operation can be considered safe enough. The decision rules integrates the earlier operational history, safety concerns and economic considerations. Also other applications are proposed to be treated within the modeling framework. One specific problem is the selection of the most suitable time instant for a major repair or retrofitting at a plant. The time horizon of the model can be selected either short-term, stretching only over a few weeks, or long-term, to encompass the complete life time of a depository of spent nuclear fuel

Parameters Calculation of Solar- and Wind-Electric Water Lifting Systems

This paper describes results of the application of stochastic programming to parameters calculation of solar- and wind-electric water lifting systems. It provides an example of both a realistic problem with inherent stochasticity, and a valuable test problem for the algorithms under development. It also gives some insights into the nature of solutions of certain classes of stochastic programming problems, and the justification for the consideration of randomness in decision models