Y. Ermoliev

Numerical techniques for stochastic optimization

This is a comprehensive and timely overview of the numerical techniques that have been developed to solve stochastic programming problems. After a brief introduction to the field, where accent is laid on modeling questions, the next few chapters lay out the challenges that must be met in this area. They also provide the background for the description of the computer implementations given in the third part of the book. Selected applications are described next. Some of these have directly motivated the development of the methods described in the earlier chapters. They include problems that come from facilities location, exploration investments, control of ecological systems, energy distribution and generation. Test problems are collected in the last chapter. This is the first book devoted to this subject. It comprehensively covers all major advances in the field (both Western and Soviet). It is only because of the recent developments in computer technology, that we have now reached a point where our computing power matches the inherent size requirements faced in this area. The book demonstrates that a large class of stochastic programming problems are now in the range of our numerical capacities.

stochastic quasigradient methods and their application to system optimization

This paper systematically surveys the development of stochastic quasigradient (SQG) methods.These metods make it possible to solve optimization problems without calculating the precise valuesw of objectives and constraints (le alone of their derivatives). For deterministic nonlinear optimization problems, these methods can be regarded as methods of randomsearch. For stochastic programming problems. SQG methods generalize the well-known stochastic approximation methods for uncnstrained optimization of the expectation of a random function to problems involving general constraints.

Stochastic quasigradient methods

An apparatus and associated method for conditioning or reconditioning oil well drilling mud, comprising multiple hoppers for holding dry additive. Each hopper has a mechanism permitting the additive to be fed into an entrainment chamber at controlled, pre-selected rates. The additives are thus mixed at controlled rates with a free jet of mud forced across the entrainment chamber, to quickly achieve the desired density, viscosity and other properties of the mud. An augur is used to force the dry additive through an adjustable gate valve to control the additive flow rate. Or, the augur may be rotated at selected rates, and a fixed area additive flow passage employed between the hopper and the entrainment chamber.

Stochastic Optimization of Insurance Portfolios for Managing Exposure to Catastrophic Risks

A catastrophe may affect different locations and produce losses that are rare and highly correlated in space and time. It may ruin many insurers if their risk exposures are not properly diversified among locations. The multidimentional distribution of claims from different locations depends on decision variables such as the insurers coverage at different locations, on spatial and temporal characteristics of possible catastrophes and the vulnerability of insured values. As this distribution is analytically intractable, the most promising approach for managing the exposure of insurance portfolios to catastrophic risks requires geographically explicit simulations of catastrophes. The straightforward use of so-called catastrophe modeling runs quickly into an extremely large number of “what-if” evaluations. The aim of this paper is to develop an approach that integrates catastrophe modeling with stochastic optimization techniques to support decision making on coverages of losses, profits, stability, and survival of insurers. We establish connections between ruin probability and the maximization of concave risk functions and we outline numerical experiments.

The Minimization of Semicontinuous Functions: Mollifier Subgradients

To minimize discontinuous functions that arise in the context of systems with jumps, for example, we propose a new approach based on approximation via averaged functions (obtained by convolution with mollifiers). The properties of averaged functions are studied, after it is shown that they can be used in an approximation scheme consistent with minimization. A new notion of subgradient is introduced based on approximations generated by mollifiers and is exploited in the design of minimization procedures.

On Optimal Allocation of Indivisibles Under Uncertainty

The optimal allocation of indivisible resources is formalized as a stochastic optimization problem involving discrete decision variables. A general stochastic search procedure is proposed, which develops the concept of the branch-and-bound method. The main idea is to process large collections of possible solutions and to devote more attention to the most promising groups. By gathering more information to reduce the uncertainty and by narrowing the search area, the optimal solution can be found with probability one. Special techniques for calculating stochastic lower and upper bounds are discussed. The results are illustrated by a computational experiment.

Stochastic Optimization Problems with Incomplete Information on Distribution Functions

The main purpose of this paper is to discuss numerical optimization procedures, based on duality theory, for stochastic extremal problems in which the distribution function is only partially known. We formulate such problems as minimax problems in which the “inner” problem involves optimization with respect to probability measures. The latter problem is solved using generalized linear programming techniques. Then we state the dual problem to the initial stochastic optimization problem. Numerical procedures that avoid the difficulties associated with solving the “inner” problem are proposed.

Generalized urn schemes and technological dynamics

Abstract Adaptive (path dependent) processes of growth modeled through urn schemes find important applications to economic dynamics (and also to other disciplines, such as biology, physics, chemistry). The paper presents some further properties of generalized urn schemes and studies dynamic stochastic processes characterized by both positive and, possibly, negative feedbacks of a functional form as ‘badly behaved’ as possible. Two applicantions to technological diffusion are considered. One of the models tackles the case when there is a separation within the pool of adopters which can be interpreted as the outcome of adaptive learning on the features of the new technologies by imperfectly informed agents. Other examples deal with dependence of final market shares of two technologies on the pricing policies of the firms which produce them. The stochasticity of the processes is caused by some mixed strategies used by the adopters or/and imperfectness of the information which they possess.

A system approach to management of catastrophic risks

There are two main strategies in dealing with rare and dependent catastrophic risks: the use of risk reduction measures (preparedness programs, land use regulations, etc.) and the use of risk spreading mechanisms, such as insurance and financial markets. These strategies are not separable. The risk reduction measures increase the insurability of risks. On the other hand, the insurance policies on premiums may enforce risk reduction measures. The role of system approaches, models and accompanying decision support systems becomes of critical importance for managing catastrophic risks. The paper discusses some methodological challenges concerning the design of such models and decision support systems.

Markets for Tradeable Emission and Ambient Permits: A Dynamic Approach

This paper discusses trade mechanisms in pollutionpermit markets. Proofs are given, that sequential,bilateral trade in tradeable emissions permitsconverges to a market equilibrium with minimal totalcosts of pollution control. If ambient or depositionpermits are traded, the convergence of bilateraltransactions occurs only in the case of a singlereceptor. For multiple receptors, the proof ofconvergence for tradeable emissions and ambientpermits is given for two trade mechanisms: sequential,multilateral trade and a Walrasian auction.