This paper systematically surveys the development of stochastic quasigradient (SQG) methods. These methods make it possible to solve optimization problems without calculating the precise values of objectives and constraints (let alone of their derivatives). For deterministic nonlinear optimization problems, these methods can be regarded as methods of random search. For stochastic programming problems, SQG methods generalize the wellknown stochastic approximation method for unconstrained optimization of the expectation of a random function to problems involving general constraints.

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. iii iv Contents 1 Models of discrete stochastic optimization 2 The stochastic branch and bound method 2.1 Outline of the method : :

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.

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