Subtraction of Convex Sets and Its Application in E-Subdifferential Calculus

A new approach to the subtraction of convex sets is presented which inverts in some limited sense the vector addition of sets as defined by Minkovski. It is shown that the notion of subtraction put forward in this paper has a number of useful algebraic properties and can be used to simplify the formulation and proof of some advanced results in convex analysis

Pi-Approximation and Decomposition of Large-Scale Problems

Partial or complete dualization of extremum problems often allows the decomposition of initially large-scale problems into smaller ones with some coordinating program of a moderate size. This idea underlies many known schemes of decomposition and the common difficulty often encountered is the problem of restoring the solution of the primal problem. The main idea of this paper is to present an algorithm for providing an easy way of obtaining the solution of the initial primal problem keeping all advantages of the dual one. The algorithm described here is based on the particular approximation of the aggregated function representing the decomposed way of solving the extremum problem. This approximation looks like a dual problem and its remarkably simple structure makes it possible to solve a corresponding extremum problem in a few iterations

Decomposition of Two-Block Optimization Problems

This paper is concerned with the problem of balancing an optimization model consisting of two submodels. The submodels are represented by separate linear programming problems and are linked by dependence on common resources, or by the presence of the same variables in both of them. The method for coordinating the activities of submodels, in order to reach an overall optimum, is based on the direct exchange of proposals between submodels. Computational improvements in comparison with the conventional master-subproblems scheme are shown

Decomposition Algorithm Based on the Primal-Dual Approximation

A decomposition algorithm based on the simultaneous approximation of the primal and dual forms of an optimization problem is proposed. This approach makes maximum use of the primal-dual information available during solution of the decomposed problem, speeds up the convergence, and provides upper and lower bounds for the optimum

Numerical Experiments with Decomposition of LP on a Small Computer

Results of numerical experiments with decomposition of linear programming problems are reporte

Convergence and Numerical Expirements with a Decomposition Algorithm

This paper gives a proof of convergence of a decomposition algorithm for solution of an optimization model consisting of two submodels. The submodels are represented by separate mathematical programming problems and are linked by dependence on common variables. The method for coordinating the activities of submodels, in order to reach an overall optimum, is based on the approximation of the original problem, which can be interpreted as the direct exchange of proposals between submodels. Computational improvements in comparison with the conventional master-subproblems schemas are shown

On epsilon-Differential Mappings and their Applications in Nondifferentiable Optimization

In Section 1 we give some review of the recent developments in nondifferential optimization and discuss the difficulties of the application of subgradient methods. It is shown that the use of epsilon-subgradient methods may bring computational advantages. Section 2 contains the technical results on continuity of epsilon-subdifferentials. The principal result of this section consists in establishing Lipschitz continuity of epsilon-subdifferential mappings. Section 3 gives some results on convergence of weighted sums of multifunctions. These results will be used in the study of the convergence of epsilon-subgradient method with sequential averages given in Section 4. Section 4 gives the convergence theory for several modifications of this method. It is shown that in some cases it is possible to neglect accuracy control for the solution of internal maximum problems in the minmax problems. The results when this accuracy is nonzero and fixed are of great practical importance

Nondifferentiable Optimization with Epsilon Subgradient Methods

The development of optimization methods has a significant meaning for systems analysis. Optimization methods provide working tools for quantitative decision making based on correct specification of the problem and appropriately chosen solution methods. Not all problems of systems analysis are optimization problems, of course, but in any systems problem optimization methods are useful and important tools. The power of these methods and their ability to handle different problems makes it possible to analize and construct very complicated systems. Economic planning for instance would be much more limited without linear programming techniques which are very specific optimization methods. LP methods had a great impact on the theory and practice of systems analysis not only as a computing aid but also in providing a general model or structure for the systems problems. LP techniques, however, are not the only possible optimization methods. The consideration of uncertainty, partial knowledge of the systems structure and characteristics, conflicting goals and unknown exogenous models and consequently more sophisticated methods to work with these models. Nondifferentiable optimization methods seem better suited to handle these features than other techniques at the present time. The theory of nondifferentiable optimization studies extremum problems of complex structure involving interactions of subproblems, stochastic factors, multi-stage decisions and other difficulties. This publication covers one particular, but unfortunately common, situation when an estimation of the outcome from some definite decision needs a solution of a difficult auxiliary, internal, extremum problem. Solution of this auxiliary problem may be very time-consuming and so may hinder the wide analysis of different decisions. The aim of the author is to develop methods of optimal decision making which avoid direct comparison of different decisions and use only easily accessible information from the computational point of view

Conceptual Newton Method for Solving Multivalued Inclusions: Scalar Case

Due to different reasons, the actual state of economic, environmental and even mechanical systems is often known only as a set of possible values of the systems indexes. Another source of uncertainty is an unspecified reaction of the system to the changes in control or unpredicted changes in systems inputs. The theory of set-valued mapping provides the mathematical tools for analysis and construction of such systems and is of great importance to system analysis methodology. This paper is concerned with one of the basic problems of application of set valued mapping -- solving multivalued inclusions. It uses the original definition of a set valued derivative and develops the Newton-like method for solving this problem. The remarkable feature of the proposed method is a quadratical rate of convergency

Trend Analysis for Sparse Data

The major theme of this paper is to present some means for an analysis of changes in characteristics of complex systems. Such systems are characterized by a number of parameters which are interdependent. The data available on such systems are sparse in the sense that only a few such systems exist in the real world. Any attempt to choose uniform population with respect to some characteristics will decrease the number of data even more until a statistics approach becomes completely unreasonable. The alternative approach is based on pattern recognition ideas and uses an idea of separation of different classes of a complex system by multidimensional surfaces. The position of these surfaces demonstrates the trends in the system's development. The analysis of coal mines with respect to different criteria has been performed as an example