Georgy K. Kamenev

Properties of a method for polyhedral approximation of the feasible criterion set in convex multiobjective problems

The paper describes new results in the field of multiobjective optimization techniques. The Interactive Decision Maps (IDM) technique is based on approximation of Feasible Criterion Set (FCS) and subsequent visualization of the Pareto frontier of FCS by interactive displaying the bi-criteria slices of FCS. The Estimation Refinement (ER) method is now the main method for approximating convex FCS in the framework of IDM. The properties of the ER method are studied. We prove that the number of facets of the approximation constructed by ER and the number of the support function calculations of an approximated set are asymptotically optimal. These results are important from the point of view of real-life applications of ER.

Introduction to Interactive Decision Maps

In this chapter the Interactive Decision Maps (IDM) technique is introduced and its simplest application in the framework of the goal approach is described. First the screening phase of the decision making process is considered and the role of visualization in it is outlined. Then, the Pareto frontier, Edgeworth-Pareto hull, decision maps and related notions of MCDM theory are introduced using a simple regional environmental model.

Real-Life Applications

In this chapter several real-life applications of the FGM/IDM technique are considered. First, the concept of a real-life application of a decision support technique is discussed. The discussion is illustrated with the application of the FGM at the State Planning Agency of the former Soviet Union in the first part of the 1980s. Sections 2 and 3 are devoted to two decision support systems that are used by water engineers for water quality planning in river basins of Russia. Screening of water quality plans is based on application of the FGM/IDM technique. The decision support systems (DSS) helps engineers to develop water quality improvement strategies that can be used as proposals in the process of final selection of water quality improvement plans.

WATER RESOURCE CONFLICT RESOLUTION BASED ON INTERACTIVE TRADEOFFS DISPLAY

Rehabilitation of degraded water resource systems, especially large scale systems such as Danube, Mississippi or the Volga River, inevitably involve conflicts over who pays, who benefits, and how much. This paper reviews the application of computer-aided approaches to water resources conflict resolution based on the generation and interactive display of the efficient tradeoff curves among conflicting performance criteria associated with various possible decisions. Computing of efficient tradeoffs is based on a mathematical model describing the system being managed. The model is supposed to be mutually acceptable to all stakeholders. Negotiators can interactively explore the full implications of solutions that represent efficient tradeoffs among identified performance criteria. In this way they can become more informed about compromises that can be made among conflicting criteria. This approach is applied to the real-time allocation of water to agricultural and environmental users whose demand for water usually exceeds the supply. Application of these procedures using computer networks (e.g., Internet or intranets) is discussed.

Optimality of the Methods for Approximating the Feasible Criterion Set in the Convex Case

Estimation Refinement (ER) is an adaptive method for polyhedral approximations of multidimensional convex sets. ER is used in the framework of the Interactive Decision Maps (IDM) technique that provides interactive visualization of the Pareto frontier for convex sets of feasible criteria vectors. We state that, for ER, the number of facets of approximating polytopes is asymptotically multinomial of an optimal order. Furthermore, the number of support function calculations, needed to be resolved during the approximation, and which complexity is unknown beforehand since a user of IDM provides his own optimization algorithm, is bounded from above by a linear function of the number of iterations.

Perturbations of Solution Sets of Linear Systems

This chapter is devoted to the analysis of stability and perturbations of the feasible criterion set (FCS). This problem is extremely important since the data are usually disturbed to a certain extent in real-life applications. It is required that the resulting perturbations of an FCS are not too substantial. In other words, an FCS must depend on the data disturbances in a continuous way. This chapter deals with the estimation of such perturbations in the linear case. The problem is reduced to estimating perturbations of the feasible decision set given by a system of linear inequalities and equalities. To be precise, the distance between two sets, the original set and the perturbed set is estimated.

Reasonable Goals Method and Its Applications

The Reasonable Goals method (RGM) is introduced in this chapter, and several applications of the method are described. We consider a simplest form of the RGM here, which supports selecting of a small number of alternatives from given lists that contain a large, but finite, number of decision alternatives. Such lists may contain millions of alternatives. The RGM is based on representing decision alternatives in the form of criterion points and on approximating the convex hull (envelope) of a variety of points. To be precise, the EPH of the convex hull is approximated. Due to such enveloping, the IDM technique can be applied, but now the user studies proxy tradeoffs between the criteria. Application of the IDM technique for exploration of the Pareto frontier of the envelope and identifying a goal vector (so-called reasonable goal) on it are the main features of the RGM. Since the convex hull is explored instead of the variety of points itself, an identified goal may not be feasible, but only reasonable. As a result, several decision alternatives that are in line with the identified goal are selected.

Computational Methods for Dynamic Systems

In this chapter we consider methods for approximation of FCSs and their EPHs for systems described by ordinary differential equations or partial derivatives. Though the methods are based on ideas which have already been described in Chapter 6, the specific form of such systems requires special attention. Now, in contrast to Chapter 6, it is assumed that the feasible decision set X belongs to infinite-dimensional linear space W of a general nature. However, the criterion vectors z are still assumed to be elements of the linear finite-dimensional criterion space Rm.

Epilogue: New applications of the IDM technique on the Web

In our Epilogue we try to look into the future. Therefore, in contrast to the other parts of the book, where recent applications of the IDM technique are described, we discuss options that might be provided by the IDM technique sooner or later. So, this part of the book can be considered as science fiction. Nevertheless, information technology develops very fast, and so we hope that teh discussed ideas will be implemented soon. In any case, we try to prove it in this part of the book.

Computational Methods for Static Models

Computational methods developed in the framework of the IDM technique are introduced in Part II of the book. This chapter concentrates on methods for static models. Methods for dynamic models and distributed systems are provided in the next chapter. The mathematical basis of the computational methods is described here in a simplified way. Complicated mathematical issues of the theory of the methods are considered in Part III. However, in contrast to the previous chapters, certain mathematical background is required from the reader of this part of the book.