Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization. AMS subject classifications. 90A09, 90C15, 90C20PII. S0036144500376650 0. Introduction. The classical mean-variance approach for which Harry Markowitz received the 1990 Nobel Prize in Economics offered the first systematic treatment of a dilemma that each investor faces: the conflicting objectives of high profit versus low risk. In dealing with this fundamental issue Markowitz came up with a parametric optimization model that was both sufficiently general for a significant range of practical situations and simple enough for theoretical analysis and numerical solution. As the Swedish Academy of Sciences put it [154], "his primary contribution consisted of developing a rigorously formulated, operational theory for portfolio selection under uncertainty." Indeed, the subject is so complex that Markowitz's seminal work of the 1950s [134,135,137] probably raised more questions than it answered, thus initiating a tremendous amount of related research. Before placing the present paper into perspective, the following paragraphs give a coarse overview of these issues. A substantial number of references are included, but we have not attempted to compile a complete list. (The 1982 research bibliography [10] contains 400 references on just one of the topics.) However, we have tried to cite (mostly in chronological order) at least several major papers on each subject to provide some starting points for the interested reader.An important aspect of pareto-optimal (efficient) portfolios is that each determines a von Neumann-Morgenstern utility function [202] for which it maximizes the expected utility of the return on investment.

The topic of this paper is minimum cost operative planning of pressurized water supply networks over a finite horizon and under reliable demand forecast. Since this is a very hard problem, it is desirable to employ sophisticated mathematical algorithms, which in turn calls for carefully designed models with suitable properties. The paper develops a nonlinear mixed integer model and a nonlinear programming model with favorable properties for gradient-based optimization methods, based on smooth component models for the network elements. In combination with further nonlinear programming techniques (Burgschweiger et al. in ZIB Report ZR-05-31, Zuse Institute Berlin, 2005), practically satisfactory near-optimum solutions even for large networks can be generated in acceptable time using standard optimization software on a PC workstation. Such an optimization system is in operation at Berliner Wasserbetriebe.

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