This paper, which is presented in two parts, is a contribution to the theory of fractional programming, i.e., maximization of quotients subject to constraints. In Part I a duality theory for linear and concave-convex fractional programs is developed and related to recent results by Bector, Craven-Mond, Jagannathan, Sharma-Swarup, et al. Basic duality theorems of linear, quadratic and convex programming are extended. In Part II Dinkelbach's algorithm solving fractional programs is considered. The rate of convergence as well as a priori and a posteriori error estimates are determined. In view of these results the stopping rule of the algorithm is changed. Also the starting rule is modified using duality as introduced in Part I. Furthermore a second algorithm is proposed. In contrast to Dinkelbach's procedure the rate of convergence is still controllable. Error estimates are obtained too.

Fractional Programming. I, Duality

The writers propose a successive improved dynamic programming (SIDP) algorithm for hydropower reservoir operation based on an analysis of concavity, complementarity, and monotonicity of hydropower problems. For single-period hydropower generation, storage and release have diminishing marginal contributions to hydropower generation (i.e., concavity), and there is also a complementary effect between storage and release (i.e., release becomes more productive in accordance with increasing storage). For multiple-period hydropower generation, the complementarity influences the concavity of the objective function and the monotonicity of operation decisions, and is the major cause of complexity in hydropower operation. With mathematical derivations, the writers propose a concave approximation to the hydropower generation function and a SIDP algorithm for hydropower reservoir operation. The efficiency of SIDP is demonstrated with two hypothetical case studies of long-term hydropower scheduling, which shows...

Improved Dynamic Programming for Hydropower Reservoir Operation

Diminishing marginal utility is an important characteristic of water resources systems. With the assumption of diminishing marginal utility (i.e., concavity) of reservoir utility functions,...

Improved Dynamic Programming for Reservoir Operation Optimization with a Concave Objective Function

This paper addresses how much flood water can be conserved for use after the flood season through the operation of reservoir by taking into account the residual flood control capacity (the difference between flood conveyance capacity and the expected inflow in a lead time). A two-stage model for dynamic control of the flood-limited water level (the maximum allowed water level during the flood season, DC-FLWL) is established considering forecast uncertainty and acceptable flood risk. It is found that DC-FLWL is applicable when the reservoir inflow ranges from small to medium levels of the historical records, while both forecast uncertainty and acceptable risk in the downstream affect the feasible space of DC-FLWL. As forecast uncertainty increases (under a given risk level) or as acceptable risk level decreases (under a given forecast uncertainty level), the minimum required safety margin for flood control increases, and the chance for DC-FLWL decreases. The derived hedging rules from the modeling framework illustrate either the dominant role of water conservation or flood control or the trade-off between the two objectives under different levels of forecast uncertainty and acceptable risk. These rules may provide useful guidelines for conserving water from flood, especially in the area with heavy water stress. The analysis is illustrated via a case study with a real-world reservoir in northeastern China.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2015WR017127

An analytical framework for flood water conservation considering forecast uncertainty and acceptable risk

Discussions of decentralized planning procedures are not new in the economic literature: they date back to Lange [6], who seems to have found inspiration in the even earlier writings of Walras [13]. However, it is during the last decade that a number of the most important contributions have been made, by Arrow, Hurwicz and Uzawa [2], Arrow and Hurwicz [1] and Malinvaud [8]. Excellent though this work is, it leaves room for extension of the subject in a number of ways

Planning without Prices

The Dual in Nonlinear Programming and its Economic Interpretation

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The Dual in Nonlinear Programming and its Economic Interpretation