Risk management in power markets: The Hedging value of production flexibility

Since the 1990s power markets are being restructured worldwide and nowadays electrical power is traded as a commodity. The liberalization and with it the uncertainty in gas, fuel and electrical power prices requires an effective management of production facilities and financial contracts. Thereby derivatives build essential instruments to exchange volume as well as price risks. The challenge for participants in the newly competitive market environment is how to design, price and hedge derivative contracts in particular combination with the flexibility embedded in dispatch strategies of production assets. Accordingly, an adequate basis for management and investment decisions is needed which responds to the highly complex market situation.     

Stochastic second-order cone programming: Applications models

Second-order cone programs are a class of convex optimization problems. We refer to them as deterministic second-order cone programs (DSCOPs) since data defining them are deterministic. In DSOCPs we minimize a linear objective function over the intersection of an affine set and a product of second-order (Lorentz) cones. Stochastic programs have been studied since 1950s as a tool for handling uncertainty in data defining classes of optimization problems such as linear and quadratic programs. Stochastic second-order cone programs (SSOCPs) with recourse is a class of optimization problems that defined to handle uncertainty in data defining DSOCPs. In this paper we describe four application models leading to SSOCPs.     

Fast value iteration: an application of Legendre-Fenchel duality to a class of deterministic dynamic programming problems in discrete time*

ABSTRACT

We propose an algorithm, which we call ‘Fast Value Iteration’ (FVI), to compute the value function of a deterministic infinite-horizon dynamic programming problem in discrete time. FVI is an efficient algorithm applicable to a class of multidimensional dynamic programming problems with concave return (or convex cost) functions and linear constraints. In this algorithm, a sequence of functions is generated starting from the zero function by repeatedly applying a simple algebraic rule involving the Legendre-Fenchel transform of the return function. The resulting sequence is guaranteed to converge, and the Legendre-Fenchel transform of the limiting function coincides with the value function.     

Robust aspects of solutions in deterministic multiple objective linear programming

We study questions of robustness of linear multiple objective problems in the sense of post-optimal analysis, that is, we study conditions under which a given efficient solution remains efficient when the criteria/objective matrix undergoes some alterations. We consider addition or removal of certain criteria, convex combination with another criteria matrix, or small perturbations of its entries. We provide a necessary and sufficient condition for robustness in a verifiable form and give two formulae to compute the radius of robustness.     

Nonnormal deterministic equivalents and a transformation in stochastic mathematical programming

This paper considers random variables of the continuous type in a stochastic programming problem and presents (1) a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits, and (2) a deterministic transformation of a stochastic programming problem with random variables in the objective function. Deterministic equivalents are developed for constraints containing uniform random variables, but the approach used can be applied to other types of continuous random variables, as well. When the random variables appear in the objective function, a deterministic transformation of the stochastic programming problem is obtained to yield a closed-form solution without resort to a Monte Carlo computer simulation. Extension of this approach to stochastic problems with discrete random variables and integer decision variables is discussed briefly. A numerical example is presented.     

Stochastic utility-efficient programming of organic dairy farms

Opportunities to make sequential decisions and adjust activities as a season progresses and more information becomes available characterise the farm management process. In this paper, we present a discrete stochastic two-stage utility-efficient programming model of organic dairy farms, which includes risk aversion in the decision maker’s objective function as well as both embedded risk (stochastic programming with recourse) and non-embedded risk (stochastic programming without recourse). Historical farm accountancy data and subjective judgements were combined to assess the nature of the uncertainty that affects the possible consequences of the decisions. The programming model was used within a stochastic dominance framework to examine optimal strategies in organic dairy systems in Norway.     

Stochastic and deterministic simulations of a delayed genetic oscillation model: Investigating the validity of reductions

Quasi-stationary approximations are commonly used in order to simplify and reduce the number of equations of genetic circuit models. Protein/protein and protein/DNA binding reactions are considered to occur on much shorter time scale than protein production and degradation processes and often tacitly assumed at a quasi-equilibrium. Taking a biologically inspired, typical, small, abstract, negative feedback, genetic circuit model as study case, we investigate in this paper how different quasi-stationary approximations change the system behaviour both in deterministic and stochastic frameworks. We investigate the consistence between the deterministic and stochastic behaviours of our time-delayed negative feedback genetic circuit model with different implementations of quasi-stationary approximations. Quantitative and qualitative differences are observed among the various reduction schemes and with the underlying microscopic model, for biologically reasonable ranges and combinations of the microscopic model kinetic rates. The different reductions do not behave in the same way: correlations and amplitudes of the stochastic oscillations are not equally captured and the population behaviour is not always in consistence with the deterministic curves.     

Stochastic semidefinite programming: a new paradigm for stochastic optimization

Semidefinite programs are a class of optimization problems that have been studied extensively during the past 15 years. Semidefinite programs are naturally related to linear programs, and both are defined using deterministic data. Stochastic programs were introduced in the 1950s as a paradigm for dealing with uncertainty in data defining linear programs. In this paper, we introduce stochastic semidefinite programs as a paradigm for dealing with uncertainty in data defining semidefinite programs.The work of this author was supported in part by the U.S. Army Research Office under Grant DAAD 19-00-1-0465. The material in this paper is part of the doctoral dissertation of this author in preparation at Washington State University.     

Maximum principle,dynamic programming,and their connection in deterministic control

Two major tools for studying optimally controlled systems are Pontryagin's maximum principle and Bellman's dynamic programming, which involve the adjoint function, the Hamiltonian function, and the value function. The relationships among these functions are investigated in this work, in the case of deterministic, finite-dimensional systems, by employing the notions of superdifferential and subdifferential introduced by Crandall and Lions. Our results are essentially non-smooth versions of the classical ones. The connection between the maximum principle and the Hamilton-Jacobi-Bellman equation (in the viscosity sense) is thereby explained by virtue of the above relationship.This research was supported by the Natural Science Fund of China.This paper was written while the author visited Keio University, Japan. The author is indebted to Professors H. Tanaka and M. Nisio for their helpful suggestions and discussions. Thanks are also due to Professor X. J. Li for his comments and criticism.     

Use of goal programming and integer programming for water quality management—A case study of Gaza Strip

This paper describes a project dealing with achieving an optimum mix of water from different underground wells, each having different amounts of nitrates and chlorides. The amounts of chlorides and nitrates in each of the wells may be higher or lower than the World Health Organization (WHO) standards. Therefore, the optimum mix would be the one that meets WHO standard which is 250 mg/l for chlorides and 50 mg/l for nitrates. A goal programming model was developed to identify the combination of wells along with the amounts of water from each well that upon mixing would result in minimizing the deviation of the amounts of chlorides and nitrates from the standards set by WHO. The output of the goal programming model along with the coordinates of the wells identified above was then used for a second model that determines the locations of the mixing points “reservoirs” in such a way that minimizes the total weighted distances from the corresponding wells. Finally, an easy-to-use pumping schedule was developed using integer programming. Results indicate that for the case study, there exist several optima which make it easier for the decision maker to consider other intangible factors if there are any.     

Optimal value function in semi-infinite programming

In this paper, we provide a systematic approach to the main topics in linear semi-infinite programming by means of a new methodology based on the many properties of the sub-differential mapping and the closure of a given convex function. In particular, we deal with the duality gap problem and its relation to the discrete approximation of the semi-infinite program. Moreover, we have made precise the conditions that allow us to eliminate the duality gap by introducing a perturbation in the primal objective function. As a by-product, we supply different extensions of well-known results concerning the subdifferential mapping.     

Optimal value functions in parametric programming

Optimal value functions in parametric programming are studied as compositions of objective functions and point-to-set mappings which define the constrained sets. Sufficiency conditions for the common regularity properties of optimal value functions are derived.

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Zusammenfassung Der Optimalwert eines parametrischen Programms wird aufgefaßt als Verknüpfung der Zielfunktion mit einer mengenwertigen Abbildung, die den zulässigen Bereich definiert. Es werden hinreichende Bedingungen für einige häufig benutzte Regularitätseigenschaften der Optimalwertfunktion hergeleitet.

     

Infinite-horizon deterministic dynamic programming in discrete time: a monotone convergence principle and a penalty method

We consider infinite-horizon deterministic dynamic programming problems in discrete time. We show that the value function of such a problem is always a fixed point of a modified version of the Bellman operator. We also show that value iteration converges increasingly to the value function if the initial function is dominated by the value function, is mapped upward by the modified Bellman operator and satisfies a transversality-like condition. These results require no assumption except for the general framework of infinite-horizon deterministic dynamic programming. As an application, we show that the value function can be approximated by computing the value function of an unconstrained version of the problem with the constraint replaced by a penalty function.     

Stochastic programming: Potential hazards when random variables reflect market interaction

There are two types of random phenomena modeled in stochastic programs. One type is what we may term “external” or “natural” random variables, such as temperature or the roll of a dice. But in many other cases, random variables are used to reflect the behavior of other market participants. This is the case for such as price and demand of a product. Using simple game theoretic models, we demonstrate that stochastic programming may not be appropriate in these cases, as there may be no feasible way to replace the decisions of others by a random variable, and arrive at the correct decision. Hence, this simple note is a warning against certain types of stochastic programming models. Stochastic programming is unproblematic in pure forms of monopoly and perfect competition, and also with respect to external random phenomena. But if market power is involved, such as in oligopolies, the modeling may not be appropriate.     

Two-machine shop scheduling: Compromise between flexibility and makespan value

In this paper, we consider the problem of providing flexibility to solutions of two-machine shop scheduling problems. We use the concept of group-scheduling to characterize a whole set of schedules so as to provide more choice to the decision-maker at any decision point. A group-schedule is a sequence of groups of permutable operations defined on each machine where each group is such that any permutation of the operations inside the group leads to a feasible schedule. Flexibility of a solution and its makespan are often conflicting, thus we search for a compromise between a low number of groups and a small value of makespan. We resolve the complexity status of the relevant problems for the two-machine flow shop, job shop and open shop. A number of approximation algorithms are developed and their worst-case performance is analyzed. For the flow shop, an effective heuristic algorithm is proposed and the results of computational experiments are reported.     

Robustness in deterministic multi-objective linear programming with respect to the relative interior and angle deviation

This paper deals with the robustness issue in deterministic multi-objective linear programming from two new standpoints. It is shown that a robustness notion recently reported in the literature is equivalent to strict efficiency. Corresponding to an efficient solution, a new quantity, robustness order (RO) is defined with respect to the interiority order of the cost matrix in the binding cone. A linear programming problem is provided to calculate the RO of a given efficient solution. The second part of the paper is devoted to investigating the robustness with respect to the eligible angle deviation of the cost matrix in the binding cone. Theoretical results are given to obtain the maximum eligible angle deviation. Finally, the relationship between two above-mentioned robustness standpoints is established. To have a better geometrical view, we prove the results for single-objective LP problems at first, and then we extend them to the multi-objective case. In addition to the theoretical results, some clarifying examples are given.     

Goal programming: Application in the management of the miombo woodland in Mozambique

Community-based management of natural resources (CBNRM) is a priority in Mozambique's policy on forestry and wildlife resources. In essence the government's policy is to manage the natural resources in partnership with the rural communities and the private sector. This represents a change in policy in the agricultural and natural resources sectors, and has potential for significant impact in economic development. This paper demonstrates the potential for employing goal programming as a planning tool in participatory natural resource management in Mozambique. The focus is on the miombo woodlands, which are the main natural forest resources in the country and which most of the local communities, the forestry and tourist industries depend on for a variety of forest products and services.     

Stochastic approximation and the final value theorem

The aim here is to show how to obtain many of the well-known limit results (i.e., central limit theorem, law of the iterated logarithm, invariance principle) of stochastic approximation (SA) by a shorter argument and under weaker conditions. The idea is to introduce an artificial sequence, related to the SA scheme, and which clearly obeys the limit law. This sequence is subtracted from the SA scheme and then simple deterministic limit theory is used to show the remainder is negligible. As a consequence of this approach proofs are shorter and the meaning of conditions becomes clearer. Because the difference equations are not summed up it is simple to state results for general a_n, c_n sequences.