Convex Cost Function Research Articles

Network-constrained economic dispatch using real-coded genetic algorithm

A genetic algorithm (GA) solution to the network-constrained economic dispatch problem is presented. A real-coded GA has been implemented to minimize the dispatch cost while satisfying generating unit and branch power flow limits. A binary-coded GA was also developed to provide a means of comparison. GA solutions do not impose any convexity restrictions on the dispatch problem. The proposed method was applied on the electrical grid of Crete Island with satisfactory results. Various tests with both convex and nonconvex unit cost functions demonstrate that the proposed GA locates the optimum solution, while it is more efficient than the binary-coded GA.

On Inventory Control Models with a Convex Cost Function

An inventory model, used to control a one-type product stock with an A-convex cost function is analyzed. The replenishment amount is taken as a control parameter. Strategies that are optimal in the sense of an average expected cost and a total revalued cost with a revaluation coefficient β are found and examined.

Infinite-Horizon Discrete-Time Optimal Control Problems

The study of optimal control problems defined on infinite intervals has recently been a rapidly growing area of research. These problems arise in engineering [1,2], in models of economic growth [5,16,26–28,40], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [3,31], and in the theory of thermodynamical equilibrium for materials [7,14,17–20,34,35,37]. In this survey, we consider discrete-time optimal control problems. Sections 2 and 3 are devoted to autonomous discretetime control systems on compact metric spaces. In Sec. 2, we present two fundamental tools in the theory of optimal control on an infinite horizon: the reduction to finite cost and the representation formula established in [9]. In Sec. 3, we present a number of results obtained in [32,33] which establish the existence of weakly optimal solutions on an infinite horizon and describe their structure. The turnpike theorem for an infinite-dimensional control system with a convex cost function [39] is considered in Sec. 4. In Sec. 5, we discuss the generalization of this turnpike result to a class of nonautonomous infinite-dimensional control systems with nonconvex cost function [41]. Section 6 is devoted to Lagrange multipliers for the discrete-time problems with a periodic cost function [15]. In Sec. 7, we study optimal control of finite-state Markov decision processes [11,13].

Duality in the optimal control for damped hyperbolic systems with positive control

We study the duality theory for damped hyperbolic equations. These systems have positive controls and convex cost functionals. Our main results lie in the application of duality theorem, that is, inf J = sup K, on various cost functions.

PERT scheduling with convex cost functions

This paper deals with the problem of finding a minimum cost schedule for a set of dependent activities when a convex cost function is attached to the starting time of each activity. A first optimality necessary and sufficient condition bearing on the head and tail blocks of a schedule is first established. A second such condition that uses the spanning active equality trees of a schedule leads to design a generic algorithm for the general case. When the cost function is the usual earliness–tardiness linear function with assymetric and independent penalty coefficients, the problem is shown to be solved in O(n max{n,m}) . Finally, the special cases when the precedence graph is an intree or a family of chains are then also shown to be solved by efficient polynomial algorithms.

Production-planning horizon, production smoothing, and convexity of the cost functions

One of the productions planning models economists have used for the past four decades is the production-smoothing model. The model's conclusion is simple: if cost functions are convex, under certain conditions, firm's production plans may show smaller variability than the firm's sales. This is a result of the first-order condition for cost minimization over the production-planning horizon. In a 1986 paper Blinder raised doubts regarding the ability of the production-smoothing model to explain the observed data. Some researchers (e.g. Ramey, Journal of Political Economy 99(2) (1991) 306) explained the inconsistency between the theory and the data by providing evidence that the cost functions are non-convex. In this paper, I point out that the shapes of the cost functions depend on the length of the time horizon under consideration. It is possible that while short-run cost functions are upward sloping, long-run cost functions may be downward sloping. Because the planning horizon for firms facing seasonal demand has been shown to be one seasonal cycle, the short-run cost function, whose shape determines the firm's production plan over the horizon, need to be estimated using single planning horizon data, i.e., a single seasonal cycle. Researchers who use long time series data are, in effect estimating long-run cost functions, the slopes of which may be non-positive. I show, further, that convexity of the production cost function is neither necessary nor sufficient condition for production smoothing to be an optimal plan.

A naive but optimal route to Walrasian behavior in oligopolies

A paradoxical phenomenon where irrationality or ignorance lead to higher profits is shown to prevail in an oligopolistic market where three firms produce a homogeneous product with identical technologies. Under the conventional assumption on the convexity of cost function, the profit made by a naive price-taker will always be higher than or at least as much as the other two rivals, no matter what kind of strategies the latter may take and whether they form a coalition or not. By sticking to the price-taking strategy, a firm always gains a positive profit improvement, should any one of its rivals try to upgrade from price-taker to the Cournotor. With these incentives, a firm prefers to remain as a naive price-taker.

Marginal cost price rule for homogeneous cost functions

Mirman and Tauman (1982) show that axioms of cost sharing, additivity, rescaling invariance, monotonicity, and consistency uniquely determine a price rule on the class of continuously differentiable cost problems as the Aumann-Shapley price mechanism. Here we prove that standard versions of these axioms determine uniquely the marginal cost price rule on the class of homogeneous and convex cost functions, which are, in addition, continuously differentiable. This result persists even if the cost functons are not required to be convex.

On environmental Kuznets curves arising from stock externalities

We develop a stock externality model in an optimal growth framework and examine the relationship between economic growth and the environment. We find that as incomes rise over the growth path, both the marginal benefits of pollution control and the marginal costs of pollution control rise. If at a given income level the marginal benefits rise more than the marginal costs, the emissions-income relationship has a negative slope at that income level and vice versa. We are thus able to generate examples of emissions-income relationships of varying shapes by varying pollution specific parameters which determine marginal costs and benefits, such as the convexity of the cost function, the rate of decay of the stock of pollution, and the share of consumption in the composite good. An inverted U-shape relationship (an “environmental Kuznets curve”) requires only minimal assumptions. A convex cost function and that environmental quality is a normal good imply an inverted U-shape relationship for an open set of parameters. We also examine how environment-income relationship varies according to what measure is used for the environment (emissions, stock of pollution, or emissions control).

Network-Constrained Economic Dispatch Using Real-Coded Genetic Algorithm

A genetic algorithm (GA) solution to the network-constrained economic dispatch problem is presented. A real-coded GA has been implemented to minimize the dispatch cost while satisfying generating unit and branch power flow limits. A binary-coded GA was also developed to provide a means of comparison. GA solutions do not impose any convexity restrictions on the dispatch problem. The proposed method was applied on the electrical grid of Crete Island with satisfactory results. Various tests with both convex and nonconvex unit cost functions demonstrate that the proposed GA locates the optimum solution, while it is more efficient than the binary-coded GA.

Optimal hydrothermal scheduling with variable production coefficient

We consider the optimal operation of a hydroelectric plant supplemented by a set of thermal plants. The initial model gives rise to a discrete minimization problem with a convex cost function, submitted to both concave and convex restrictions. The geometry of the water reservoir is taken into account by a production coefficient, which is a function of the volume of available water. A slightly different formulation of the problem allows for a continuous limit, in which both the geometry of the restrictions and the optimal operation modes admit a simple description. Optimal operations correspond to juxtapositions of arcs in the boundary of the admissible set and pieces of geodesic-like trajectories in a 1-1 space-time. For the general problem, we show existence of optimal operations, and, with stronger hypothesis, also uniqueness within a special class of thrifty operations. A numerical example, with data obtained from a concrete situation, is solved by making use of the characterization of optimal modes.

Bid Costs and Endogenous Bid Caps

We study contests where several privately informed agents bid for a price. All bidders bear a cost of bidding that is an increasing function of their bids, and, moreover, bids may be capped. We show that, regardless of the number of bidders, if agents have linear or concave cost functions then setting a bid cap is not profitable for a designer who wishes to maximize the average bid. On the other hand, if agents have convex cost functions (i.e. an increasing marginal cost) then affectively capping the bids is profitable for a designer facing a sufficiently large number of bidders.

Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization

This paper presents a polynomial time algorithm for solving submodular flow problems with a class of discrete convex cost functions. This class of problems is a common generalization of the submodular flow and valuated matroid intersection problems. The algorithm adopts a new scaling technique that scales the discrete convex cost functions via the conjugacy relation. The algorithm can be used to find a pair of optima in the form of the Fenchel-type duality theorem in discrete convex analysis.

Optimal forestry control with variable forest area

This paper investigates the optimal forestry control problem when the forest area is variable. A forester is considered who is endowed with effectively infinite area of land and whose objective is to maximize the discounted value of profit by controlling his planting and harvesting rates under a convex cost function. Necessary and sufficient conditions for optimality are established. The uniqueness of the solution is examined. Changes in the timber price are shown to have possibly no effect upon the optimal steady-state harvesting age but upon the forest area. A case exhibiting monotonic convergence of the optimal paths is provided.

An enhanced Lagrangian neural network for the ELD problems with piecewise quadratic cost functions and nonlinear constraints

This paper presents a new Lagrangian artificial neural network (ANN) and its application to the power system economic load dispatch (ELD) problems with piecewise quadratic cost functions (PQCFs) and nonlinear constraints. By restructuring the dynamics of the modified Lagrangian ANN [IEEE ICNN, 1 (1996) 537], stable convergence characteristics are obtained even with the nonlinear constraints. The convergence speeds are enhanced by employing the momentum technique and providing a criteria for choosing the learning rate parameters. Instead of having one convex cost function for each unit, which is normally the case in typical ELD problem formulations, more realistic multiple quadratic cost functions are used to reflect the effects of valve point loadings and possible fuel changes. In addition, the B matrix approach is employed for more accurate estimation of the transmission losses than treating them as a constant, which necessitate the inclusion of a nonlinear equality constraint. The effectiveness of the proposed ANN applied to the ELD problem is demonstrated through extensive simulation tests.

Optimal partition of QoS requirements with discrete cost functions

The future Internet is expected to support applications with quality of service (QoS) requirements. To this end, several mechanisms are suggested in the IETF; the most promising among them is DiffServ. An important problem in this framework is how to partition the QoS requirements of an application along a selected path. The problem which is, in general, NP-complete, was solved for continuous convex cost functions by Lorenz and Orda (see IEEE/ACM Trans. Networking. vol.6, p.768-78, 1998 and Proc. IEEE INFOCOM'99, p.246-53, 1999). This paper concentrates on discrete cost functions, which better model the existing and upcoming mechanisms in the Internet. We present efficient exact and approximated solutions for various conditions of the problem. We also show that although the more complex problem of QoS sensitive routing with discrete cost functions is hard, it has a fully polynomial approximation scheme.

A fast linear programming algorithm for blind equalization

A fast implementation of a special non-MSE cost function for blind equalization is presented here. This baud-rate equalization algorithm is based on a convex cost function coupled with a simple linear constraint on the equalizer parameters. For a generic class of channels with persistently exciting quadrature amplitude modulation input signals, this new algorithm allows the convergence of equalizer parameters to a unique global minimum achieving intersymbol interference suppression and carrier phase recovery.

Stability and evolution of overlapping genes.

When the same sequence of nucleotides codes for regions of more than one functional polypeptide, this sequence contains overlapping genes. Overlap is most common in rapidly evolving genomes with high mutation rates such as viruses, bacteria, and mitochondria. Overlap is thought to be important as: (1) a means of compressing a maximum amount of information into short sequences of structural genes; and (2) as a mechanism for regulating gene expression through translational coupling of functionally related polypeptides. The stability of overlapping codes is examined in relation to the information cost of overlap and the mutation rate of the genome. The degree of overlap in a given population will tend to become monomorphic. Evolution toward partial overlap of genes is shown to depend on a convex cost function of overlap. Overlap does not evolve when expression of overlapping genes is mutually exclusive and produced by rare mutations to the wild-type genome. Assuming overlap increases coupling between functionally related genes, the conditions favoring overlap are explored in relation to the kinetics of gene activation and decay. Coupling is most effective for genes in which the gene overlapping at its 5' end (leading gene) decays rapidly, while the gene overlapping at the 3' end (induced gene) decays slowly. If gene expression can feedback on itself (autocatalysis), then high rates of activation favor overlap.

Duality in the optimal control of hyperbolic equations with positive controls

We study the duality theory for hyperbolic equations. Also, we consider distributed control systems with positive control and convex cost functionals.

STABILITY AND EVOLUTION OF OVERLAPPING GENES

When the same sequence of nucleotides codes for regions of more than one functional polypeptide, this sequence contains overlapping genes. Overlap is most common in rapidly evolving genomes with high mutation rates such as viruses, bacteria, and mitochondria. Overlap is thought to be important as: (1) a means of compressing a maximum amount of information into short sequences of structural genes; and (2) as a mechanism for regulating gene expression through translational coupling of functionally related polypeptides. The stability of overlapping codes is examined in relation to the information cost of overlap and the mutation rate of the genome. The degree of overlap in a given population will tend to become monomorphic. Evolution toward partial overlap of genes is shown to depend on a convex cost function of overlap. Overlap does not evolve when expression of overlapping genes is mutually exclusive and produced by rare mutations to the wild-type genome. Assuming overlap increases coupling between functionally related genes, the conditions favoring overlap are explored in relation to the kinetics of gene activation and decay. Coupling is most effective for genes in which the gene overlapping at its 5′ end (leading gene) decays rapidly, while the gene overlapping at the 3′ end (induced gene) decays slowly. If gene expression can feedback on itself (autocatalysis), then high rates of activation favor overlap.Corresponding Editor: M. Riley