Showing papers on "Concave function published in 1999"

Coherent dispersion criteria for optimal experimental design

Abstract: We characterize those coherent design criteria which depend only on the dispersion matrix (assumed proper and nonsingular) of the “state of nature,” which may be a parameter-vector or a set of future observables, and describe the associated decision problems. Connections are established with the classical approach to optimal design theory for the normal linear model, based on concave functions of the information matrix. Implications of the theory for more general models are also considered.

Exact solutions to the transportation problem on the line

Abstract: Given distributions ;w of production and v of consumption on the line, the Monge–Kantorovich problem is to decide which producer should supply each consumer in order to minimize the total transportation costs. Here cost will be assumed to be a strictly concave function of the distance, which translates into an economy of scale for longer trips and may encourage cross–hauling. The resulting solutions display a hierarchical structure that reflects a striking separation into local and global scales also found in the real world. Moreover, this structure can be exploited to reduce the infinite–dimensional linear problem to a convex minimization in m variables, where 2 m + 2 counts the number of times that ;w – v changes sign. A combinatorial algorithm is then derived which yields exact solutions by optimizing a certain finite sequence of convex, separable network flows.

A production-transportation problem with stochastic demand and concave production costs

Abstract: The problem of transporting goods from a set of supply points (factories) to a set of demand points (customers) so as to minimize linear transportation costs is well known and very efficient solution methods exist. In the well-known facility location problem one also includes fixed charges for the supply points. On the other hand, in ordinary transportation problems, stochastic demand has been introduced and modelled by convex costs, yielding the Stochastic Transportation Problem, [STP], solved by for example the methods in Cooper and LeBlanc (1977) and Holmberg and Jornsten (1984). However, the simultaneous use of these generalizations has recieved little attention until now. Only a few suggestions for solution methods can be found, LeBlanc (1977), Franca and Luna (1982). In this paper we consider generalizing this problem even further, by introducing general concave costs at the supply points, as well as convex costs at the demand points. Fixed charges are not the only cost structures that occur when producing goods at a factory. Economies of scale very often yield other concave cost functions.

Structural properties of singularities of semiconcave functions

Abstract: A semiconcave function on an open domain of R" is a function that can be locally represented as the sum of a concave function plus a smooth one. The local structure of the singular set (non-differentiability points) of such a function is studied in this paper. A new technique is presented to detect singularities that propagate along Lipschitz arcs and, more generally, along sets of higher dimension. This approach is then used to analyze the singular set of the distance function from a closed subset of R^n

Loss of convexity of simple closed curves moved by surface diffusion

Abstract: We rigorously prove that there exists a simple, strictly convex, smooth closed curve which loses convexity but stays simple without developing singularities when it moves by its surface diffusion for a short time.

On self-concordant convex–concave functions

Abstract: In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of “sufficiently well-behaved” convex-concave functions—those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an exterior penalty polynomial time method for Semidefinite Programming and to the problem of inscribing the largest volume ellipsoid into a given polytope.

Extremal points of a functional on the set of convex functions

Abstract: We investigate the extremal points of a functional f f(Vu), for a convex or concave function f. The admissible functions u : Q C RN ->F R are convex themselves and satisfy a condition u2 < u < ul. We show that the extremal points are exactly u1 and u2 if these functions are convex and coincide on the boundary 9Q. No explicit regularity condition is imposed on f, u1, or U2. Subsequently we discuss a number of extensions, such as the case when ul or u2 are non-convex or do not coincide on the boundary, when the function f also depends on u, etc.

Minimum-Aggregate-Concave-Cost Multicommodity Flows in Strong-Series-Parallel Networks

Abstract: This work develops a polynomial-time dynamic-programming algorithm for minimum-aggregate-concave-cost multicommodity flow problems in strong-series-parallel networks. Many important problems from production and inventory management, capacity planning, network design and transportation can be formulated in these terms. Our results include a characterization of extreme flows in strong-series-parallel networks and an algorithm based on this characterization that searches extreme flows efficiently to find an optimal one. The algorithm runs in time proportional to A(N + K), where A, N and K are respectively the numbers of arcs, nodes and commodities in the network, and appears to be the first to solve the problem in polynomial time. When applied to the dynamic economic-order-quantity problem, the algorithm matches the performance of that of Wagner and Whitin (1958). Moreover, our algorithm has broader applications, including the multi-division capacity expansion problem and the generalization of the dynamic economic-order-quantity problem to series-parallel production processes in which subassemblies are assembled into a finished product.

The core of a class of non-atomic games which arise in economic applications*

Abstract: We study the core of a non-atomic game v which is uniformly con- tinuous with respect to the DNA-topology and continuous at the grand co- alition. Such a game has a unique DNA-continuous extension v on the space B1 of ideal sets. We show that if the extension v is concave then the core of the game v is non-empty iv is homogeneous of degree one along the diagonal of B1. We use this result to obtain representation theorems for the core of a non- atomic game of the form va f m where m is a finite dimensional vector of measures and f is a concave function. We also apply our results to some non- atomic games which occur in economic applications.

Finiteness of conical algorithms with ω-subdivisions

Abstract: In this paper the problem of finding the global optimum of a concave function over a polytope is considered. A well-known class of algorithms for this problem is the class of conical algorithms. In particular, the conical algorithm based on the so called ω-subdivision strategy is considered. It is proved that, for any given accuracy e>0, this algorithm stops in a finite time by returning an e-optimal solution for the problem, while it is convergent for e=0.

On Optimization Properties of Functions, with a Concave Minorant

Abstract: We give a definition of the class of functions with a concave minorant and compare these functions with other classes of functions often used in global optimization, e.g. weakly convex functions, d.c. functions, Lipschitzian functions, continuous and lower semicontinuous functions. It is shown that the class of functions with a concave minorant is closed under operations mainly used in optimization and how a concave minorant can be constructed for a given function.

Generalized a-valid cut procedure for concave minimization

Abstract: The concept of a γ-valid cutting plane has been used in many types of algorithms for solving concave minimization problems. Unfortunately, the procedures proposed to date for constructing these cuts are valid only under certain assumptions that often may not hold in practice. Chief among these is the requirement that the feasible region of the concave minimization problem in question have full dimension, and that the objective function of this problem be concave rather than quasiconcave. In this article, we propose, validate, and show how to implement a more general γ-valid cutting plane procedure which eliminates these restrictions.

Necessary and Sufficient Conditions for Dominance Using Generalized Lorenz Curves

Abstract: Almost all functions used in economics are concave, but there are important examples of convex functions, e. g., cost functions and poverty or concentration indices. In these cases one can choose between working with convex or concave functions (e.g., poverty indices or welfare functions).

Quasi-concave functions and convex convergence to infinity

Abstract: By a convex mode of convergence to infinity 〈Ck〉, we mean a sequence of nonempty closed convex subsets of a normed linear space X such that for each k, Ck+1 ⊆ int Ck and and a sequence 〈xn〉 is X is declared convergent to infinity with respect to 〈Ck〉 provided each Ck contains xn eventually. Positive convergence to infinity with respect to a pointed cone with nonempty interior as well as convergence to infinity in a fixed direction fit within this framework. In this paper we study the representation of convex modes of convergence to infinity by quasi-concave functions and associated remetrizations of the space.

Continuity of Optimal Control in Differential Games and Certain Properties of Weakly and Strongly Convex Functions

Abstract: In the first part of the paper, the equivalence of Lipschitzian differentiability of a function and a set of conditions of weak convexity and weak concavity of this function is proved, as well as sufficient conditions for the continuous dependence of the saddle point on a strongly convex-concave function of a parameter are given. In the second part, it is proved that the value function of a game is smooth and the optimal positional and programmed strategies of the players are continuous in zero-sum nonlinear differential games with strongly convex-concave Lagrangian.

Fluid stochastic event graphs for optimisation of failure-prone manufacturing systems

Abstract: This paper addresses the performance evaluation and optimisation of failure-prone manufacturing systems. For this purpose, we propose a fluid stochastic event graph model that is a decision-free Petri net and can be used to model transfer lines, kanban systems, etc. In this model, tokens are considered as continuous flows. A transition can be in operating state or in failure state. A transition in operating state can fire at its maximal speed and a transition in failure state cannot fire. Transitions between failure and operating states are independent of the firing conditions and the sojourn time in each state is a random variable of general distribution. The system is hybrid in the sense that it has both continuous dynamic as a result of continuous material flow and discrete events including failures and repairs. For the purpose of performance evaluation, a set of evolution equations that determines continuous state variables at epochs of discrete events is established. Based on the evolution equations, we prove that the cumulative firing of transitions are Lipschitz continuous, non-decreasing and concave functions of some system parameters including maximal firing rates and the initial marking. Gradient estimators of the cumulative firings with respect to the system parameters are derived. Ergodic properties of the stochastic process are established. Unbiasedness and strong consistency of the gradient estimators are proved. Finally, an optimisation problem of the initial marking that maximises a concave function of throughput rate and the initial marking is addressed.