Showing papers on "Concave function published in 1995"

Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations

Abstract: This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint, and (ii) it uses a two-sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented that have no gap between necessity and sufficiency. Conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil. A simple dual program is introduced that involves the maximization of a strictly concave function on an interval. This dual program simplifies the theory and algorithms for trust region subproblems. It also illustrates that the trust region subproblems are implicit convex programming problems, and thus explains why they are so tractable.The duality theory also provides connections to eigenvalue perturbation theory. Trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems, and adding a linear term in the objective function is seen to correspond to a p...

Concave Minimization: Theory, Applications and Algorithms

Abstract: The purpose of this chapter is to present the essential elements of the theory, applications, and solution algorithms of concave minimization. Concave minimization problems seek to globally minimize real-valued concave functions over closed convex sets. As in other global optimization problems, in concave minimization problems there generally exist many local optima which are not global. Concave minimization problems are NP-hard. Even seemingly-simple cases can possess an exponential number of local minima. However, in spite of these difficulties, concave minimization problems are more tractable than general global optimization problems. This is because concave functions and minimizations display some special mathematical properties. Concave minimization problems also have a surprisingly-diverse range of direct and indirect applications. The special mathematical properties and diverse range of applications of concave minimization have motivated the construction of a rich and varied set of approaches for their solution. Three fundamental classes of solution approaches can be distinguished. These are the enumerative, successive approximation, and successive partitioning (branch and bound) approaches. After giving a brief introduction, the chapter describes and discusses some of the most important special mathematical properties of concave functions and of concave minimization problems. Following this, it describes several of the most important categories of direct and indirect applications of concave minimization. The chapter then describes each of the three fundamental solution approaches for concave minimization. For each approach, the essential elements are explained in a unified framework. Following this, for each approach, most of the well-known individual concave minimization algorithms that use the approach are explained and compared. The scope of the presentation of the solution approaches and algorithms is limited to those that use deterministic (rather than stochastic) methods. The chapter concludes with some suggested topics for further research.

Characterizing properties of stochastic objective functions

Abstract: This paper develops tools for analyzing properties of stochastic objective functions that take the form ( , ) ( , ) ( ; ) Vu dF ≡ ∫s xx s s θ θ . The paper analyzes the relationship between properties of the primitive functions, the utility function u and probability distribution F, and properties of the stochastic objective. The methods apply when the utility function is restricted to lie in a set of functions which is a "closed convex cone" (e.g., nondecreasing functions, concave functions, or supermodular functions). Approaches previously applied to characterize monotonicity of V (that is, stochastic dominance theorems) can be used to establish other properties of V as well. The first part of the paper establishes necessary and sufficient conditions for V to satisfy "closed convex cone properties," such as supermodularity, in the parameter θ. Then, we consider necessary and sufficient conditions for monotone comparative statics predictions. A new property of payoff functions is introduced, called l-supermodularity, which is shown to be necessary and sufficient for comparative statics predictions. The results are illustrated with applications.

Extension of the method of generalized quasilinearization for second order boundary value problems

Abstract: In this paper the method of generalized quasilinearization has been extended to second order nonlinear boundary value problem. We develop the method of quasilinearization when the right hand side function is the sum of a convex and a concave function. Yet like monotone method, we have two sided moriotone iteraterates which converges quadratically to the unique solution of the boundary value problem.

Minimax and Its Applications

Abstract: Consider the problem min x∈X max i∈I f i (x) where X is a convex set, I is a finite set of indices and the f i (x)’s are continuous concave functions of x. In this article, we study a characterization of x ∈ X at which the minimax value is achieved. We also study some applications of the characterization.

Multifractal decomposition of the Sierpinski carpet

Abstract: We analyse the multifractal decomposition of a measure defined on a general Sierpinski carpet. We compute the dimension spectrum f(α) and we show that this function is real-analytic on a certain domain when it is not degenerated. Actually, we prove that f is the Legendre-Fenchel transformation of a free energy function F which is also real-analytic. These two functions have the typical behaviours. In particular, F is strictly increasing and is in general strictly concave (respectively linear in the degenerate case), and f, on its domain, has a typical shape of a strictly concave function (respectively defined in one point in the degenerate case). We associate also to the singularity sets Cα measures which are singular with respect to each other, and we see that these measures are very well fitted to these singularity sets. This work completes with (D. Simpelaere, Chaos, Solitons and Fractals 4(12), 2223–2235 (1994)) the study of the multifractal analysis of the Sierpinski carpets.

Nonconvex optimization over a polytope using generalized capacity improvement

Abstract: Capacity improvement involves the reduction of the size of the feasible region of an optimization problem without ‘cutting-off’ the optimal solution point(s). Capacity improvement can be used in a branch-and-bound procedure to produce tighter relaxations to subproblems in the enumeration tree. Previous capacity improvement work has concentrated on tightening the simple lower and upper bounds on variables. In this paper, the capacity improvement procedure is generalized to apply toall constraints that form the feasible region. For the minimization of a separable concave function over a bounded polytope, the method of calculating the capacity improvement parameters is very straightforward. Computational results for fixed-charge and quadratic concave minimization problems demonstrate the effectiveness of this procedure.

Properties of the modulus of continuity for monotonous convex functions and applications.

Abstract: For a monotone convex function f ∈ C [ a , b ] we prove that the modulus of continuity w ( f ; h ) is concave on [ a , b ] as function of h . Applications to approximation theory are obtained.

Subdifferentiability and lipschitz continuity in experimental design problems

Abstract: The calculus of concave functions is a widely accepted tool for optimum experimental design problems. However, as a function of the support points and the weights the design problem fails to be concave. In this paper we make use of generalized gradients in the sense of Rockafellar (1980) and Clarke (1983). A chain rule is presented for the subdifferential of the composition of an information function with the moment matrix mapping. Lipschitz continuity of the global design function is proved and conditions for strict differentiability are given.

Linearly constrained convex programming as unconstrained differentiable concave programming

Abstract: Recently, Fang proposed approximating a linear program in Karmarkar's standard form by adding an entropic barrier function to the objective function and using a certain geometric inequality to transform the resulting problem into an unconstrained differentiable concave program. We show that, by using standard duality theory for convex programming, the results of Fang and his coworkers can be strengthened and extended to linearly constrained convex programs and more general barrier functions.

Minimum-concave-cost flows in series-parallel networks

Abstract: This work develops polynomial-time dynamic-programming algorithms for two classes of minimum-concave-cost flow problems in series-parallel networks. Significant problems from production and inventory management, capacity planning, network design and transportation can be formulated in these terms. A directed graph is series-parallel if it can be constructed from a single directed arc by a finite sequence of expansions each involving replacement of an arc either by arcs in series or by arcs in parallel. A graph is strong-series-parallel if it is series-parallel and the series and parallel replacements in its construction preserve the direction of the arcs they replace. The first class of problems is the minimum-aggregate-concave-cost multicommodity uncapacitated flow problem in a strong-series-parallel network. In this problem, the cost of a flow is the sum of concave functions, each depending on the aggregate flow in an arc. Our results for this problem 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 second class of problems considered in this work is the minimum-concave-cost T-period capacitated dynamic economic-order-quantity problem in which there are at most K different capacities. This problem can be reduced to one of uncapacitated network flows. Although the resulting network is series-parallel, it is not strongly so. Thus the above algorithm does not apply. Our method generalizes the Wagner-Whitin algorithm for the uncapacitated problem to the capacitated case and runs in time proportional to $T\sp{2K{+}2}.$ When $K = 1,$ this running time matches that of an algorithm of Florian and Klein (1971). When $T = K,$ our algorithm runs in exponential time, reflecting the fact that this instance is known to be NP-hard. Thus, the algorithm bridges the gap between a class of known polynomially solvable problems and a class of NP-hard problems.

Solving Groundwater Management Problems Using a New Methodology

Abstract: Recently the problem of groundwater management has been approached by several optimization techniques including the classical linear/nonlinear programming methods, simulated annealing, neural networks, genetic algorithms and the outer approximation method The ‘outer approximation’ method is a global optimization technique for the minimization of a concave function over a compact set of constraints The concept of this method, as well as applications of the method to groundwater management, problems, was first presented by the authors for problems with a convex set of constraints (Karatzas and finder, [1993]), and in a later work for a non-convex set of constraints (baratzas and finder, [1991])