Showing papers on "Concave function published in 1994"

Worst-Case Analysis of Heuristics for the Bin Packing Problem with General Cost Structures

Abstract: We consider the famous bin packing problem where a set of items must be stored in bins of equal capacity. In the classical version, the objective is to minimize the number of bins used. Motivated by several optimization problems that occur in the context of the storage of items, we study a more general cost structure where the cost of a bin is a concave function of the number of items in the bin. The objective is to store the items in such a way that total cost is minimized. Such cost functions can greatly alter the way the items should be assigned to the bins. We show that some of the best heuristics developed for the classical bin packing problem can perform poorly under the general cost structure. On the other hand, the so-called next-fit increasing heuristic has an absolute worst-case bound of no more than 1.75 and an asymptotic worst-case bound of 1.691 for any concave and monotone cost function. Our analysis also provides a new worst-case bound for the well studied next-tit decreasing heuristic when...

Estimating a Monotone Density from Censored Observations

Abstract: We study the nonparametric maximum likelihood estimator (NPMLE) for a concave distribution function $F$ and its decreasing density $f$ based on right-censored data. Without the concavity constraint, the NPMLE of $F$ is the product-limit estimator proposed by Kaplan and Meier. If there is no censoring, the NPMLE of $f$, derived by Grenander, is the left derivative of the least concave majorant of the empirical distribution function, and its local and global behavior was investigated, respectively, by Prakasa Rao and Groeneboom. In this paper, we present a necessary and sufficient condition, a self-consistency equation and an analytic solution for the NPMLE, and we extend Prakasa Rao's result to the censored model.

Method and system for scheduling using a facet ascending algorithm or a reduced complexity bundle method for solving an integer programming problem

Abstract: A method and system for scheduling using a facet ascending algorithm or a reduced complexity bundle method for solving an integer programming problem is presented. A Lagrangian dual function of an integer scheduling problem is maximized (for a primal minimization problem) to obtain a good near-feasible solution, and to provide a lower bound to the optimal value of the original problem. The dual function is a polyhedral concave function made up of many facets. The facet ascending algorithm of the present invention exploits the polyhedral concave nature of the dual function by ascending facets along intersections of facets. At each iteration, the algorithm finds the facets that intersect at the current dual point, calculates a direction of ascent along these facets, and then performs a specialized line search which optimizes a scaler polyhedral concave function in a finite number of steps. An improved version of the facet ascending algorithm, the reduced complexity bundle method, maximizes a nonsmooth concave function of variables. This is accomplished by finding a hyperplane separating the origin and the affine manifold of a polyhedron. The hyperplane also separates the origin and the polyhedron since the polyhedron is a subset of its affine manifold. Then an element of the bundle is projected onto the subspace normal to the affine manifold to produce a trial direction normal. If the projection is zero (i.e., indicating the affine manifold contains the origin), a re-projection onto the subspace normal to the affine manifold of an appropriate face of the polyhedron gives a trial direction. This reduced complexity bundle method always finds an e-ascent trial direction or detects an e-optimal point, thus maintaining global convergence. The method can be used to maximize the dual function of a mixed-integer scheduling problem.

A finite concave minimization algorithm using branch and bound and neighbor generation

Abstract: In this article we present a new finite algorithm for globally minimizing a concave function over a compact polyhedron. The algorithm combines a branch and bound search with a new process called neighbor generation. It is guaranteed to find an exact, extreme point optimal solution, does not require the objective function to be separable or even analytically defined, requires no nonlinear computations, and requires no determinations of convex envelopes or underestimating functions. Linear programs are solved in the branch and bound search which do not grow in size and differ from one another in only one column of data. Some preliminary computational experience is also presented.

Concave gauge functions and applications

Abstract: Many problems of optimization involve the minimization of an objective function on a convex cone. In this respect we define a concave gauge function which will be used in interior point methods. Application are given in particular on the space of real symmetric matrices.

A generalization of the construction of test problems for nonconvex optimization

Abstract: In this paper we adopt and generalize the basic idea of the method presented in [3] and [4] to construct test problems that involve arbitrary, not necessarily quadratic, concave functions, for both Concave Minimization and Reverse Convex Programs

A penalty for concave minimization derived from the tuy cutting plane

Abstract: A wide variety of optimization problems have been approached with branch-and-bound methodology, most notably integer programming and continuous nonconvex programming. Penalty calculations provide a means to reduce the number of subproblems solved during the branch-and-bound search. We develop a new penalty based on the Tuy cutting plane for the nonconvex problem of globally minimizing a concave function over linear constraints and continuous variables. Computational testing with a branch-and-bound algorithm for concave minimization indicates that, for the problems solved, the penalty reduces solution time by a factor ranging from 1.2 to 7.2. © 1994 John Wiley & Sons, Inc.

Log-Concave Utility and its Applications

Abstract: We have found several propositions in the economics of information which depend on the log of the cumulative distribution distribution of a random variable being a concave function. In this paper we present several theorems and applications of the property of log-concavity.

Fixed-point theorems of -concave (convex)operator and application

Abstract: In this paper, the definition of -concave (-convex) operator is given, andsome new results are obtained. Compared with some previous results, these new results weaken some conditions (e.g. take off compactness, continuity etc.)and strengthen the conclusions.

About saddle values of a class of convex-concave functions

Abstract: The infimal convolution of convex functions is extended to the saddle case and a class of well-behaved convex-concave functions is introduced

Testing for and against a concavity restriction with normal errors

Abstract: Tests of linearity in regression functions are frequently applied in practice. If a researcher hasapriori information about the shape of the regression function, then incorporating this information into the test typically increases the power at alternatives that satisfy the hypothesized shape restriction. We study likelihood ratio tests of linearity with the alternative constrained to be concave (or convex) as well as tests of concavity as the null hypothesis. To complete the development of the test statistics and their null distributions, we only need to study the level probabilities. A numerical example and a discussion of the use of modifications of these tests as alternatives to the likelihood ratio test for homogeneity versus a unimodal (umbrella) ordering are included