Showing papers on "Concave function published in 1988"

Coordinate ascent for maximizing nondifferentiable concave functions

Abstract: We present a coordinate ascent method for maximizing concave (not necessarily differentiable) functions possessing a certain separable structure. This method, when applied to the dual of a linearly constrained convex program, includes as special cases a successive projection algorithm of Han [11], the method of multipliers [10, 12, 20], and a number of dual coordinate ascent methods [5-7, 13, 15, 17-18, 25, 30]. We also generalize the results of Auslender [1, §6] and of Bertsekas-Tsitsiklis [3, §3.3.5] on the convergence of this method. One primal application of this method is the proximal minimization algorithm [23].

A note on the solution of bilinear programming problems by reduction to concave minimization

Abstract: We present a new algorithm for solving the bilinear programming problem by reduction to concave minimization. This algorithm is finite, does not assume the boundedness of the constraint set, and uses an efficient procedure for checking whether a concave function is bounded below on a given halfline. Some preliminary computational experience with a computer code for implementing the algorithm on a microcomputer is also reported.

On existence and uniqueness of maximum likelihood estimates in quantal and ordinal response models

Abstract: For quantal and ordinal response models, conditions on existence and uniqueness of maximum likelhood estimates are presented. Results are derived from general results on direction sets and spaces associated with a proper concave function. If each summand of the log likelihood is in any direction either strictly concave or affine, necessary and sufficient conditions are obtained. If all cell counts are strictly positive, then it is shown that estimates always exist, and that they are unique if all parameters are identifiable. If estimates exist without being unique, results on uniquely estimable linear functions are given, paralleling corresponding results in linear regression. An extension of the maximum likelihood principle is outlined yielding similar results even if the likelihood does not attain its supremum. The logit model, the linear probability model, cumulative and sequential models and binomial response models are considered in detail.

Stochastic Scheduling on Parallel Processors and Minimization of Concave Functions of Completion Times

Abstract: We consider a stochastic scheduling problem in which n jobs are to be scheduled on m identical processors which operate in parallel. The processing times of the jobs are not known in advance but they have known distributions with hazard rates ρ 1, (t), …, ρ n (t). It is desired to minimize the expected value of к(C), where C i is the time at which job i is completed C = (C 1, …, C n ), and к(C) is increasing and concave in C. Suppose processor i first becomes available at time τ i . We prove that if there is a single static list priority policy which is optimal for every τ = (τ 1, …, τ m ), then the minimal expected cost must be increasing and concave in τ. Moreover, if к(C) is supermodular in C then this cost is also supermodular in τ. This result is used to prove that processing jobs according to the static list priority order (1,2,…,n) minimizes the expected value of ∑w i h(C i ), when h(·) is a nondecreasing, concave function, w 1 ≥ … ≥ w n , and ρ 1 (t 1)w 1 ≥ … ≥ ρ n (t n )w n for all t 1, …, t n .

Automatic continuity of concave functions

Abstract: A necessary and sufficient condition is given that a semicontinuous, nonnegative, concave function on a finite dimensional closed convex set X necessarily be continuous at a point xo E X. Application of this criterion at all points of X yields a characterization, due to Gale, Klee and Rockafellar, of convex polyhedra in terms of continuity of their convex functions. Let V be a real vector space of dimension n 0, being closed. Observe also that the L+(0, s) are convex. We say X is polyhedral if it is specified by a finite number of linear inequalities (3) X = {v E V: Ai(v) ? bi, Ai E V*, bi E R, 1 0 we have a closed convex set of Y C V x [0, t] such that (4) (a) Yn(V x {0}) =Xx {0} (b) If (x, r) E Y, then (x, r') E Y for 0 < r' < r. Received by the editors December 18, 1986 and, in revised form, May 4, 1987. 1980 Mathernatic8s Subject (lassification (1985 R?evision). Primary 52A20.

Global Optimization of Concave Functions Subject to Separable Quadratic Constraints and of All-Quadratic Separable Problems

Abstract: : This paper proposes different methods for finding the global minimum of concave function subject to quadratic separable constraints. THe first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

Minimizing a quasi-concave function subject to a reverse convex constraint

Abstract: Any constraintg(x)≥0 is called a reverse convex constraint ifg: Rn→ R1 is a continuous convex function. This paper establishes a finite method for finding an optimal solution to a concave program with an additional reverse convex constraint. The method presented is a new approach to global optimization problems since it combines the idea of the branch and bound method with the idea of the cutting plane method.

Rate of convergence for the saddle points of convex-concave functions

Abstract: In the lines of H. Attouch and R. Wets, two kinds of variational metrics are introduced between closed proper classes of convex-concave functions. The comparison between these two distances gives rise to a metric stability result for the associated saddle-points.

単一作業者が受け持つべき機械台数に関する研究 : 第1報,最適受持台数の存在のための条件

Abstract: This paper considers a man-machine system consisting of some numerically -controlled (NC) machines and a single worker. This system processed many kinds of items. The role of worker is loading the workpieces on and unloading the workpieces from machines. In such a system design, it is an important problem to determine the number of machine5 to be operated by the worker. First. we formulate the system as a stochastic model to analyze the above optimal problem. The desicion variable is the number of machines and the optimal criteria are 'the maximum production rate', 'the minimum total cost', 'the maximum profit rate' and 'the maximum return's rate of operating cost'. It is shown that the optimal numbers under these criteria depend on only the utilization of the worker. and hence the minimum total cost, and the maximum return's rate criteria are the same. It is proved that there exist the optimal numbers under these criteria if the worker's utilization is a concave function of the number of machines used. Furthermore, an ordering between the optimal numbers under the minimum total cost and the maximum profit rate criteria is suggested theoretically. The validity of the concavity and the ordering is established by many numerical experiments. The applicability of the results to practical problem is discussed via a case study.

Branch-and-bound approach for a stochastic production planning problem with capacity constraints

Abstract: A stochastic production planning problem with a fmite number of planning periods is analyzed where cumulative demands up to each period are independent random variables with continuous probability distributions. In the problem, backlogging is permitted and production is restricted by its capacity. Dynamic but linear costs of inventory holding and backlogging, and of production with setup charge are considered. A branch·and-bound algorithm is developed to find an optimal plan within fmite searching steps, and its computational effectiveness is evaiuated. 1. I ntroducti on This paper considers a production planning problem with known stochastic demands where the planning horizon is composed of a finite number of planning periods and cumulative demands up to each period have a continuous probability distribution. Capacity restrictions are imposed on production. and unsatis­ fied demands are backlogged. Furthermore. dynamic but linear costs of inventory holding and backlogging. and of production with setup charge are included. The production problem can then be interpreted as in a similar form to a stochastic programming problem with simple recourse (Ziemba (12)). In fact. the stochastic production problem shall be transformed to an equivalent deterministic problem from which an optimal solution to the original problem ~s obtained. The equivalent deterministic problem has an objective function which is neither convex nor concave. Rather. it is a mixture of convex and concave functions which makes it difficult to solve the problem by using usual convex programming or concave programming algorithms. Therefore. this paper suggests

A Mathematical Programming Approach for Finding the Stochastically Most Relevant Failure Mechanism

Abstract: In calculating the failure probability of structural systems, the most important operation is the search for the stochasticly most relevant failure mechanism. The nodal and mesh description for the modelling of a flexural frame with fully plastic behaviour and slabs discretized into triangular finite elements whose behaviour conforms the yield line theory are considered. The mathematical programming method arising from these models can be formulated as the minimization of a quadratic concave function over a linear domain.