Lot-sizing with non-stationary cumulative capacities

We study a new class of capacitated economic lot-sizing problems. We show that the problem is NP-hard in general and derive a fully polynomial-time approximation algorithm under mild conditions on the cost functions. Furthermore, we develop a polynomial-time algorithm for the case where all cost functions are concave.     

Quasi-concave functions on meet-semilattices

This paper deals with maximization of set functions defined as minimum values of monotone linkage functions. In previous research, it has been shown that such a set function can be maximized by a greedy type algorithm over a family of all subsets of a finite set. In this paper, we extend this finding to meet-semilattices.We show that the class of functions defined as minimum values of monotone linkage functions coincides with the class of quasi-concave set functions. Quasi-concave functions determine a chain of upper level sets each of which is a meet-semilattice. This structure allows development of a polynomial algorithm that finds a minimal set on which the value of a quasi-concave function is maximum. One of the critical steps of this algorithm is a set closure. Some examples of closure computation, in particular, a closure operator for convex geometries, are considered.     

Bin packing with general cost structures

Following the work of Anily et?al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously.     

Numerical Methods for Single Machine Scheduling with Non-linear Cost Functions to Minimize Total Cost

In this paper we are concerned with the problem of sequencing a given set of jobs without preemption on a single machine so as to minimize total cost, where associated with each job is a processing time and a differentiable cost function defined on the completion time of the job. The problem, in general, is NP-complete and, therefore, there is unlikely to be an algorithm to solve the problem in reasonable time, thus a heuristic algorithm is desirable. We present two heuristic algorithms to solve the problem. The first algorithm is based on the differential of the cost functions, and the second algorithm is based on the least square approximation of the cost functions. Computational experiences for the case of quadratic, cubic, and exponential cost functions are presented.     

Approximation algorithm for a class of global optimization problems

In this paper we develop and derive the computational cost of an ${\varepsilon}$ -approximation algorithm for a class of global optimization problems, where a suitably defined composition of some ratio functions is minimized over a convex set. The result extends a previous one about a class of Linear Fractional/Multiplicative problems.     

Computing all solutions of linear generalized Nash equilibrium problems

In this paper we consider linear generalized Nash equilibrium problems, i.e., the cost and the constraint functions of all players in a game are assumed to be linear. Exploiting duality theory, we design an algorithm that is able to compute the entire solution set of these problems and that terminates after finite time. We present numerical results on some academic examples as well as some economic market models to show effectiveness of our algorithm in small dimensions.     

The problem of the optimal biobjective spanning tree

This paper studies the problem of finding the set of optimal spanning trees of a connected graph, considering two cost functions defined on the set of edges. This problem is NP-hard and the solution is described through an algorithm that builds the family of efficient trees. This algorithm needs two procedures that solve the following uniobjective problems: the construction of all the spanning trees of a connected graph and the construction of the whole set of minimum cost spanning trees. The computational results obtained are shown in Section 5.     

Online scheduling with general machine cost functions

For most scheduling problems the set of machines is fixed initially and remains unchanged for the duration of the problem. Recently online scheduling problems have been investigated with the modification that initially the algorithm possesses no machines, but that at any point additional machines may be purchased. In all of these models the assumption has been made that each machine has unit cost. In this paper we consider the problem with general machine cost functions. Furthermore we also consider a more general version of the problem where the available machines have speed, the algorithm may purchase machines with speed 1 and machines with speed s. We define and analyze some algorithms for the solution of these problems and their special cases. Moreover we prove some lower bounds on the possible competitive ratios.     

Quadratic problems defined on a convex hull of points

In this paper we consider an algorithm for a class of quadratic problems defined on a polytope which is described as the convex hull of a set of points. The algorithm is based on simplex partitions using convex underestimating functions.     

Clique partitioning with value-monotone submodular cost

We consider the problem of partitioning a graph into cliques of bounded cardinality. The goal is to find a partition that minimizes the sum of clique costs where the cost of a clique is given by a set function on the nodes. We present a general algorithmic solution based on solving the problem variant without the cardinality constraint. We obtain constant factor approximations depending on the solvability of this relaxation for a large class of submodular cost functions which we call value-monotone submodular functions. For special graph classes we give optimal algorithms.     

Adjustable robust optimization with objective uncertainty

In this work, we study optimization problems where some cost parameters are not known at decision time and the decision flow is modeled as a two-stage process within a robust optimization setting. We address general problems in which all constraints (including those linking the first and the second stages) are defined by convex functions and involve mixed-integer variables, thus extending the existing literature to a much wider class of problems. We show how these problems can be reformulated using Fenchel duality, allowing to derive an enumerative exact algorithm, for which we prove asymptotic convergence in the general case, and finite convergence for cases where the first-stage variables are all integer.An implementation of the resulting algorithm, embedding a column generation scheme, is then computationally evaluated on a variant of the Capacitated Facility Location Problem with uncertain transportation costs, using instances that are derived from the existing literature. To the best of our knowledge, this is the first approach providing results on the practical solution of this class of problems.     

Nonconvex functions and variational inequalities

In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.     

Classification and generation of disturbance vectors for collision attacks against SHA-1

The main contribution of this paper is to provide a classification of disturbance vectors used in differential collision attacks against ${\tt{SHA}-1}$ . We show that all published disturbance vectors can be classified into two types of vectors, type-I and type-II. We present a deterministic algorithm which produce efficient disturbance vectors with respect to any given cost function. We define two simple cost functions to evaluate the efficiency of a candidate disturbance vector. Using our algorithm and those cost function we retrieved all previously known vectors and found that the most efficient disturbance vector is the one first reported as Codeword2 by Jutla and Patthak, A matching lower bound on the minimum weight of SHA-1 expansion code. Cryptology ePrint Archive, Report 2005/266, (2005). We also present a statistical evaluation of local collisions?? holding probabilities and show that the common assumption of local collision independence is flawed.     

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function \(f_i(z)\). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a (\(7.88+\epsilon \))-approximation local search algorithm for this problem.     

Risk averse elastic shape optimization with parametrized fine scale geometry

Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.     

The -digraph optimization problem and the greedy algorithm

In this paper we present a new optimization problem and a general class of objective functions for this problem. We show that optimal solutions to this problem with these objective functions are found with a simple greedy algorithm. Special cases include matroids, Huffman's data compression problem, a special class of greedoids, a special class of min cost max flow problems (related to Monge sequences), a special class of weighted f-factor problems, and some new problems.     

Discrete time/cost trade-offs in project scheduling with time-switch constraints

The discrete time/cost trade-off problem assumes the duration of project activities to be discrete, non-increasing functions of the amount of a single non-renewable resource. The problem has been studied under three possible objectives. The so-called deadline problem involves the scheduling of project activities in order to minimize the total cost of the project while meeting a given deadline. The budget problem aims at minimizing the project duration without exceeding a given budget. A third objective involves the generation of the complete efficient time/cost profile over the set of feasible project durations. In this paper we describe a solution procedure for the deadline problem in which three special cases of time-switch constraints are involved. These constraints impose a specified starting time on the project activities and force them to be inactive during specified time periods. We propose a branch-and-bound algorithm and a heuristic procedure which both make use of a lower bound calculation for the discrete time/cost trade-off problem (without time-switch constraints). The procedures have been coded in Visual C++, version 6.0 under Windows 2000 and have been validated on a randomly generated problem set. We also discuss an illustrative example based on a real-life situation.     

Multi-objective optimization method for learning thresholds in a decision-theoretic rough set model

For decision-theoretic rough sets, a key issue is determining the thresholds for the probabilistic rough set model by setting appropriate cost functions. However, it is not easy to obtain correct cost functions because of a lack of prior knowledge and few previous studies have addressed the determination of learning thresholds and cost functions from datasets. In the present study, a multi-objective optimization model is proposed for threshold learning. In our model, we integrate an objective function that minimizes the decision cost with another that decreases the size of the boundary region. The ranges of the thresholds and two types of F_measure are used as constraints. In addition, a multi-objective genetic algorithm is employed to obtain the Pareto optimal set. We used 12 UCI datasets to validate the performance of our method, where the experimental results demonstrated the trade-off between the two objectives as well as showing that the thresholds obtained by our method were more intuitive than those obtained using other methods. The classification abilities of the solutions were improved by the F_measure constraints.     

A smoothing Newton method for NCPs with the P0-property

In this paper, we first investigate a two-parametric class of smoothing functions which contains the penalized smoothing Fischer-Burmeister function and the penalized smoothing CHKS function as special cases. Then we present a smoothing Newton method for the nonlinear complementarity problem based on the class of smoothing functions. Issues such as line search rule, boundedness of the level set, global and quadratic convergence are studied. In particular, we give a line search rule containing the common used Armijo-type line search rule as a special case. Also without requiring strict complementarity assumption at the P₀-NCP solution or the nonemptyness and boundedness of the solution set, the proposed algorithm is proved to be globally convergent. Preliminary numerical results show the efficiency of the algorithm and provide efficient domains of the two parameters for the complementarity problems.