A reduction result for location problems with polyhedral barriers

In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facilities where a set of polyhedral barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the median and center objective functions). An exact algorithm and a heuristic solution procedure based on this reduction result are developed.     

Planar Weber location problems with barriers and block norms

The Weber problem for a given finite set of existing facilities Ex={Ex₁,Ex₂,...,Ex_M}⊂∝² with positive weights w_m (m=1,...,M) is to find a new facility X^*∈∝² such that Σ _m=1 ^M w_md(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by block norms. A variation of this problem is obtained if barriers are introduced which are convex polyhedral subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers, like lakes, military regions, national parks or mountains, are frequently encountered in practice. From a mathematical point of view barrier problems are difficult, since the presence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a discretization result: one of the grid points in the grid defined by the existing facilities and the fundamental directions of the polyhedral distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm. This revised version was published online in June 2006 with corrections to the Cover Date.     

The rectilinear distance Weber problem in the presence of a probabilistic line barrier

This paper considers the problem of locating a single facility in the presence of a line barrier that occurs randomly on a given horizontal route on the plane. The objective is to locate this new facility such that the sum of the expected rectilinear distances from the facility to the demand points in the presence of the probabilistic barrier is minimized. Some properties of the problem are reported, a solution algorithm is provided with an example problem, and some future extensions to the problem are discussed.     

On the use of the Varignon frame for single facility Weber problems in the presence of convex barriers

This paper presents a new experimental approach to the Weber problem in the presence of convex barriers by using the Varignon frame. The Varignon frame is a mechanical system of strings, weights and a board with holes that has been used to identify an optimal location for the classical Weber problem. We show through analytical results that the same analog can also be used for some of the Weber problems in the presence of barriers. Some examples from the literature are revisited through experiments. Findings are compared to those found in the literature. Practical use of the analog is discussed as it provides rapid solutions, allows for flexibility, and enables one to visualize the problem.     

An efficient solution method for Weber problems with barriers based on genetic algorithms

In this paper we consider the problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers. It is assumed that a barrier is a region where neither facility location nor travelling are permitted. The resulting non-convex optimization problem can be reduced to a finite series of convex subproblems, which can be solved by the Weiszfeld algorithm in case of the Weber objective function and Euclidean distances. A solution method is presented that, by iteratively executing a genetic algorithm for the selection of subproblems, quickly finds a solution of the global problem. Visibility arguments are used to reduce the number of subproblems that need to be considered, and numerical examples are presented.     

A new local search for continuous location problems

This paper presents a new local search approach for solving continuous location problems. The main idea is to exploit the relation between the continuous model and its discrete counterpart. A local search is first conducted in the continuous space until a local optimum is reached. It then switches to a discrete space that represents a discretisation of the continuous model to find an improved solution from there. The process continues switching between the two problem formulations until no further improvement can be found in either. Thus, we may view the procedure as a new adaption of formulation space search. The local search is applied to the multi-source Weber problem where encouraging results are obtained. This local search is also embedded within Variable Neighbourhood Search producing excellent results.     

Planar point-objective location problems with nonconvex constraints: A geometrical construction

The planar point-objective location problem has attracted considerable interest among Location Theory researchers. The result has been a number of papers giving properties or algorithms for particular instances of the problem. However, most of these results are only valid when the feasible region where the facility is to be located is the whole space ², which is a rather inaccurate approximation in many real world location problems.In this paper, the feasible region is allowed to be any closed, not necessarily convex, setS in ². The special structure of this nonconvex vector-optimization problem is exploited, leading to a geometrical resolution procedure when the feasible regionS can be decomposed into a finite number of (not necessarily disjoint) polyhedra.     

On the entropic regularization method for solving min-max problems with applications

Consider a min-max problem in the form of min _xX max_1im {f _i (x)}. It is well-known that the non-differentiability of the max functionF(x) max_1im {f _i (x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximationF _p(x) that uniformly converges toF(x) overX with a difference bounded by ln(m)/p, forp > 0. In this way, withp being sufficiently large, minimizing the smooth functionF _p(x) overX provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems.This research work was supported in part by the 1995 NCSC-Cray Research Grant and the National Textile Center Research Grant S95-2.     

SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

The single facility location problem with average-distances

This paper considers a location problem in ℝ ⁿ , where the demand is not necessarily concentrated at points but it is distributed in hypercubes following a Uniform probability distribution. The goal is to locate a service facility minimizing the weighted sum of average distances (measured with ℓ _p norms) to these demand hypercubes. In order to do that, we present an iterative scheme that provides a sequence converging to an optimal solution of the problem for p∈[1,2]. For the planar case, analytical expressions of this iterative procedure are obtained for p=2 and p=1, where two different approaches are proposed. The paper ends with a computational analysis of the proposed methodology, comparing its efficiency with a standard minimizer.      

A global optimization procedure for the location of a median line in the three-dimensional space

A global optimization procedure is proposed to find a line in the Euclidean three-dimensional space which minimizes the sum of distances to a given finite set of three-dimensional data points.Although we are using similar techniques as for location problems in two dimensions, it is shown that the problem becomes much harder to solve. However, a problem parameterization as well as lower bounds are suggested whereby we succeeded in solving medium-size instances in a reasonable amount of computing time.     

Analytic solution for American barrier options with two barriers

This paper concerns American barrier options with two barriers. Standard American Options are difficult to price but there exist good numerical or analytical approximation methods. The situation is different for American barrier options. These options cease to exist or come into being if some price barrier is hit during the option's life. The paper studies analytic valuation of American barrier options with two barriers where the barriers become active by turns. In this paper, analytic valuation formulas for these options are derived by using both constant and exponential barriers for optimal early exercise policies.     

A library for continuous convex separable quadratic knapsack problems

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.     

Minisum location problem with farthest Euclidean distances

The paper formulates an extended model of Weber problem in which the customers are represented by convex demand regions. The objective is to generate a site in R ² that minimizes the sum of weighted Euclidean distances between the new facility and the farthest points of demand regions. This location problem is decomposed into a polynomial number of subproblems: constrained Weber problems (CWPs). A projection contraction method is suggested to solve these CWPs. An algorithm and the complexity analysis are presented. Three techniques of bound test, greedy choice and choice of starting point are adopted to reduce the computational time. The restricted case of the facility is also considered. Preliminary computational results are reported, which shows that with the above three techniques adopted the algorithm is efficient.The authors were supported by the dissertation fund of Nari-Relays Corporation.     

Graph algorithm for partially-linear problems obtained by absolute value functions

The increasing interest in partially-linear models is caused by their role in economics, technology and mathematics. Every continuous partially-linear function with a finite number of linear parts can be formulated by means of linear function and absolute value functions. If we presebt every absolote value function as vertex of a graph we obtain the graph interpretaion of this optimization problem. The Partially-linear problems are often not only nondifferentiable but nonconvex as well,which makes their solving by standard optimization methods difficult. In this work we give the necessary and sufficient conditions for local optimality. An algorithm for solving such problems, based on the constructive optimization methods is suggested.     

Novel constraints satisfaction models for optimization problems in container terminals

The growth of containerization and transporting goods in containers has created many problems for ports. In this paper, we systematically survey a literature over problems in container terminals. The operations that take place in container terminals are described and then the problems are classified into five scheduling decisions. For each of the decisions, an overview of the literature is presented. After that, each of the decisions is formulated as Constraint Satisfaction and Optimization Problems (CSOPs). The literature also includes simulations and performance in container terminals. To evaluate any solution methods to the decisions and to measure its efficiency, several indicators are suggested.     

Special Pair of Dual Parametric Nonlinear Optimization Problems with Common Set of Optimal Points

On the base of a given strictly convex function defined on the Euclidean space E n ( n S 2) we can-without the assumption that it is differentiable - introduce some manifolds in topologic sense. Such manifolds are sets of all optimal points of a certain parametric non-linear optimization problem. This paper presents above all certain generalization of some results of [F. No ? i ) ka and L. Grygarová (1991). Some topological questions connected with strictly convex functions. Optimization , 22 , 177-191. Akademie Verlag, Berlin] and [L. Grygarová (1988). Über Lösungsmengen spezieller konvexer parametrischer Optimierungsaufgaben . Optimization 19 , 215-228. Akademie Verlag Berlin], under less strict assumptions. The main results are presented in Sections 3 and 4, in Section 3 the geometrical characterization of the set of optimal points of a certain parametric minimization problem is presented; in Section 4 we study a maximization non-linear parametric problem assigned to it. It seems that it is a certain pair of parametric optimization problems with the same set of their optimal points, so that this pair of problems can be denoted as a pair of dual parametric non-linear optimization problems. This paper presents, most of all in Section 2, a number of interesting geometric facts about strictly convex functions. From the point of view of non-smooth analysis the present article is a certain complement to Chapter 4.3 of the book [B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer (1982). Nonlinear Parametric Optimization . Akademie Verlag, Berlin] where a convex parametric minimization problem is considered under more general and stronger conditions (but without any assumptions concerning strict convexity and without geometrical aspects).     

The stochastic generalised assignment problem with Bernoulli demands

This paper deals with the Stochastic Generalised Assignment problem. It presents several models for the special case when demands are independent and Bernoulli distributed. Each model designs an assignment structure before the demands are known. Two policies are considered to handle infeasibilities in particular instances of the demands vector. Model performances are compared under both policies.     

Solving nonlinear programming problems with very many constraints

For solving the smooth constrained nonlinear programming problem, sequential quadratic programming (SQP) methods are considered to be the standard tool, as long as they are applicable. However one possible situation preventing the successful solution by a standard SQP-technique, arises if problems with a very large number of constraints are to be solved. Typical applications are semi-infinite or min-max optimization, optimal control or mechanical structural optimization. The proposed technique proceeds from a user defined number of linearized constraints, that is to be used internally to determine the size of the quadratic programming subproblem. Significant constraints are then selected automatically by the algorithm. Details of the numerical implementation and some experimental results are presented