Showing papers in "Mathematical Programming in 1982"
TL;DR: Borders on the value of the stochastic solution are presented, that is, the potential benefit from solving the stoChastic program over solving a deterministic program in which expected values have replaced random parameters.
Abstract: Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.
310 citations
TL;DR: This paper studies both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems and several convergence results are obtained for some nonlinear approximation methods.
Abstract: In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.
310 citations
TL;DR: This paper describes polynomial heuristics for three important hard problems—the discrete fixed cost median problem (the plant location problem), the continuous fixed cost Median problem in a Euclidean space, and the network fixedcost median problem with convex costs.
Abstract: We describe in this paper polynomial heuristics for three important hard problems--the discrete fixed cost median problem (the plant location problem), the continuous fixed cost median problem in a Euclidean space, and the network fixed cost median problem with convex costs. The heuristics for all the three problems guarantee error ratios no worse than the logarithm of the number of customer points. The derivation of the heuristics is based on the presentation of all types of median problems discussed as a set covering problem.
289 citations
TL;DR: In this paper, a stochastic method for global optimization is described and evaluated, which involves a combination of sampling, clustering and local search, with a range of confidence intervals on the value of the global optimum.
Abstract: A stochastic method for global optimization is described and evaluated. The method involves a combination of sampling, clustering and local search, and terminates with a range of confidence intervals on the value of the global optimum. Computational results on standard test functions are included as well.
263 citations
TL;DR: Solutions to this family of convex programs are shown to be Nash equilibria in the formal sense of N person games, which leads to a mathematical programming-based algorithm for determining an oligopolistic market equilibrium.
Abstract: During the past several years it has become increasingly common to use mathematical programming methods for deriving economic equilibria of supply and demand. Well-defined approaches exist for the case of a single firm (monopoly) and for the case of many firms (perfect competition). In this paper a certain family of convex programs is formulated to determine equilibria for the case of a few firms (oligopoly). Solutions to this family of convex programs are shown to be Nash equilibria in the formal sense ofN person games. This equivalence leads to a mathematical programming-based algorithm for determining an oligopolistic market equilibrium.
234 citations
TL;DR: A characterization of all minimum cuts, separating a source from a sink in a network, and applications arise in sensitivity and parametric analyses of networks, the vertex packing and maximum closure problems, in unconstrained pseudo-boolean optimization and project selection, as well as in other areas of application of minimum cuts.
Abstract: This paper presents a characterization of all minimum cuts, separating a source from a sink in a network. A binary relation is associated with any maximum flow in this network, and minimum cuts are identified with closures for this relation. As a consequence, finding all minimum cuts reduces to a straightforward enumeration. Applications of this results arise in sensitivity and parametric analyses of networks, the vertex packing and maximum closure problems, in unconstrained pseudo-boolean optimization and project selection, as well as in other areas of application of minimum cuts.
192 citations
TL;DR: A sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases that includes interchanges that avoid the use of any eliminations when revising the factorization at an iteration.
Abstract: We describe a sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases. It includes interchanges that, whenever this is possible, avoid the use of any eliminations (with consequent fill-ins) when revising the factorization at an iteration. Test results on some medium scale problems are presented and comparisons made with the algorithm of Forrest and Tomlin.
154 citations
TL;DR: This work approaches the problem of estimating Hessian matrices by differences from a graph theoretic point of view and shows that both direct and indirect approaches have a natural graph coloring interpretation.
Abstract: Large scale optimization problems often require an approximation to the Hessian matrix. If the Hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem. The problem of estimating Hessian matrices by diferences can be phrased as follows: Given the sparsity structure of a symmetric matrix $A$, obtain vectors $d_{1},d_{2},\ldots,d_{p}$ such that $Ad_{1},Ad_{2},\ldots,Ad_{p}$ determine $A$ uniquely with $p$ as small as possible. We approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation. The complexity of the problem is analyzed and efficient practical heuristic procedures are developed. Numerical results illustrate the differences between the various approaches.
154 citations
TL;DR: This work provides a precise characterization of the closed-form expression for any value function in integer programs in which the objective function and constraint matrix are fixed while the right-hand side varies.
Abstract: We consider integer programs in which the objective function and constraint matrix are fixed while the right-hand side varies. The value function gives, for each feasible right-hand side, the criterion value of the optimal solution. We provide a precise characterization of the closed-form expression for any value function. The class of Gomory functions consists of those functions constructed from linear functions by taking maximums, sums, non-negative multiples, and ceiling (i.e., next highest integer) operations. The class of Gomory functions is identified with the class of all possible value functions by the following results: (1) for any Gomory functiong, there is an integer program which is feasible for all integer vectorsv and hasg as value function; (2) for any integer program, there is a Gomory functiong which is the value function for that program (for all feasible right-hand sides); (3) for any integer program there is a Gomory functionf such thatf(v)≤0 if and only ifv is a feasible right-hand side. Applications of (1)–(3) are also given.
140 citations
TL;DR: The purpose of this paper is to derive, in a unified way, second order necessary and sufficient optimality criteria, for four types of nonsmooth minimization problems: therete minimax problem, therete l1-approximation, the minimization of the exact penalty function and the minimizations of the classical exterior penalty function.
Abstract: The purpose of this paper is to derive, in a unified way, second order necessary and sufficient optimality criteria, for four types of nonsmooth minimization problems: thediscrete minimax problem, thediscrete l1-approximation, the minimization of theexact penalty function and the minimization of theclassical exterior penalty function. Our results correct and supplement conditions obtained by various authors in recent papers.
124 citations
TL;DR: 2-step superlinear convergence is proved to be the final stage of a ‘global’ method to solve the nonlinear programming problem and is compared (theoretically) to the popular successive quadratic programming approach.
Abstract: In this paper we consider the final stage of a ‘global’ method to solve the nonlinear programming problem. We prove 2-step superlinear convergence. In the process of analyzing this asymptotic behavior, we compare our method (theoretically) to the popular successive quadratic programming approach.
TL;DR: The convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function is defined, and the properties of that index are studied.
Abstract: Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX1, ź,Xm.
Assume thatf is quasiconvex and is the sum of nonconstant functionsf1, ź,fm defined on the respective factor sets. Then everyfi is continuous; with at most one exception every functionfi is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function.
We define the convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2, or, equivalently, in terms of the separation of the graphs off1 andf2 by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.
TL;DR: A first order criterion for pseudo-Convexity and second order criteria for quasi-convexness and pseudo-concexity are given for twice differentiable functions on open convex sets.
Abstract: A first order criterion for pseudo-convexity and second order criteria for quasi-convexity and pseudo-convexity are given for twice differentiable functions on open convex sets. The relationships between these second order criteria and other known criteria are also analysed. Finally, the numbers of operations required to verify these criteria are calculated and compared.
TL;DR: A cyclically controlled method of subgradient projections (CSP) for the convex feasibility problem of solving convex inequalities is presented and a particular application to an image reconstruction problem of emission computerized tomography is mentioned.
Abstract: A cyclically controlled method of subgradient projections (CSP) for the convex feasibility problem of solving convex inequalities is presented. The features of this method make it an efficient tool in handling huge and sparse problems. A particular application to an image reconstruction problem of emission computerized tomography is mentioned.
TL;DR: The generalized Benders' decomposition is a suitable approach for solving mixed-integer quadratic programs in which the objective function is quadratics in the integer and in the continuous variables, and the constraints are linear in the variables of both types.
Abstract: This paper considers mixed-integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables, and the constraints are linear in the variables of both types. The generalized Benders' decomposition is a suitable approach for solving such programs. However, the program does not become more tractable if this method is used, since Benders' cuts are quadratic in the integer variables. A new equivalent formulation that renders the program tractable is developed, under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables. Benders' cuts that are derived from the new formulation are linear in the integer variables, and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems. The new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm.
TL;DR: This work relates the simple plant location problem to the vertex packing problem and derives several classes of facets of their associated integer polytopes.
Abstract: We relate the simple plant location problem to the vertex packing problem and derive several classes of facets of their associated integer polytopes.
TL;DR: An algorithm is motivated and described that is based on an exact penalty function and possesses both global and superlinear convergence properties and the superlinear nature is proven here.
Abstract: In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.
TL;DR: A Newton-like method is presented for minimizing a function ofn variables and is a variant of the discrete Newton algorithm that uses only function and gradient values and requires fewer operations than the standard method whenn > 39.
Abstract: A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn > 39, and storage is proportional ton rather thann2.
TL;DR: In this article, the authors discuss several forms of thep-center location problems on an undirected tree network and obtain polynomial algorithms for solving the model based on utilizing results for rigid circuit graphs.
Abstract: We discuss several forms of thep-center location problems on an undirected tree network. Our approach is based on utilizing results for rigid circuit graphs to obtain polynomial algorithms for solving the model. Duality theory on perfect graphs is used to define and solve the dual location model.
TL;DR: A numerical method for the unconstrained minimization of a convex nonsmooth function of several variables and global convergence of the method to the minimum is proved.
Abstract: A numerical method for the unconstrained minimization of a convex nonsmooth function of several variables is presented. It is closely related to the ‘bundle type’ approach and to the conjugate subgradient method. A way is suggested to reduce the amount of information to be stored during the computational procedure. Global convergence of the method to the minimum is proved.
TL;DR: It is demonstrated that the standard implementation of theQR decomposition doesn't yield an orthonormal basis Z(x) whose elements vary continuously withx, and three possible remedies are suggested.
Abstract: A highly regarded method to obtain an orthonormal basis, $Z$, for the null space of a matrix $A^{T}$ is the $QR$ decomposition of $A$, where $Q$ is the product of Householder matrices. In several optimization contexts $A(x)$ varies continuously with $x$ and it is desirable the $Z(x)$ vary continuously also. In this note we demonstrate that the standard implementation of the $QR$ decomposition does not yield an orthonormal basis $Z(x)$ whose elements vary continuously with $x$. We suggest three possible remedies.
TL;DR: ‘inversion’ routines within the simplex method are looked at, particularly those for sparse triangular factorization of a basis by Gaussian elimination and for solution of triangular linear systems.
Abstract: This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or ‘staircase’, structure. The present paper looks at ‘inversion’ routines within the simplex method, particularly those for sparse triangular factorization of a basis by Gaussian elimination and for solution of triangular linear systems. The succeeding paper examines ‘pricing’ routines. Both papers describe extensive (though preliminary) computational experience, and can point to some quite promising results.
TL;DR: Some modifications of the ellipsoid algorithm for linear programming are given and a numerically stable implementation of this algorithm is described, concerned with practical problems where user-supplied bounds can usually be provided.
Abstract: We give some modifications of the ellipsoid algorithm for linear programming and describe a numerically stable implementation. We are concerned with practical problems where user-supplied bounds can usually be provided. Our implementation allows constraint dropping and updates bounds on the optimal value, and should be able to terminate with an indication of infeasibility or with a provably good feasible solution in a moderate number of iterations.
TL;DR: A basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman are established and a new concept ‘primal—dual pair of subdivided manifolds’ is introduced which will serve as a foundation for constructing a wide class of variable Dimension algorithms.
Abstract: In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept ‘primal—dual pair of subdivided manifolds’ and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed.
TL;DR: It is shown that an optimization algorithm may be constructed on the basis of a statistical model and some ideas of the rational choice theory as well as related algorithms and reports on investigations of their efficiency are given.
Abstract: This paper summarizes the results of axiomatic constructing statistical models of complicated multimodal functions. It is shown that an optimization algorithm may be constructed on the basis of a statistical model and some ideas of the rational choice theory. A brief review of related algorithms and reports on investigations of their efficiency is given.
TL;DR: A method is described that circumvents the massive degeneracy inherent in these constraints of the form xj≤ xk and can be implemented using triangular basis factorizations.
Abstract: Special methods for dealing with constraints of the form xj-
TL;DR: The concern is with solving as linear or convex quadratic programs special cases of the optimal containment and meet problems of a set for which some translation contains a set or meets each element in a collection of sets, respectively.
Abstract: The concern is with solving as linear or convex quadratic programs special cases of the optimal containment and meet problems. The optimal containment or meet problem is that of finding the smallest scale of a set for which some translation contains a set or meets each element in a collection of sets, respectively. These sets are unions or intersections of cells where a cell is either a closed polyhedral convex set or a closed solid ball.
TL;DR: It is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube.
Abstract: The definition of random polytope adopted in this paper restricts consideration to those probability measures satisfying two properties. First, the measure must induce an absolutely continuous distribution over the positions of the bounding hyperplanes of the random polytope; and second, it must result in every point in the space being equally as likely as any other point of lying within the random polytope. An efficient Monte Carlo method for their computer generation is presented together with analytical formulas characterizing their aggregate properties. In particular, it is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube. The implied upper bound of 2n wheren is the dimensionality of the space is significantly less than McMullen's attainable bound on the maximal number of vertices even for a moderate number of constraints.
TL;DR: The existence is proved when the set of decisions satisfies some compactness conditions and the efficient set is characterized in terms of the exposed efficient decisions when certain convexity and Compactness conditions are satisfied.
Abstract: Existence and characterization theorems for the efficient (nondominated) set of decisions inR
n
are presented. The existence is proved when the set of decisions satisfies some compactness conditions. The efficient set is characterized in terms of the exposed efficient decisions when certain convexity and compactness conditions are satisfied.
TL;DR: An efficient algorithm for finding an optimum weight-sum matching forest is presented, which is a careful application of known branching and matching algorithms and the maximum cardinality matching forest problem is solved as a special case.
Abstract: We introduce the concept of matching forests as a generalization of branchings in a directed graph and matchings in an undirected graph. Given special weights on the edges of a mixed graph, we present an efficient algorithm for finding an optimum weight-sum matching forest. The algorithm is a careful application of known branching and matching algorithms. The maximum cardinality matching forest problem is solved as a special case.