Showing papers in "Mathematical Programming in 1980"
TL;DR: It is established that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.
Abstract: We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.
146 citations
TL;DR: This paper presents a characterization of the solutions of a singly constrained quadratic program and uses this characterization in the development of a polynomially bounded algorithm for this class of problems.
Abstract: This paper presents a characterization of the solutions of a singly constrained quadratic program. This characterization is then used in the development of a polynomially bounded algorithm for this class of problems.
132 citations
TL;DR: This paper suggests approximations for smoothing out the kinks caused by the presence of “max” or “min” operators in many non-smooth optimization problems, particularly the continuous-discrete min—max optimization problem.
Abstract: In this paper, we suggest approximations for smoothing out the kinks caused by the presence of “max” or “min” operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR
n
. Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.
126 citations
TL;DR: A network investment application is given which includes as a special case a coal transportation problem which is exploited to solve two stage linear programs under uncertainty where the first stage variables are 0–1.
Abstract: Stochastic programs with continuous variables are often solved using a cutting plane method similar to Benders' partitioning algorithm. However, mixed 0–1 integer programs are also solved using a similar procedure along with enumeration. This similarity is exploited in this paper to solve two stage linear programs under uncertainty where the first stage variables are 0–1. Such problems often arise in capital investment. A network investment application is given which includes as a special case a coal transportation problem.
91 citations
TL;DR: By using directions of negative curvature when they exist, it is proved that the sequences of iterates produced by these algorithms converge to stationary points at which the Hessian matrix of the objective function is positive semidefinite.
Abstract: The Armijo and Goldstein step-size rules are modified to allow steps along a curvilinear path of the formx(α) + x + αs + α2d, wherex is the current estimate of the minimum,s is a descent direction andd is a nonascent direction of negative curvature By using directions of negative curvature when they exist, we are able to prove, under fairly mild assumptions, that the sequences of iterates produced by these algorithms converge to stationary points at which the Hessian matrix of the objective function is positive semidefinite
90 citations
TL;DR: A finite algorithm is presented in this study for solving Bilinear programs by developing a suitable cutting plane which deletes at least a face of a polyhedral set.
Abstract: A finite algorithm is presented in this study for solving Bilinear programs. This is accomplished by developing a suitable cutting plane which deletes at least a face of a polyhedral set. At an extreme point, a polar cut using negative edge extensions is used. At other points, disjunctive cuts are adopted. Computational experience on test problems in the literature is provided.
84 citations
TL;DR: Two algorithms for solving the surrogate dual, suggested by analogy with Lagrangean optimisation, are described and proofs of their convergence given.
Abstract: Various theoretical properties of the surrogate dual of a mathematical programming problem are discussed, including some connections with the Lagrangean dual. Two algorithms for solving the surrogate dual, suggested by analogy with Lagrangean optimisation, are described and proofs of their convergence given. A simple example is solved using each method.
83 citations
TL;DR: A new class of facets for knapsack polytopes is obtained and a combinatorial inequality is obtained from Fulkerson's max—max inequality.
Abstract: A new class of facets for knapsack polytopes is obtained. This class of inequalities is shown to define a polytope with zero–one vertices only. A combinatorial inequality is obtained from Fulkerson's max—max inequality.
76 citations
TL;DR: For solving the Euclidean distance Weber problem Weiszfeld proposed an iterative method that can be applied to generalized Weber problems in Banach spaces and Fermat's principle in geometrical optics.
Abstract: For solving the Euclidean distance Weber problem Weiszfeld proposed an iterative method. This method can also be applied to generalized Weber problems in Banach spaces. Examples for generalized Weber problems are: minimal surfaces with obstacles, Fermat's principle in geometrical optics and brachistochrones with obstacles.
72 citations
TL;DR: A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points.
Abstract: A new method is presented for finding a local optimum of the equality constrained nonlinear programming problem. A nonlinear autonomous system is introduced as the base of the theory instead of usual approaches. The relation between critical points and local optima of the original optimization problem is proved. Asymptotic stability of the critical points is also proved. A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points. Some numerical results are given to show the efficiency of the method.
71 citations
TL;DR: It is shown that McCormick's second order sufficient optimality conditions are also necessary for a solution to a quadratic program to be locally unique and hence these conditions completely characterize a locally unique solution of any quadRatic program.
Abstract: It is shown that McCormick's second order sufficient optimality conditions are also necessary for a solution to a quadratic program to be locally unique and hence these conditions completely characterize a locally unique solution of any quadratic program. This result is then used to give characterizations of a locally unique solution to the linear complementarity problem. Sufficient conditions are also given for local uniqueness of solutions of the nonlinear complementarity problem.
TL;DR: This paper describes a method to solve large sparse maximum entropy problems with linear equality constraints using Newtons and the conjugate gradient method.
Abstract: This paper describes a method to solve large sparse maximum entropy problems with linear equality constraints using Newtons and the conjugate gradient method. A numerical example is given to introduce the reader to possible applications of entropy models and this method. Some experience from large scale problems is also reported.
TL;DR: A unified approach to stochastic feasible direction methods is developed and an abstract point-to-set map description of the algorithm is used and a general convergence theorem is proved.
Abstract: A unified approach to stochastic feasible direction methods is developed. An abstract point-to-set map description of the algorithm is used and a general convergence theorem is proved. The theory is used to develop stochastic analogs of classical feasible direction algorithms.
TL;DR: Bounds on the error resulting from the process of aggregation in a class of linear minimum-cost network-flow problems, where groups of nodes in a large problem are replaced by aggregate nodes are developed.
Abstract: It is often necessary or desirable in practice to replace a large, detailed optimization model with a smaller, approximate model. It would be useful to have bounds on the error resulting from this process of aggregation. This paper develops such bounds for a class of linear minimum-cost network-flow problems, where groups of nodes in a large problem are replaced by aggregate nodes. Two types of bounds are derived: A priori bounds are available before solving the aggregated problem; a posteriori bounds, which are generally tighter, may be computed afterwards.
TL;DR: A variable dimension algorithm is presented for the linear complementarity problems that solves a sequence of subproblems of different dimensions, the sequence being possibly nonmonotonic in the dimension of the subproblem solved.
Abstract: A variable dimension algorithm is presented for the linear complementarity problems − Mz = q; s,z ≥ 0; sizi = 0 fori = 1,2, ⋯ ,n. The algorithm solves a sequence of subproblems of different dimensions, the sequence being possibly nonmonotonic in the dimension of the subproblem solved. Every subproblem is the linear complementarity problem defined by a leading principal minor of the matrixM. Index-theoretic arguments characterize the points at which nonmonotonic behavior occurs.
TL;DR: For a nonlinear programming problem with locally Lipschitz objective and inequality constraint functions and continuously differentiable equality constraint functions, a necessary and sufficient condition is presented for the set of multiplier vectors to be nonempty and bounded.
Abstract: For a nonlinear programming problem with locally Lipschitz objective and inequality constraint functions and continuously differentiable equality constraint functions, a necessary and sufficient condition is presented for the set of multiplier vectors to be nonempty and bounded.
TL;DR: First and in particular second order necessary and sufficient conditions for a class of nondifferentiable optimization problems in which there are both objective and constraint functions defined in terms of a norm are given.
Abstract: This paper gives first and in particular second order necessary and sufficient conditions for a class of nondifferentiable optimization problems in which there are both objective and constraint functions defined in terms of a norm. The conditions are expressed in terms of a Lagrangian function and its derivatives, and use the ideas of feasible directional sequence and subgradients. Certain regularity assumptions are required and for the second order necessary conditions it is shown that the assumption is realistic for polyhedral norms. Illustrative examples are discussed.
TL;DR: First- and second-order conditions are given to characterize a local minimizer of an exact penalty function that gives support to the claim that the exactly penalty function and the nonlinear programming problem are closely related.
Abstract: In this paper we give first- and second-order conditions to characterize a local minimizer of an exact penalty function. The form of this characterization gives support to the claim that the exact penalty function and the nonlinear programming problem are closely related. In addition, we demonstrate that there exist arguments for the penalty function from which there are no descent directions even though these points are not minimizers.
IBM1
TL;DR: This paper presents an alternative approach which achieves fast execution times for degenerate simplex method iterations, especially when used with multiple pricing.
Abstract: For general sparse linear programs two of the most efficient implementations of the LU factorization with Bartels—Golub updating are due to Reid and Saunders. This paper presents an alternative approach which achieves fast execution times for degenerate simplex method iterations, especially when used with multiple pricing. The method should have wide applicability since the simplex method performs a high proportion of degenerate iterations on most practical problems. A key feature of Saunders' method is combined with the updating strategy of Reid so as to make the scheme suitable for implementation out of core. Its efficiency is confirmed by experimental results.
TL;DR: It is shown that the Steiner triple systems give rise to a series of problems that are probably hard to solve by implicit enumeration, and that for ann variable problem, branch and bound algorithms using a linear programming relaxation, and/or elimination by dominance require the examination of a super-polynomial number of partial solutions.
Abstract: Fulkerson et al. have given two examples of set covering problems that are empirically difficult to solve. They arise from Steiner triple systems and the larger problem, which has a constraint matrix of size 330 × 45 has only recently been solved. In this note, we show that the Steiner triple systems do indeed give rise to a series of problems that are probably hard to solve by implicit enumeration. The main result is that for ann variable problem, branch and bound algorithms using a linear programming relaxation, and/or elimination by dominance require the examination of a super-polynomial number of partial solutions
TL;DR: The paper presents a sufficient condition for quasiconvexity in terms of Hessian, hereby extending an earlier result by Katzner in 1970 and a slight modification of the assumptions.
Abstract: The paper presents a sufficient condition for quasiconvexity in terms of Hessian, hereby extending an earlier result by Katzner in 1970, and (by a slight modification of the assumptions) a necessary and sufficient condition for quasiconvexity.
TL;DR: The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem.
Abstract: The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem In addition, the “badly behaved set” of constraints, ie the set of constraints which causes problems in the Kuhn—Tucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented
TL;DR: In this paper, a new measure based on the ratio of the length of the sides and the content of the subsimplices of the triangulation is introduced, which is appropriate for computing fixed points using simplicial subdivisions.
Abstract: We consider measures for triangulations ofRn. A new measure is introduced based on the ratio of the length of the sides and the content of the subsimplices of the triangulation. In a subclass of triangulations, which is appropriate for computing fixed points using simplicial subdivisions, the optimal one according to this measure is calculated and some of its properties are given. It is proved that for the average directional density this triangulation is optimal (within the subclass) asn goes to infinity. Furthermore, we compare the measures of the optimal triangulation with those of other triangulations. We also propose a new triangulation of the affine hull of the unit simplex. Finally, we report some computational experience that confirms the theoretical results.
TL;DR: This analysis is based on converting the scaling problem into a linear program and describes the extreme points of the polytope which occurs in the linear program in terms of cyclic products associated with a graph corresponding to the matrix.
Abstract: A scaling of a non-negative, square matrixA ≠ 0 is a matrix of the formDAD−1, whereD is a non-negative, non-singular, diagonal, square matrix. For a non-negative, rectangular matrixB ≠ 0 we define a scaling to be a matrixCBE−1 whereC andE are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scalingX is the maximal ratio of non-zero elements ofX. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.
TL;DR: A strong convergence theorem is proven to hold for the general algorithm of the branch and bound type for solving nonconvex programming problems given in [1].
Abstract: A strong convergence theorem is proven to hold for the general algorithm of the branch and bound type for solving nonconvex programming problems given in [1].
TL;DR: The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent and effective for fixed point problems.
Abstract: The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent. It is efficient and accurate for fixed point problems. L.T. Watson, T.Y. Li, and C.Y. Wang have adapted the method for zero finding problems, the nonlinear complementarity problem, and nonlinear two-point boundary value problems. Here theoretical justification is given for applying the method to some mathematical programming problems, and computational results are presented.
TL;DR: A triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inRn, which allows for factors of incrementation of more than two and can be accelerated without using restart methods.
Abstract: In this paper a triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inR
n
. This triangulation allows for factors of incrementation of more than two. The factor may be of any size and even different at each level. Also the starting point on a new level may be any gridpoint of the last found completely labelled subsimplex on the last level. So, the decision which new factor of incrementation and which starting point is used, can be made on the ground of previous approximations. Doing so, the convergence rate can be accelerated without using restart methods.
TL;DR: Results relating to the expected properties of polytopes indicate that n-polytopes defined by “large” numbers of constraints are difficult to obtain by random processes, and the expected value of the number of vertices of polytope is considerably less than Klee's least upper bound.
Abstract: This paper provides answers to several questions raised by V. Klee regarding the efficacy of Mattheiss' algorithm for finding all vertices of convex polytopes. Several results relating to the expected properties of polytopes are given which indicate thatn-polytopes defined by “large” numbers of constraints are difficult to obtain by random processes, the expected value of the number of vertices of polytope is considerably less than Klee's least upper bound the expected performance of Mattheiss' algorithm is far better than Klee's upper bound would suggest.
TL;DR: This study combines least-index pivot selection rules with Keller's algorithm for quadratic programming to obtain a finite method for processing degenerate problems.
Abstract: In this study, we combine least-index pivot selection rules with Keller's algorithm for quadratic programming to obtain a finite method for processing degenerate problems.