Showing papers in "Mathematical Programming in 1973"
TL;DR: The solution of the Chinese postman problem using matching theory is given and the convex hull of integer solutions is described as a linear programming polyhedron, used to show that a good algorithm gives an optimum solution.
Abstract: The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more generalb-matching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.
963 citations
TL;DR: This paper shows that the cliques of the intersection graph provide a first set of facets for the polyhedron in question, and it is shown that the cycles without chords of odd length of the intersections graph give rise to a further set of facet.
Abstract: In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form "ź". This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.
616 citations
TL;DR: This paper has shown that any maximizing sequence for the dual can be made to yield, in a general way, an asymptotically minimizingsequence for the primal which typically converges at least as rapidly.
Abstract: Several recent algorithms for solving nonlinear programming problems with equality constraints have made use of an augmented “penalty” Lagrangian function, where terms involving squares of the constraint functions are added to the ordinary Lagrangian. In this paper, the corresponding penalty Lagrangian for problems with inequality constraints is described, and its relationship with the theory of duality is examined. In the convex case, the modified dual problem consists of maximizing a differentiable concave function (indirectly defined) subject to no constraints at all. It is shown that any maximizing sequence for the dual can be made to yield, in a general way, an asymptotically minimizing sequence for the primal which typically converges at least as rapidly.
387 citations
TL;DR: The relevance of differentiable functions of three variables, having the property that if they are treated by the minimization algorithm that searches along the coordinate directions in sequence, then the search path tends to a closed loop, is discussed.
Abstract: Some examples are given of differentiable functions of three variables, having the property that if they are treated by the minimization algorithm that searches along the coordinate directions in sequence, then the search path tends to a closed loop. On this loop the gradient of the objective function is bounded away from zero. We discuss the relevance of these examples to the problem of proving general convergence theorems for minimization algorithms that use search directions.
344 citations
TL;DR: This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm for the General Fermat Problem.
Abstract: The General Fermat Problem asks for the minimum of the weighted sum of distances fromm points inn-space. Dozens of papers have been written on variants of this problem and most of them have merely reproduced known results. This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm. Although the same algorithm has been rediscovered at least three times, there seems to be no completely correct treatment of its properties in the literature. Such a treatment, including a proof of convergence, is the sole object of this note. Other aspects of the problem are given scant attention.
331 citations
TL;DR: Five options within this revised simplex algorithm for the enumeration of the set of efficient extreme points were tested on a variety of problems, providing indications for effective use of the algorithm.
Abstract: For linear multiple-objective problems, a necessary and sufficient condition for a point to be efficient is employed in the development of a revised simplex algorithm for the enumeration of the set of efficient extreme points. Five options within this algorithm were tested on a variety of problems. Results of these tests provide indications for effective use of the algorithm.
244 citations
BP1
TL;DR: After a fresh look at the iteration process, the author introduces dynamic column weighting factors as a means of estimating gradients for the purpose of selecting a maximum gradient column.
Abstract: Pivot column and row selection methods used by the Devex code since 1965 are published here for the first time. After a fresh look at the iteration process, the author introduces dynamic column weighting factors as a means of estimating gradients for the purpose of selecting a maximum gradient column. The consequent effect of this column selection on rounding error is observed. By allowing that a constraint may not be positioned so exactly as its precise representation in the computer would imply, a wider choice of pivot row is made available, so making room for a further selection criterion based on pivot size. Three examples are given of problems having between 2500 and 5000 rows, illustrating the overall time and iteration advantages over the standard simplex methods used today. The final illustration highlights why these standard methods take so many iterations. These algorithms were originally coded for the Atlas computer and were re-coded in 1969 for the Univac 1108.
228 citations
TL;DR: Sufficient conditions are given for the existence of exact penalty functions for inequality constrained problems more general than concave and several classes of such functions are presented.
Abstract: In this paper some new theoretic results on piecewise differentiable exact penalty functions are presented. Sufficient conditions are given for the existence of exact penalty functions for inequality constrained problems more general than concave and several classes of such functions are presented.
202 citations
TL;DR: The theory presented in this paper is based to a large extent on recent results of the author concerning logarithmic concave measures on two stochastic programming decision models, where the solvability of the second stage problem only with a prescribed (high) probability is required.
Abstract: Two stochastic programming decision models are presented. In the first one, we use probabilistic constraints, and constraints involving conditional expectations further incorporate penalties into the objective. The probabilistic constraint prescribes a lower bound for the probability of simultaneous occurrence of events, the number of which can be infinite in which case stochastic processes are involved. The second one is a variant of the model: two-stage programming under uncertainty, where we require the solvability of the second stage problem only with a prescribed (high) probability. The theory presented in this paper is based to a large extent on recent results of the author concerning logarithmic concave measures.
182 citations
TL;DR: A direct application of the linear programming duality theorem leads to a new necessary condition for the existence of hamiltonian circuits; this condition appears to be stronger than the ones previously known.
Abstract: Jack Edmonds developed a new way of looking at extremal combinatorial problems and applied his technique with a great success to the problems of the maximal-weight degreeconstrained subgraphs. Professor C. St. J.A. Nash-Williams suggested to use Edmonds' approach in the context of hamiltonian graphs. In the present paper, we determine a new set of inequalities (the “comb inequalities”) which are satisfied by the characteristic functions of hamiltonian circuits but are not explicit in the straightforward integer programming formulation. A direct application of the linear programming duality theorem then leads to a new necessary condition for the existence of hamiltonian circuits; this condition appears to be stronger than the ones previously known. Relating linear programming to hamiltonian circuits, the present paper can also be seen as a continuation of the work of Dantzig, Fulkerson and Johnson on the traveling salesman problem.
157 citations
IBM1
TL;DR: The Edmonds—Karp Scaling Method becomes the only known “good” (in the sense of Edmonds) algorithm for computing minimum cost flows.
Abstract: For any integern, a modified transportation problem with 2n + 2 nodes is constructed which requires 2
n
+ 2
n−2−2 iterations using all but one of the most commonly used minimum cost flow algorithms. As a result, the Edmonds—Karp Scaling Method [3] becomes the only known “good” (in the sense of Edmonds) algorithm for computing minimum cost flows.
TL;DR: This work considers a two person max—min problem in which the maximizing player moves first and the minimizing player has perfect information of the outcome of this move.
Abstract: We consider a two person max—min problem in which the maximizing player moves first and the minimizing player has perfect information of the outcome of this move. The move of the maximizing player influences not only the objective function but also the constraints of the minimizing player. The joint constraints as well as the objective function are assumed to be linear.
TL;DR: It is shown how, given a nonlinear programming problem with inequality constraints, it is possible to construct an exact penalty function with a local unconstrained minimum at any local minimum of the constrained problem.
Abstract: It is shown how, given a nonlinear programming problem with inequality constraints, it is possible to construct an exact penalty function with a local unconstrained minimum at any local minimum of the constrained problem. The unconstrained minimum is sufficiently smooth to permit conventional optimization techniques to be used to locate it. Numerical evidence is presented on five well-known test problems.
TL;DR: It is shown that in general the solution of the problem of minimizingQ(x) = 1/2xTKx − kTx subject toGx ≦ g andDx = d behaves whenK, k, G, g, D andd are perturbed, assuming thatK is positive definite.
Abstract: This paper studies how the solution of the problem of minimizingQ(x) = 1/2x
T
Kx − k
T
x subject toGx ≦ g andDx = d behaves whenK, k, G, g, D andd are perturbed, say by terms of size∈, assuming thatK is positive definite. It is shown that in general the solution moves by roughly∈ ifG, g, D andd are not perturbed; whenG, g, D andd are in fact perturbed, much stronger hypotheses allow one to show that the solution moves by roughly∈. Many of these results can be extended to more general, nonquadratic, functionals.
TL;DR: The flexibility of this technique is examined through experiments with different branching and subproblem selection strategies, and the efficacy of these various heuristics is assessed.
Abstract: The branch and bound method of solving the mixed integer linear programming problem is summarized. The flexibility of this technique is examined through experiments with different branching and subproblem selection strategies, and the efficacy of these various heuristics is assessed.
TL;DR: Two criteria for fitting a linear function to a set of points are considered, viz., least sum of absolute deviations and the least maximum absolute deviation, both of which give rise to a linear program.
Abstract: The problem considered here is that of fitting a linear function to a set of points. The criterion normally used for this is least squares. We consider two alternatives, viz., least sum of absolute deviations (called the L1 criterion) and the least maximum absolute deviation (called the Chebyshev criterion). Each of these criteria give rise to a linear program. We develop some theoretical properties of the solutions and in the light of these, examine the suitability of these criteria for linear estimation. Some of the estimates obtained by using them are shown to be counter-intuitive.
TL;DR: A class of matrices is introduced such that for anyM in this class a solution to the linear complementarity problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility.
Abstract: The linear complementarity problem is the problem of finding solutionsw, z tow = q + Mz, w≥0,z≥0, andw
T
z=0, whereq is ann-dimensional constant column, andM is a given square matrix of dimensionn. In this paper, the author introduces a class of matrices such that for anyM in this class a solution to the above problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility. This class is a refinement of that introduced and characterized by Eaves. It is also shown that for someM in this class, there is an even number of solutions for all nondegenerateq, and that matrices for general quadratic programs and matrices for polymatrix games nicely relate to these matrices.
TL;DR: A saddle point theory in terms of extended Lagrangian functions is presented for nonconvex programs and the results parallel those for convex programs conjoined with the usuallagrangian formulation.
Abstract: A saddle point theory in terms of extended Lagrangian functions is presented for nonconvex programs. The results parallel those for convex programs conjoined with the usual Lagrangian formulation.
TL;DR: This result is generalised to show that ‘greedy-type’ algorithms select an optimum forall linear functions c·x; x in some compact set U if and only if the convex closure ofU is essentially a polymatroid.
Abstract: If is a collection of subsets ofS andw is a nonnegative weight function onS, the problem of selecting a subset belonging to
which has maximum weight is solved by a ‘greedy-type’ algorithm forall w if and only if
is the set of independent sets of a matroid. This result is then generalised to show that ‘greedy-type’ algorithms select an optimum forall linear functionsc·x; x in some compact set
$$U \subseteq R^n $$
andc > 0 if and only if the convex closure ofU is essentially a polymatroid. A byproduct of this is a new characterization of polymatroids.
TL;DR: It is shown that any generalized network problem whose matrix does not have full row rank can be transformed into an equivalent pure network problem and a constructive method for doing this is given.
Abstract: The purpose of this paper is to show that any generalized network problem whose matrix does not have full row rank can be transformed into an equivalent pure network problem and to give a constructive method for doing this.
TL;DR: This article deals mainly with a comparison of certain computational techniques used for the solution of non-linear constrained mathematical programming problems.
Abstract: This article deals mainly with a comparison of certain computational techniques used for the solution of non-linear constrained mathematical programming problems.
TL;DR: A principal pivoting algorithm is given for finding local minimizing points for general quadratic minimization problems and it is shown that the method is a generalization of algorithms of Dantzig, and Van de Panne and Whinston for convex quadratically minimized problems.
Abstract: A principal pivoting algorithm is given for finding local minimizing points for general quadratic minimization problems. The method is a generalization of algorithms of Dantzig, and Van de Panne and Whinston for convex quadratic minimization problems.
TL;DR: The innovative subroutines are shown to be efficient to compute and effective in finding good integer programming solutions and providing strong lower bounds for the branch and bound search.
Abstract: This paper gives specific computational details and experience with a group theoretic integer programming algorithm. Included among the subroutines are a matrix reduction scheme for obtaining group representations, network algorithms for solving group optimization problems, and a branch and bound search for finding optimal integer programming solutions. The innovative subroutines are shown to be efficient to compute and effective in finding good integer programming solutions and providing strong lower bounds for the branch and bound search.
TL;DR: Several algorithms are presented for solving the non-linear programming problem, based on “variable-metric” projections of the gradient of the objective function into a local approximation to the constraints.
Abstract: Several algorithms are presented for solving the non-linear programming problem, based on “variable-metric” projections of the gradient of the objective function into a local approximation to the constraints. The algorithms differ in the nature of this approximation. Inequality constraints are dealt with by selecting at each step a subset of “active” constraints to treat as equalities, this subset being the smallest necessary to ensure that the new point remains feasible. Some numerical results are given for the Colville problems.
TL;DR: Under the condition thatf is continuously differentiable, it is proved that for a generic set of such anf, the problem has a discrete solution set.
Abstract: Givenf: R+n→ Rn, the complementarity problem is to find a solution tox ≥ 0,f(x) ≥ 0, and 〈x, f(x)〉 = 0. Under the condition thatf is continuously differentiable, we prove that for a generic set of such anf, the problem has a discrete solution set. Also, under a set of generic nondegeneracy conditions and a condition that implies existence, we prove that the problem has an odd number of solutions.
TL;DR: This paper shows that the linear programming formulation of the two-commodity network flow problem leads to a direct derivation of the known results concerning this problem.
Abstract: This paper shows that the linear programming formulation of the two-commodity network flow problem leads to a direct derivation of the known results concerning this problem. An algorithm for solving the problem is given which essentially consists of two applications of the Ford—Fulkerson max flow computation. Moreover, the algorithm provides constructive proofs for the results. Some new facts concerning feasible integer flows are also given.
TL;DR: Using the insight provided by the general theory, Geoffrion's Generalized Benders Decomposition procedure (GBD) is modified to permit the deletion of old constraints without upsetting the infinite convergence property.
Abstract: Eaves and Zangwill [2] have developed a very general theory of the convergence of cutting plane algorithms. This theory is applied to prove the convergence of Geoffrion's Generalized Benders Decomposition procedure (GBD) [5]. Using the insight provided by the general theory, GBD is then modified to permit the deletion of old constraints without upsetting the infinite convergence property. Finally, certain approximations of GBD are presented and the robustness of the convergence results is indicated.
TL;DR: The present approach uses the support planes of the constraint region to transform the standard convex program into an equivalent linear program, and the duality theory of infinite linear programming shows how to construct a new dual program of bilinear type.
Abstract: The theme of this paper is the application of linear analysis to simplify and extend convex analysis. The central problem treated is the standard convex program — minimize a convex function subject to inequality constraints on other convex functions. The present approach uses the support planes of the constraint region to transform the convex program into an equivalent linear program. Then the duality theory of infinite linear programming shows how to construct a new dual program of bilinear type. When this dual program is transformed back into the convex function formulation it concerns the minimax of an unconstrained Lagrange function. This result is somewhat similar to the Kuhn—Tucker theorem. However, no constraint qualifications are needed and yet perfect duality maintains between the primal and dual programs.
TL;DR: Two examples are presented, whereby O(n3) flow augmentations using tie-breaking variants of either the Edmonds—Karp shortest path or fewest reverse arcs in path maximum flow algorithms are required and it is not possible to substantially improve the performance of either algorithm by resolving ties.
Abstract: Two examples are presented. The first example is “bad” for a large subset of the “primal” minimum cost flow algorithms, namely those algorithms which start with the required amount ofs − t flow distributed in a feasible, but nonoptimal manner, and which get optimal by sending flow about negative cycles. In particular, the example is bad for the “primal” method which always sends flow about a cycle which yields the largest decrease in the objective function. The second example requires O(n
3) flow augmentations using tie-breaking variants of either the Edmonds—Karp shortest path or fewest reverse arcs in path maximum flow algorithms. This example implies that it is not possible to substantially improve the performance (in a worst case sense) of either algorithm by resolving ties.
TL;DR: In this article, a general integer programming algorithm consisting of problem relaxation, solution of the relaxed problem parametrically by dynamic programming, and generation of kth best solutions until a feasible solution is found is presented.
Abstract: When regarded as a shortest route problem, an integer program can be seen to have a particularly simple structure. This allows the development of an algorithm for finding thekth best solution to an integer programming problem with max{O(kmn), O(k logk)} operations. Apart from its value in the parametric study of an optimal solution, the approach leads to a general integer programming algorithm consisting of (1) problem relaxation, (2) solution of the relaxed problem parametrically by dynamic programming, and (3) generation ofkth best solutions until a feasible solution is found. Elementary methods based on duality for reducingk for a given problem relaxation are then outlined, and some examples and computational aspects are discussed.