Showing papers in "Mathematical Programming in 1984"
TL;DR: It is shown that the ascent procedure can be viewed as a generalization of both the Chu-Liu-Edmonds directed spanning tree algorithm and the Bilde-Krarup-Erlenkotter ascent algorithm for the plant location problem.
Abstract: The Steiner tree problem on a directed graph (STDG) is to find a directed subtree that connects a root node to every node in a designated node setV. We give a dual ascent procedure for obtaining lower bounds to the optimal solution value. The ascent information is also used in a heuristic procedure for obtaining feasible solutions to the STDG. Computational results indicate that the two procedures are very effective in solving a class of STDG's containing up to 60 nodes and 240 directed/120 undirected arcs. The directed spanning tree and uncapacitated plant location problems are special cases of the STDG. Using these relationships, we show that our ascent procedure can be viewed as a generalization ofboth the Chu-Liu-Edmonds directed spanning tree algorithm and the Bilde-Krarup-Erlenkotter ascent algorithm for the plant location problem. The former comparison yields a dual ascent interpretation of the steps of the directed spanning tree algorithm.
369 citations
TL;DR: The main result is that the four gaps associated with the four relaxations are equal, and a class of gap-free functions (properly including the supermodular ones) is exhibited.
Abstract: The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx
i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.
344 citations
TL;DR: By associating systematically to this problem an equivalent one posed in ann-fold cartesian product space, by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former are obtained.
Abstract: When an optimization problem is posed in a product space it is classical to decompose this problem. The goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space. By associating systematically to this problem an equivalent one posed in ann-fold cartesian product space, we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former. Applications to three rather classical mathematical programming problems are given.
273 citations
TL;DR: This work considers how large these approximations have to be, if they prevent convergence when the objective function is bounded below and continuously differentiable, and obtains a useful convergence result in the case when there is a bound on the second derivative approximation that depends linearly on the iteration number.
Abstract: Many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large [2]. We consider how large these approximations have to be, if they prevent convergence when the objective function is bounded below and continuously differentiable. Thus we obtain a useful convergence result in the case when there is a bound on the second derivative approximations that depends linearly on the iteration number.
171 citations
TL;DR: This paper presents an alternative approach to that by Murphy, Sherali and Soyster for computing market equilibria with mathematical programming methods based upon a variational inequality representation of the problem and the use of a diagonalization/relaxation algorithm.
Abstract: This paper presents an alternative approach to that by Murphy, Sherali and Soyster [13] for computing market equilibria with mathematical programming methods. This approach is based upon a variational inequality representation of the problem and the use of a diagonalization/relaxation algorithm.
171 citations
TL;DR: An efficient procedure for generating fair cost allocations which, in some instances, coincide with the nucleolus is developed and it is shown that by employing Sterns' transfer scheme the authors can generate a sequence of cost vectors which converges to theucleolus.
Abstract: We develop two efficient procedures for generating cost allocation vectors in the core of a minimum cost spanning tree (m.c.s.t.) game. The first procedure requires O(n
2) elementary operations to obtain each additional point in the core, wheren is the number of users. The efficiency of the second procedure, which is a natural strengthening of the first procedure, stems from the special structure of minimum excess coalitions in the core of an m.c.s.t. game. This special structure is later used (i) to ease the computational difficulty in computing the nucleolus of an m.c.s.t. game, and (ii) to provide a geometric characterization for the nucleolus of an m.c.s.t. game. This geometric characterization implies that in an m.c.s.t. game the nucleolus is the unique point in the intersection of the core and the kernel. We further develop an efficient procedure for generating fair cost allocations which, in some instances, coincide with the nucleolus. Finally, we show that by employing Sterns' transfer scheme we can generate a sequence of cost vectors which converges to the nucleolus.
155 citations
TL;DR: Some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem, and the results will be applied to a certain class of approximation problems.
Abstract: In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.
137 citations
TL;DR: The delicate question of predicting the direction of the change in the traffic pattern and the incurred travel costs resulting from changes in the travel cost functions and travel demands is focused on and certain counter intuitive phenomena such as ‘Braess' paradox’ are elucidated.
Abstract: We consider the asymmetric continuous traffic equilibrium network model with fixed demands where the travel cost on each link of the transportation network may depend on the flow on this as well as other links of the network and we perform stability and sensitivity analysis. Assuming that the travel cost functions are monotone we first show that the traffic equlibrium pattern depends continuously upon the assigned travel demands and travel cost functions. We then focus on the delicate question of predicting the direction of the change in the traffic pattern and the incurred travel costs resulting from changes in the travel cost functions and travel demands and attempt to elucidate certain counter intuitive phenomena such as ‘Braess' paradox’. Our analysis depends crucially on the fact that the governing equilibrium conditions can be formulated as a variational inequality.
135 citations
TL;DR: For any multilinear inequality in 0–1 variables, an equivalent family of linear inequalities is defined, which contains the well-known system of generalized covering inequalities, as well as other linear equivalents of the multil inear inequality that are more compact, i.e., of smaller cardinality.
Abstract: Any real-valued nonlinear function in 0---1 variables can be rewritten as a multilinear function. We discuss classes of lower and upper bounding linear expressions for multilinear functions in 0---1 variables. For any multilinear inequality in 0---1 variables, we define an equivalent family of linear inequalities. This family contains the well-known system of generalized covering inequalities, as well as other linear equivalents of the multilinear inequality that are more compact, i.e., of smaller cardinality. In a companion paper [7]. we discuss dominance relations between various linear equivalents of a multilinear inequality, and describe a class of algorithms for multilinear 0---1 programming based on these results.
110 citations
TL;DR: An algorithm for solving the linear program associated with the multiple choice knapsack problem is described, and is shown to work in time linear in the number of variables.
Abstract: An algorithm for solving the linear program associated with the multiple choice knapsack problem is described The algorithm is shown to work in time linear in the number of variables This improves the previously known best bound for this problem, and is optimal to within a constant factor
93 citations
TL;DR: Applications are made to a variety of economic models, including the transferable utility trading economies of Shapley and Shubik and a multishipper one-commodity transshipment model with convex cost functions and concave revenue functions.
Abstract: A cooperative game in characteristic-function form is obtained by allowing a number of individuals to esercise partial control over the constraints of a (generally nonlinear) mathematical programming problem, either directly or through committee voting. Conditions are imposed on the functions defining the programming problem and the control system which suffice to make the game totally balanced. This assures a nonempty core and hence a stable allocation of the full value of the programming problem among the controlling palyers. In the linear case the core is closely related to the solutions of the dual problem. Applications are made to a variety of economic models, including the transferable utility trading economies of Shapley and Shubik and a multishipper one-commodity transshipment model with convex cost functions and concave revenue functions. Dropping the assumption of transferable utility leads to a class of controlled multiobjective or ‘Pareto programming’ problems, which again yield totally balanced games.
TL;DR: This paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable, and a convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below.
Abstract: This paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable. The algorithm can be implemented and may be considered a modification of the e-subgradient algorithm and Lemarechal's descent algorithm. Also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems. A convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below. Limited computational experience with the algorithm is also reported.
TL;DR: For a nonlinear programming problem involving pseudolinear functions only, it is proved that every efficient solution is properly efficient under some mild conditions.
Abstract: First order and second order characterizations of pseudolinear functions are derived. For a nonlinear programming problem involving pseudolinear functions only, it is proved that every efficient solution is properly efficient under some mild conditions.
TL;DR: A dual in the form of a Max-Min problem is introduced and duality relations are established for this Min-Max problem, where the maximum of finitely many linear ratios is to be minimized subject to linear constraints.
Abstract: We consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints For this Min-Max problem, a dual in the form of a Max-Min problem is introduced and duality relations are established
TL;DR: A general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem and results have been established for the positive definite Hessian case and the positive semi-definite case.
Abstract: We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case.
TL;DR: This paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages.
Abstract: In contrast to traditional sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent variations in a number of parameters. A primary focus of this approach is to determine a maximum tolerance percentage for selected right-hand-side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value. Similarly, the approach yields a maximum tolerance percentage for selected objective function coefficients. This paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages.
TL;DR: The theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz are extended and a tight lower bound on the parameter value is provided.
Abstract: In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.
TL;DR: In this paper, a family of linear inequalities that contains more compact linearizations of a multilinear 0-1 program than the one based on generalized covering inequalities was defined.
Abstract: A nonlinear 0–1 program can be restated as a multilinear 0–1 program, which in turn is known to be equivalent to a linear 0–1 program with generalized covering (g.c.) inequalities. In a companion paper [6] we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0–1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time to an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12–15, and the difference in their performance grows with the number of such terms.
TL;DR: It is shown that for any submodular functionf on a crossing family with, if the polyhedron is nonempty, then there exist a unique distributive lattice with and a unique sub modular function with such that B(f) coincides with the base polyhedronic system associated with the subModular system.
Abstract: The present paper shows that for any submodular functionf on a crossing family
with
, if the polyhedron
is nonempty, then there exist a unique distributive lattice
with
and a unique submodular function
with
such thatB(f) coincides with the base polyhedron
associated with the submodular system
. Here, iff is integer-valued, thenf
1 is also integer-valued. Based on this fact, we also show the relationship between the independent-flow problem considered by the author and the minimum cost flow problem considered by J. Edmonds and R. Giles.
TL;DR: A convex (or concave) conjugate function of a submodular (or supermodular) function is defined and a Fenchel-type min-max theorem for sub modular and super modular functions is shown.
Abstract: We consider submodular programs which are problems of minimizing submodular functions on distributive lattices with or without constraints. We define a convex (or concave) conjugate function of a submodular (or supermodular) function and show a Fenchel-type min-max theorem for submodular and supermodular functions. We also define a subgradient of a submodular function and derive a necessary and sufficient condition for a feasible solution of a submodular program to be optimal, which is a counterpart of the Karush-Kuhn-Tucker condition for convex programs.
TL;DR: Stability of the optimal solution of stochastic programs with recourse with recourse under assumption of strict complementarity known from the theory of nonlinear programming is studied.
Abstract: In this paper, stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied Provided that the set of admissible solutions is defined by equality constraints only, asymptotical normality of the optimal solution follows by standard methods If nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming (Theorem 1) The general results are applied to the simple recourse problem with random right-hand sides under various assumptions on the underlying distribution (Theorems 2–4)
TL;DR: It is shown that given anyw's there is an integral approximation ofx's so that the errorsEi are small-roughly that $$E_i = O(\sqrt i In i )$$ simultaneously for alli.
Abstract: Letn linear formsL
i
onm variables be given, normalized so that all coefficients have absolute value at most unity. Letw
1, ...,w
m
be real numbers andx
1, ...,x
m
be integers. We sayE
i
=L
i
(w
1, ...,w
m
)-L
i
(x
1, ...,x
m
) is the error in approximating thew's by thex's with respect to formL
i
It is shown that given anyw's there is an integral approximation ofx's so that the errorsE
i
are small-roughly that
$$E_i = O(\sqrt i In i )$$
simultaneously for alli.
TL;DR: An O(n4K) algorithm for the max-cut problem for graphs ing(K), which is recursive and is based on some properties of longest and longest odd cycles of graphs.
Abstract: Given a graphG=[V, E] with positive edge weights, the max-cut problem is to find a cut inG such that the sum of the weights of the edges of this cut is as large as possible. Letg(K) be the class of graphs whose longest odd cycle is not longer than2K+1, whereK is a nonnegative integer independent of the numbern of nodes ofG. We present an O(n
4K) algorithm for the max-cut problem for graphs ing(K). The algorithm is recursive and is based on some properties of longest and longest odd cycles of graphs.
TL;DR: In the sparse case, when eachNi is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDLT factorization without fill-in.
Abstract: The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsfi whose Hessians have nontrivial nullspacesNi, Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionfi is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalfi such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureNi1 have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachNi is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDLT factorization without fill-in.
TL;DR: Acceptability criteria for the linesearch stepsize are introduced which require only function values and some modifications of criteria based on the knowledge of the directional derivative are illustrated.
Abstract: In this paper acceptability criteria for the linesearch stepsize are introduced which require only function values. Simple algorithm models based on these criteria are presented. Some modifications of criteria based on the knowledge of the directional derivative are also illustrated.
TL;DR: Tarski's fixed point theorem is extended to the case of set-valued mappings, and is applied to a class of complementarity problems defined by isotone set- valued operators in a complete vector lattice.
Abstract: Tarski's fixed point theorem is extended to the case of set-valued mappings, and is applied to a class of complementarity problems defined by isotone set-valued operators in a complete vector lattice.
TL;DR: A polynomial approximation scheme is presented for the Subset-Sum Problem and it is proved that its worst-case performance dominates that of Johnson's well-known scheme.
Abstract: Given a set ofn positive integers and another positive integerW, the Subset-Sum Problem is to find that subset whose sum is closest to, without exceeding,W. We present a polynomial approximation scheme for this problem and prove that its worst-case performance dominates that of Johnson's well-known scheme.
TL;DR: This paper describes a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set that is applicable to both primal and dual quadratic programming algorithms that use an active-set strategy.
Abstract: Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy.
TL;DR: An iterative, Newton-type method for solving a class of linear complementarity problems, discovered by Mangasarian, which is shown to converge to the least element solutions.
Abstract: This paper presents an iterative, Newton-type method for solving a class of linear complementarity problems. This class was discovered by Mangasarian who had established that these problems can be solved as linear programs. Cottle and Pang characterized solutions of the problems in terms of least elements of certain polyhedral sets. The algorithms developed in this paper are shown to converge to the least element solutions. Some applications and computational results are also discussed.
TL;DR: The relaxation method for linear inequalities is studied and new bounds on convergence obtained and an asymptotically tight estimate is given for the case when the inequalities are processed in a cyclical order.
Abstract: The relaxation method for linear inequalities is studied and new bounds on convergence obtained. An asymptotically tight estimate is given for the case when the inequalities are processed in a cyclical order. An improvement of the estimate by an order of magnitude takes place if strong underrelaxation is used. Bounds on convergence usually involve the so-called condition number of a system of linear inequalities, which we estimate in terms of their coefficient matrix.