Showing papers in "Mathematical Programming in 1994"
TL;DR: Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
Abstract: We report on improved practical algorithms for lattice basis reduction. We present a variant of the L3-algorithm with “deep insertions” and a practical algorithm for blockwise Korkine-Zolotarev reduction, a concept extending L3-reduction, that has been introduced by Schnorr (1987). Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC 2 computer.
1,390 citations
TL;DR: This paper establishes that the interior-reflective Newton approach is globally and quadratically convergent, and develops a specific example of interior- reflective Newton methods which can be used for large-scale and sparse problems.
Abstract: We consider a new algorithm, an interior-reflective Newton approach, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths (reflection paths). The interior-reflective approach does not require identification of an "activity set". In this paper we establish that the interior-reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of interior-reflective Newton methods which can be used for large-scale and sparse problems.
1,101 citations
TL;DR: This work derives compact representations of BFGS and symmetric rank-one matrices for optimization and presents a compact representation of the matrices generated by Broyden's update for solving systems of nonlinear equations.
Abstract: We derive compact representations of BFGS and symmetric rank-one matrices for optimization. These representations allow us to efficiently implement limited memory methods for large constrained optimization problems. In particular, we discuss how to compute projections of limited memory matrices onto subspaces. We also present a compact representation of the matrices generated by Broyden's update for solving systems of nonlinear equations.
657 citations
TL;DR: An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems.
Abstract: A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems. Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I. The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving rise to a new quadratic outer approximation algorithm. An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems. It is possible to develop the theory in an elegant way for a large class of nonsmooth MINLPs based on the use of convex composite functions and subdifferentials, although an interpretation for thel
1 norm is also given.
643 citations
TL;DR: This paper presents a decomposition method for solving convex minimization problems that preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation.
Abstract: This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.
420 citations
225 citations
TL;DR: A general principle is derived demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated.
Abstract: We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.
177 citations
TL;DR: A new iterative scheme for the numerical solution of a class of linear variational inequalities and the convergence proof are introduced.
Abstract: In this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities. Each iteration of the method consists essentially only of a projection to a closed convex set and two matrix-vector multiplications. Both the method and the convergence proof are very simple.
159 citations
TL;DR: Polyhedra with O(2n) facets having an O(n2 logn) separation algorithm for CLS and O( n3) for BLS are exhibited, as well as extended formulations with O (n2) constraints in both cases, O(N2) variables for CLS
Abstract: We examine the single-item lot-sizing problem with Wagner-Whitin costs over an n period horizon, i.e. p(t) + h(t) greater than or equal to p(t+1) for t = 1, ..., n-1, where p(t), h(t) are the unit production and storage costs in period t respectively, so it always pays to produce as late as possible. We describe integral polyhedra whose solution as linear programs solve the uncapacitated problem (ULS), the uncapacitated problem with backlogging (BLS), the uncapacitated problem with startup costs (ULSS) and the constant capacity problem (CLS), respectively. The polyhedra, extended formulations and separation algorithms are much simpler than in the general cost case. In particular for models ULS and ULSS the polyhedra in the original space have only O(n(2)) facets as opposed to O(2(n)) in the general case. For CLS and BLS no explicit polyhedral descriptions are known for the general case in the original space. Here we exhibit polyhedra with O(2(n)) facets having an O(n(2) log n) separation algorithm for CLS and O(n(3)) for BLS, as well as extended formulations with O(n(2)) constraints in both cases, O(n(2)) variables for CLS and O(n) variables for BLS.
146 citations
TL;DR: The primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution, which converges exponentially fast to a particular dual optimal solution.
Abstract: We consider the linear program min{cźx: Axźb} and the associated exponential penalty functionfr(x) = cźx + rΣexp[(Aix ź bi)/r]. Forr close to 0, the unconstrained minimizerx(r) offr admits an asymptotic expansion of the formx(r) = x* + rd* + ź(r) wherex* is a particular optimal solution of the linear program and the error termź(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectoryź(r) which converges exponentially fast to a particular dual optimal solution. These results are completed by an asymptotic analysis whenr tends to ź: the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution.
140 citations
TL;DR: This paper gives some integer programming formulations for the Steiner tree problem on undirected and directed graphs and study the associated polyhedra and gives some families of facets for the undirecting case along with some compositions and extensions.
Abstract: In this paper we give some integer programming formulations for the Steiner tree problem on undirected and directed graphs and study the associated polyhedra. We give some families of facets for the undirected case along with some compositions and extensions. We also give a projection that relates the Steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph. This is used to show that the LP-relaxation of the directed formulation is superior to the LP-relaxation of the undirected one.
TL;DR: This paper shows that a modification of the Kojima—Megiddo—Mizuno algorithm “solves” the pair of problems in polynomial time without assuming the existence of the LP solution, and develops anO(nL)-iteration complexity result for a variant of the algorithm.
Abstract: Kojima, Megiddo, and Mizuno investigate an infeasible-interior-point algorithm for solving a primal--dual pair of linear programming problems and they demonstrate its global convergence. Their algorithm finds approximate optimal solutions of the pair if both problems have interior points, and they detect infeasibility when the sequence of iterates diverges. Zhang proves polynomial-time convergence of an infeasible-interior-point algorithm under the assumption that both primal and dual problems have feasible points. In this paper, we show that a modification of the Kojima--Megiddo--Mizuno algorithm "solves" the pair of problems in polynomial time without assuming the existence of the LP solution. Furthermore, we develop anO(nL)-iteration complexity result for a variant of the algorithm.
TL;DR: An error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions is established and it is shown that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx.
Abstract: Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush--Kuhn--Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0---1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.
TL;DR: This work considers the vertex-weighted version of the undirected Steiner tree problem, and obtains some very large classes of facet-defining valid inequalities for the Steiners tree polytope.
Abstract: We consider the vertex-weighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is series—parallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the Steiner tree problem. By projecting this formulation, we obtain some very large classes of facet-defining valid inequalities for the Steiner tree polytope.
TL;DR: This paper develops and generalizes Auchmuty's results, and relates his class of merit functions to other works done in this field, and investigates differentiability and convexity properties, and presents characterizations of the set of solutions to variational inequalities.
Abstract: Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.
TL;DR: This paper presents approximate Newton or SQP methods for solving nonlinear programming problems whose objective and constraint functions have locally Lipschitzian derivatives, and establishes Q-superlinear convergence of these methods under the assumption that these derivatives are semismooth.
Abstract: In the literature, the proof of superlinear convergence of approximate Newton or SQP methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions. Sometimes, the second-order derivatives of those functions are required to be Lipschitzian. In this paper, we present approximate Newton or SQP methods for solving nonlinear programming problems whose objective and constraint functions have locally Lipschitzian derivatives, and establishQ-superlinear convergence of these methods under the assumption that these derivatives are semismooth. This assumption is weaker than the second-order differentiability. The extended linear-quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives.
TL;DR: A solution method for location-allocation problems involving thelp norm, where 1
Abstract: We present a solution method for location-allocation problems involving thel
p
norm, where 1
TL;DR: This paper considers the inverse shortest paths problem where arc costs are subject to correlation constraints and a new method is proposed for solving this class of problems as a generalization of the algorithm presented in Burton and Toint (Mathematical Programming 53, 1992).
Abstract: This paper considers the inverse shortest paths problem where arc costs are subject to correlation constraints. The motivation for this research arises from applications in traffic modelling and seismic tomography. A new method is proposed for solving this class of problems. It is constructed as a generalization of the algorithm presented in Burton and Toint (Mathematical Programming 53, 1992) for uncorrelated inverse shortest paths. Preliminary numerical experience with the new method is presented and discussed.
TL;DR: It is shown that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank-one perturbation matrices, a class which contains the intervalMatrices, testing singularity reduces to testing whether a certain matrix is not a P-matrix.
Abstract: Recently Rohn and Poljak proved that for interval matrices with rank-one radius matrices testing singularity is NP-complete. This paper will show that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank-one perturbation matrices, a class which contains the interval matrices, testing singularity reduces to testing whether a certain matrix is not a P-matrix. It follows from this result that the problem of testing whether a given matrix is a P-matrix is co-NP-complete.
TL;DR: The algorithm of Zhang is modified to obtain anO(n2L) infeasible-interior-point algorithm for monotone linear complementarity problems that has an asymptoticQ-subquadratic convergence rate.
Abstract: We modify the algorithm of Zhang to obtain anO(n2L) infeasible-interior-point algorithm for monotone linear complementarity problems that has an asymptoticQ-subquadratic convergence rate. The algorithm requires the solution of at most two linear systems with the same coefficient matrix at each iteration.
TL;DR: This paper studies the problem of finding a two-edge connected spanning subgraph of minimum weight and shows that when the graph is series-parallel, the polytope is completely described by the trivial constraints and the so-called cut constraints.
Abstract: This paper studies the problem of finding a two-edge connected spanning subgraph of minimum weight. This problem is closely related to the widely studied traveling salesman problem and has applications to the design of reliable communication and transportation networks. We discuss the polytope associated with the solutions to this problem. We show that when the graph is series-parallel, the polytope is completely described by the trivial constraints and the so-called cut constraints. We also give some classes of facet defining inequalities of this polytope when the graph is general.
TL;DR: New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented, derived from a general second order Fermat's rule for the maximization of a function over an arbitrary subset of a Banach space.
Abstract: New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat's rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki (1988) and Cominetti (1989); but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. We point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. Our conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral we recover the classical conditions of Ben-Tal--Zowe (1982) and Cominetti (1990).
TL;DR: The application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix is discussed and polynomial upper bounds on the combinatorial diameter of an associated polyhedron are derived.
Abstract: We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron.
TL;DR: New local and global error bounds are given for both nonmonotone and monotone linear complementarity problems and a possible candidate for a “best” error bound emerges as the sum of two natural residuals.
Abstract: New local and global error bounds are given for both nonmonotone and monotone linear complementarity problems. Comparisons of various residuals used in these error bounds are given. A possible candidate for a “best” error bound emerges from our comparisons as the sum of two natural residuals.
TL;DR: A numerical study on the use of eigenvalue-based techniques to find upper and lower bounds for the graph partitioning problem and shows that the techniques are very robust and consistently produce upper andLower bounds having a relative gap of typically a few percentage points.
Abstract: LetG = (N, E) be an edge-weighted undirected graph. The graph partitioning problem is the problem of partitioning the node setN intok disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvalue-based techniques to find upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.
TL;DR: This paper identifies several classes of facet defining inequalities and relate them to special classes of graphs on which the Steiner tree problem is known to be NP-hard.
Abstract: This is the second part of two papers addressing the study of the facial structure of the Steiner tree polyhedron. In this paper we identify several classes of facet defining inequalities and relate them to special classes of graphs on which the Steiner tree problem is known to be NP-hard.
TL;DR: It is shown that asymptotic optimality can be achieved with a finite master program provided that a quadratic regularizing term is included and the elimination of the cutting planes impacts neither the number of iterations required nor the statistical properties of the terminal solution.
Abstract: Stochastic decomposition is a stochastic analog of Benders' decomposition in which randomly generated observations of random variables are used to construct statistical estimates of supports of the objective function. In contrast to deterministic Benders' decomposition for two stage stochastic programs, the stochastic version requires infinitely many inequalities to ensure convergence. We show that asymptotic optimality can be achieved with a finite master program provided that a quadratic regularizing term is included. Our computational results suggest that the elimination of the cutting planes impacts neither the number of iterations required nor the statistical properties of the terminal solution.
TL;DR: An efficient implementation of theO(mn + n2 logn) time algorithm originally proposed by Nagamochi and Ibaraki (1992) for computing the minimum capacity cut of an undirected network and its running time is not significantly affected by the types of the networks being solved.
Abstract: In this paper, we present an efficient implementation of theO(mn + n2 logn) time algorithm originally proposed by Nagamochi and Ibaraki (1992) for computing the minimum capacity cut of an undirected network. To enhance computation, various ideas are added so that it can contract as many edges as possible in each iteration. To evaluate the performance of the resulting implementation, we conducted extensive computational experiments, and compared the results with that of Padberg and Rinaldi's algorithm (1990), which is currently known as one of the practically fastest programs for this problem. The results indicate that our program is considerably faster than Padberg and Rinaldi's program, and its running time is not significantly affected by the types of the networks being solved.
TL;DR: An upper bound for the rate of convergence is given in terms of the objective functions of the associated deterministic problems and it is shown how it can be applied to derivation of the Law of Iterated Logarithm for the optimal solutions.
Abstract: In this paper we study stability of optimal solutions of stochastic programming problems with fixed recourse. An upper bound for the rate of convergence is given in terms of the objective functions of the associated deterministic problems. As an example it is shown how it can be applied to derivation of the Law of Iterated Logarithm for the optimal solutions. It is also shown that in the case of simple recourse this upper bound implies upper Lipschitz continuity of the optimal solutions with respect to the Kolmogorov--Smirnov distance between the corresponding cumulative probability distribution functions.
TL;DR: The predictor—corrector method for solving linear programs from infeasible starting points is analyzed and can be combined with Ye's finite termination scheme under very general assumptions.
Abstract: A predictor--corrector method for solving linear programs from infeasible starting points is analyzed. The method is quadratically convergent and can be combined with Ye's finite termination scheme under very general assumptions. If the starting points are large enough then the algorithm hasO(nL) iteration complexity. If the ratio between feasibility and optimality at the starting points is small enough then the algorithm has O( $$\sqrt {n L} $$ ) iteration complexity. For feasible starting points the algorithm reduces to the Mizuno--Todd--Ye predictor--corrector method.