Showing papers in "Mathematical Programming in 1990"

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

Abstract: Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Approximation algorithms for scheduling unrelated parallel machines

Abstract: We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep ij . The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

Proximity control in bundle methods for convex

Abstract: Proximal bundle methods for minimizing a convex functionf generate a sequence {x k } by takingx k+1 to be the minimizer of $$\hat f^k (x) + u^k |x - x^k |^2 /2$$ , where $$\hat f^k $$ is a sufficiently accurate polyhedral approximation tof andu k > 0. The usual choice ofu k = 1 may yield very slow convergence. A technique is given for choosing {u k } adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.

A filled function method for finding a global minimizer of a function of several variables

Abstract: The concept of a filled function is introduced. We construct a particular filled function and analyze its properties. An algorithm for global minimization is generated based on the concept and properties of the filled function. Some typical examples with 1 to 10 variables are tested and computational results show that in most cases this algorithm works better than the tunneling algorithm. The advantages and disadvantages are analyzed and further research directions are discussed.

Active set algorithms for isotonic regression: a unifying framework

Abstract: In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.

Chva´tal closures for mixed integer programming problems

Abstract: Chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedronP and integral vectorw, if max(wx|x ∈ P, wx integer} =t, thenwx ⩽ t is valid for all integral vectors inP. We consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that\(\bar wx\) be integer for some other integral vector\(\bar w\). The cutting-plane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedronP, the set of vectors that satisfy every cutting plane forP with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvatal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.

MSLiP: a computer code for the multistage stochastic linear programming problem

Abstract: This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.

A recursive procedure to generate all cuts for 0-1 mixed integer programs

Abstract: We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0–1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.

An algorithm for linear programming which requires O(( m + n ) n 2 + ( m + n ) 1.5 n ) L ) arithmetic operations

Abstract: We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of $$\sqrt {m + n} $$ .

A trust region algorithm for equality constrained optimization

Abstract: A trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function. Under certain conditions global convergence and local superlinear convergence results are proved.

Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints

Abstract: In this paper we consider heuristic algorithms for a special case of the generalized bilevel mathematical programming problem in which one of the levels is represented as a variational inequality problem. Such problems arise in network design and economic planning. We obtain derivative information needed to implement these algorithms for such bilevel problems from the theory of sensitivity analysis for variational inequalities. We provide computational results for several numerical examples.

Facets of the clique partitioning polytope

Abstract: A subsetA of the edge set of a graphG = (V, E) is called a clique partitioning ofG is there is a partition of the node setV into disjoint setsW 1,⋯,W k such that eachW i induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = ∪ 1{uv|u, v ∈ W i ,u ≠ v}. Given weightsw e ∈ℝ for alle ∈ E, the clique partitioning problem is to find a clique partitioningA ofG such that ∑ e∈A w e is as small as possible. This problem—known to be -hard, see Wakabayashi (1986)—comes up, for instance, in data analysis, and here, the underlying graphG is typically a complete graph. In this paper we study the clique partitioning polytope of the complete graphK n , i.e., is the convex hull of the incidence vectors of the clique partitionings ofK n . We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofK n induce facets of . The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in Grotschel and Wakabayashi (1989).

An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds

Abstract: This paper gives an O(n) algorithm for a singly constrained convex quadratic program using binary search to solve the Kuhn-Tucker system. Computational results indicate that a randomized version of this algorithm runs in expected linear time and is suitable for practical applications. For the nonconvex case ane-approximate algorithm is proposed which is based on convex and piecewise linear approximations of the objective function.

Minimum-weight two-connected spanning networks

Abstract: We consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a setV. We assume a symmetric, nonnegative distance functiond(·) defined onV × V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal two-connected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thatany two-connected graph satisfying these conditions is theunique optimal solution for a particular choice of ‘canonical’ distances satisfying the triangle inequality. We use these properties to show that the weight of an optimal traveling salesman cycle is at most 4/3 times the weight of an optimal two-connected solution; examples are provided which approach this bound arbitrarily closely. In addition, we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points.

Feasibility issues in a primal-dual interior-point method for linear programming

Abstract: A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a "big-M", and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.

Facet identification for the symmetric traveling salesman polytope

Abstract: Several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given. An identification procedure accepts as input the support graph of a point which does not belong to the polytope, and returns as output some of the facet inducing inequalities violated by the point. A procedure which always accomplishes this task is calledexact, otherwise it is calledheuristic. We give exact procedures for the subtour elimination and the 2-matching constraints, based on the Gomory—Hu and Padberg—Rao algorithms respectively. Efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially. Exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities. These procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large-scale problem instances with sizes up to 2392 nodes to optimality.

On a subproblem of trust region algorithms for constrained optimization

Abstract: We study a subproblem that arises in some trust region algorithms for equality constrained optimization. It is the minimization of a general quadratic function with two special quadratic constraints. Properties of such subproblems are given. It is proved that the Hessian of the Lagrangian has at most one negative eigenvalue, and an example is presented to show that the Hessian may have a negative eigenvalue when one constraint is inactive at the solution.

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

Abstract: A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.

An efficient algorithm for the minimum capacity cut problem

Abstract: Given a finite undirected graph with nonnegative edge capacities the minimum capacity cut problem consists of partitioning the graph into two nonempty sets such that the sum of the capacities of edges connecting the two parts is minimum among all possible partitionings. The standard algorithm to calculate a minimum capacity cut, due to Gomory and Hu (1961), runs in O(n 4) time and is difficult to implement. We present an alternative algorithm with the same worst-case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm. We report computational results for graphs with up to 2000 nodes.

Exchange price equilibria and variational inequalities

Abstract: The aim of this paper is to show the relevance of the concept and the theory of variational inequalities in the study of economic equilibria.

Newton's method for the nonlinear complementarity problem: a B-differentiable equation problem

Abstract: This paper describes a damped-Newton method for solving the nonlinear complementarity problem when it is formulated as a system of B-differentiable equations through the use of the Minty-map. This general Newton algorithm contains a one-dimensional line search and possesses a global convergence property under certain conditions; modifications and heuristic implementations of the algorithm for the case when these conditions do not hold are also discussed. The numerical experiments show that, in general, this new scheme is more efficient and robust than the traditional Josephy-Newton algorithm.

Algebraic optimization: the Fermat-Weber location problem

Abstract: The Fermat-Weber location problem is to find a point in ℝ n that minimizes the sum of the (weighted) Euclidean distances fromm given points in ℝ n . In this work we discuss some relevant complexity and algorithmic issues. First, using Tarski's theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem, where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one-dimensional polynomial. Secondly, we exhibit an explicit solution to the strong separation problem associated with the Fermat-Weber model. This separation result shows that ane-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.

The augmented Lagrangian method for equality and inequality constraints in Hilbert spaces

Abstract: In this paper we consider an augmented Lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a Hilbert space, an inequality constraint whose image space is finite dimensional, and an affine inequality constraint whose image space is in an infinite dimensional Hilbert space. We obtain local convergence of this method without imposing strict complementarity conditions when the equality, as well as the inequality constraint with finite dimensional image space are augmented. To the author's knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional.

Concurrent stochastic methods for global optimization

Abstract: The global optimization problem, finding the lowest minimizer of a nonlinear function of several variables that has multiple local minimizers, appears well suited to concurrent computation. This paper presents a new parallel algorithm for the global optimization problem. The algorithm is a stochastic method related to the multi-level single-linkage methods of Rinnooy Kan and Timmer for sequential computers. Concurrency is achieved by partitioning the work of each of the three main parts of the algorithm, sampling, local minimization start point selection, and multiple local minimizations, among the processors. This parallelism is of a coarse grain type and is especially well suited to a local memory multiprocessing environment. The paper presents test results of a distributed implementation of this algorithm on a local area network of computer workstations. It also summarizes the theoretical properties of the algorithm.

Convergence properties of trust region methods for linear and convex constraints

Abstract: We develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem. Global convergence is established for general convex constraints while the local analysis is for linearly constrained problems. The main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations. As a consequence of the identification properties, we develop rate of convergence results by assuming that the step is a truncated Newton method. Our development is mainly geometrical; this approach allows the development of a convergence theory without any linear independence assumptions.

On the solution of concave knapsack problems

Abstract: We consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint. We characterize strict local miniimizers of concave minimization problems subject to linear constraints, and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is NP-hard, it is possible to determine a local minimizer of this problem with at most O(n logn) operations and 1+[logn] evaluations of the function. If the function is quadratic this algorithm requires at most O(n logn) operations.

Large scale nonlinear deterministic and stochastic optimization: formulations involving simulation of subsurface contamination

Abstract: Once subsurface water supplies become contaminated, designing cost-effective and reliable remediation schemes becomes a difficult task. The combination of finite element simulation of groundwater contaminant transport with nonlinear optimization is one approach to determine the best well selection and optimal fluid withdrawal and injection rates to contain and remove the contaminated water. Both deterministic and stochastic programming problems have been formulated and solved. These tend to be large scale problems, owing to the simulation component which serves as a portion of the constraint set. The overall problem of combined groundwater process simulation and nonlinear optimization is discussed along with example problems. Because the contaminant transport simulation models give highly uncertain results, quantifying their uncertainty and incorporating reliability into the remediation design results in a class of large stochastic nonlinear problems. The reliability problem is beginning to be addressed, and some strategies and formulations involving chance constraints and Monte Carlo methods are presented.

On the convergence of cross decomposition

Abstract: Cross decomposition is a recent method for mixed integer programming problems, exploiting simultaneously both the primal and the dual structure of the problem, thus combining the advantages of Dantzig—Wolfe decomposition and Benders decomposition. Finite convergence of the algorithm equipped with some simple convergence tests has been proved. Stronger convergence tests have been proposed, but not shown to yield finite convergence. In this paper cross decomposition is generalized and applied to linear programming problems, mixed integer programming problems and nonlinear programming problems (with and without linear parts). Using the stronger convergence tests finite exact convergence is shown in the first cases. Unbounded cases are discussed and also included in the convergence tests. The behaviour of the algorithm when parts of the constraint matrix are zero is also discussed. The cross decomposition procedure is generalized (by using generalized Benders decomposition) in order to enable the solution of nonlinear programming problems.

Computational schemes for large-scale problems in extended linear-quadratic programming

Abstract: Numerical approaches are developed for solving large-scale problems of extended linear-quadratic programming that exhibit Lagrangian separability in both primal and dual variables simultaneously. Such problems are kin to large-scale linear complementarity models as derived from applications of variational inequalities, and they arise from general models in multistage stochastic programming and discrete-time optimal control. Because their objective functions are merely piecewise linear-quadratic, due to the presence of penalty terms, they do not fit a conventional quadratic programming framework. They have potentially advantageous features, however, which so far have not been exploited in solution procedures. These features are laid out and analyzed for their computational potential. In particular, a new class of algorithms, called finite-envelope methods, is described that does take advantage of the structure. Such methods reduce the solution of a high-dimensional extended linear-quadratic program to that of a sequence of low-dimensional ordinary quadratic programs.

Approximate subdifferential and metric regularity: the finite-dimensional case

Abstract: The notion of graphical metric regularity is introduced and conditions ensuring this kind of regularity for systems of finite-dimensional multifunctions are given in terms of partial approximate subdifferentials.