Showing papers on "Linear programming published in 1988"

Geometric Algorithms and Combinatorial Optimization

Abstract: This book develops geometric techniques for proving the polynomial time solvability of problems in convexity theory, geometry, and - in particular - combinatorial optimization. It offers a unifying approach based on two fundamental geometric algorithms: - the ellipsoid method for finding a point in a convex set and - the basis reduction method for point lattices. The ellipsoid method was used by Khachiyan to show the polynomial time solvability of linear programming. The basis reduction method yields a polynomial time procedure for certain diophantine approximation problems. A combination of these techniques makes it possible to show the polynomial time solvability of many questions concerning poyhedra - for instance, of linear programming problems having possibly exponentially many inequalities. Utilizing results from polyhedral combinatorics, it provides short proofs of the poynomial time solvability of many combinatiorial optimization problems. For a number of these problems, the geometric algorithms discussed in this book are the only techniques known to derive polynomial time solvability. This book is a continuation and extension of previous research of the authors for which they received the Fulkerson Prize, awarded by the Mathematical Programming Society and the American Mathematical Society.

Neural networks for nonlinear programming

Abstract: The dynamics of the modified canonical nonlinear programming circuit are studied and how to guarantee the stability of the network's solution. By considering the total cocontent function, the solution of the canonical nonlinear programming circuit is reconciled with the problem being modeled. In addition, it is shown how the circuit can be realized using a neural network, thereby extending the results of D.W. Tank and J.J. Hopefield (ibid., vol.CAS-33, p.533-41, May 1986) to the general nonlinear programming problem. >

Lower bound limit analysis using finite elements and linear programming

Abstract: This paper describes a technique for computing lower bound limit loads in soil mechanics under conditions of plane strain. In order to invoke the lower bound theorem of classical plasticity theory, a perfectly plastic soil model is assumed, which may be either purely cohesive or cohesive-frictional, together with an associated flow rule. Using a suitable linear approximation of the yield surface, the procedure computes a statically admissible stress field via finite elements and linear programming. The stress field is modelled using linear 3-noded traingles and statically admissible stress discontinuities may occur at the edges of each triangle. Imposition of the stress-boundary, equilibrium and yield conditions leads to an expression for the collapse load which is maximized subject to a set of linear constraints on the nodal stresses. Since all of the requirements for a statically admissible solution are satisfied exactly (except for small round-off errors in the optimization computations), the solution obtained is a strict lower bound on the true collapse load and is therefore ‘safe’. A major drawback of the technique, as first described by Lysmer,1 is the large amount of computer time required to solve the linear programming problem. This paper shows that this limitation may be avoided by using an active set algorithm, rather than the traditional simplex or revised simplex strategies, to solve the resulting optimization problem. This is due to the nature of the constraint matrix, which is always very sparse and typically has many more rows that columns. It also proved that the procedure can, without modification, be used to derive strict lower bounds for a purely cohesive soil which has increasing strength with depth. This important class of problem is difficult to tackle using conventional methods. A number of examples are given to illustrate the effectiveness of the procedure.

A polynomial-time algorithm, based on Newton's method, for linear programming

Abstract: A new interior method for linear programming is presented and a polynomial time bound for it is proven. The proof is substantially different from those given for the ellipsoid algorithm and for Karmarkar's algorithm. Also, the algorithm is conceptually simpler than either of those algorithms.

A primal-dual interior point algorithm for linear programming

Abstract: This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1 — η/n); each iteration reduces the duality gap by at least η/n. Here n denotes the size of the problems and η a positive number depending on initial interior feasible solutions of the problems. The algorithm is based on an application of the classical logarithmic barrier function method to primal and dual linear programs, which has recently been proposed and studied by Megiddo.

A pareto race

Abstract: A dynamic and visual “free-search” type of interactive procedure for multiple-objective linear programming is presented. The method enables a decision maker to freely search any part of the efficient frontier by controlling the speed and direction of motion. The objective function values are represented in numeric form and as bar graphs on a display. The method is implemented on an IBM PC/1 microcomputer and is illustrated using a multiple-objective linear-programming model for managing disposal of sewage sludge in the New York Bight. Some other applications are also briefly discussed.

Interior path following primal-dual algorithms

Abstract: We describe a primal-dual interior point algorithm for linear programming and convex quadratic programming problems which requires a total of O(n$\sp3$L) arithmetic operations. Each iteration updates a penalty parameter and finds an approximate Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithm barrier function problem. The algorithm is based on the path following idea. We show that the duality gap is reduced at each iteration by a factor of (1 $-$ $\delta$/$\sqrt{n}$) where 0.1 $\leq$ $\delta$ $<$ 1. As a consequence, an optimal solution for our problem can be found in at most O($\sqrt{n}$L) iterations. We also describe how the primal-dual algorithm can be extended to solve a class of convex separable programming problems subject to linear constraints. In this case, it is shown that the duality gap is reduced at each iteration by a factor of (1 $-$ $\delta$/$\sqrt{n}$), where $\delta$ is positive and depends on some parameters associated with the objective function.

Polynomial Algorithms for Linear Programming

Abstract: This paper contrasts the recent polynomial algorithms for linear programming of Khachian and Karmarkar. We show that each requires the solution of a weighted least-squares subproblem at every iteration. By comparing these subproblems we obtain further insights into the two methods.

Expressing combinatorial optimization problems by linear programs

Abstract: Many combinatorial optimization problems call for the optimization of a linear function over a certain polytope. Typically, these polytopes have an exponential number of facets. We explore the problem of finding small linear programming formulations when one may use any new variables and constraints. We show that expressing the matching and the Traveling Salesman Problem by a symmetric linear program requires exponential size. We relate the minimum size needed by a LP to express a polytope to a combinatorial parameter, point out some connections with communication complexity theory, and examine the vertex packing polytope for some classes of graphs.

Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems

Abstract: The author establishes a number of properties associated with the dynamical system H=(H,(H,N)), where H and N are symmetric n-by-n matrices and (A,B)=AB-BA. The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices. Particular emphasis is placed on the role of this equation as an analog computer. For example, it is shown how to map the data associated with a linear programming problem into H(0) and N in such a way as to have H=(H(H,N)) evolve to a solution of the linear programming problem. This result can be applied to find systems that solve a variety of generic combinatorial optimization problems, and it also provides an algorithm for diagonalizing symmetric matrices. >

Feedback control of linear discrete-time systems under state and control constraints

Abstract: In this paper the problem of stabilizing linear discrete-time systems under state and control linear constraints is studied. Based on the concept of positive invariance, existence conditions of linear state feedback control laws respecting both the constraints are established. These conditions are then translated into an algorithm of linear programming.

Optimal VAr planning by approximation method for recursive mixed-integer linear programming

Abstract: The authors propose an algorithm for solving reactive power planning problems. The optimization approach is based on a recursive mixed-integer programming technique using an approximation method. A fundamental feature of this algorithm is that the number of capacitor or reactor units can be treated as a discrete variable in solving large-scale VAr (volt-ampere reactive) planning problems. Numerical results have verified the validity and efficiency of the algorithm. >

Economic-emission load dispatch through goal programming techniques

Abstract: The economic-emission load dispatch problem which accounts for minimization of both cost and emission is a multiple, conflicting-objective function problem. Goal programming techniques are most suitable for such type of problems. Here, the economic-emission load dispatch problem is solved through linear and nonlinear goal programming algorithms. The application and validity of the proposed algorithms are demonstrated for a sample system having six generators. >

Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization

Abstract: The problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case. The minimum value of the objective function can be obtained by solving a semi-infinite linear programming problem, and an iterative procedure for obtaining approximate solutions is introduced. Application of the l/sup 1/-optimal problem to robustness is discussed. A mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations. >

The constrained via minimization problem for PCB and VLSI design

Abstract: A novel via minimization approach is presented for two-layer routing of printed-circuit boards and VLSI chips. The authors have analyzed and characterized different aspects of the problem and derived an equivalent graph model for the problem from the linear-programming formulation. Based on the analysis of their unified formulation, the authors pose a practical heuristic algorithm. The algorithm can handle both grid-based and gridless routing. Also, an arbitrary number of wires is allowed to intersect at a via, and both Manhattan and knock-knee routings are allowed. >

A general bilevel linear programming formulation of the network design problem

Abstract: A formulation of the network design problem as a bilevel linear program is presented which admits both the convex and concave investment functions. It also allows a more general representation of travel cost functions than a previous formulation by LeBlanc and Boyce (1986).

A Mathematical Programming Model of Decentralized Multi-Level Systems

Abstract: In decentralized systems, the objectives at the different levels often differ. Each level controls only a subset of the decision variables but is affected by the decisions made at the other levels. Recent methods in bi-level linear programming solve such systems better than decomposition methods. Here, the bi-level LP is extended to a bi-level system with many decision-makers at the lower level, and a tri-level system with one decision-maker at each level. In both of these cases, the higher level acts as the leader and the lower level as the follower in the Stackelberg game. The resulting techniques are illustrated with simple numerical examples.

MW/voltage control in a linear programming based optimal power flow

Abstract: A functional improvement to the linear-programming-based optimal power flow technique is reported. The new feature allows the rescheduling of the active power controls to correct voltage-magnitude-constraint violations. It preserves the reliability and speed characteristics of the traditional linear programming approach. Results demonstrating the effectiveness of the method on a small and a large power system are presented. >

Lot-size models with back-logging: strong reformulations and cutting planes

Abstract: We examine mixed integer programming reformulations of the uncapacitated lot-sizing problem with backlogging. First we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables, leading to a known plant location reformulation and a shortest path reformulation. Each of these reformulations is strong in the sense that its linear programming relaxation solves the lot-sizing problem. Secondly we attempt to treat the problem in the space of the original variables. We give an implicit description of the convex hull of solutions, and show how the problem of finding a violated cutting plane can be solved as a linear program. We also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm. The efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi-item capacitated lot-sizing problems with backlogging. Near optimal solutions have been found to problems with 8 periods and up to 100 times.

Boole-Bonferroni inequalities and linear programming

Abstract: We present a method to obtain sharp lower and upper bounds for the probability that at least one out of a number of events in an arbitrary probability space will occur. The input data are some of the binomial moments of the occurrences, such as the sum of the probabilities of the individual events, or the sum of the joint probabilities of all pairs of events. We develop a special, very simple linear programming algorithm to obtain these bounds. The method allows us to compute good bounds in an optimal way, utilizing only the first few terms in the inclusion-exclusion formula. Possible applications include obtaining bounds for the reliability of a stochastic system, solving algorithmically some stochastic programming problems, and approximating multivariate probabilities in statistics. In a numerical example we approximate the probability that a Gaussian process runs below a given level in a number of consecutive epochs.

Short-term thermal unit commitment-a new method

Abstract: A method and an algorithm based on this new method are presented. The effectiveness of the algorithm is illustrated by studying a 20-unit midwestern utility system, the EPRI 174-unit synthetic utility system D, and the EPRI 155-unit synthetic utility system E. The algorithm produces the same unit commitment schedule for the 20-unit system as a frequently used DP-STC algorithm in 15 s of computation time versus 524 s, respectively. The computation time is approximately linear with the number of hours in the unit commitment horizon. For the EPRI 174-unit system the algorithm requires only 205 s of computation time on a VAX 11/780 for a 48-hour horizon. >

Practical reactive power allocation/operation planning using successive linear programming

Abstract: A method for reactive power planning is presented that it finds an optimal solution for both allocation and operation planning in large systems using linear programming (LP). The method utilizes calculated linear sensitivities including active power and voltage phase angle in the formulation. Although the overall method includes these relations, the number of constraints and variables are not augmented in its first procedure, APPROACH-1. Its second procedure, APPROACH-2, overcomes numerical problems caused by a dense constraint matrix. This is achieved by retaining untouched sparse sensitivities in the constraint matrix and by eliminating any calculations related to the inverse matrix. The results of applying this method to a practical 224-bus system and the IEEE-30 bus system verify its robustness and fast convergence. >

Impact of Hierarchical Memory Systems On Linear Algebra Algorithm Design

Abstract: Linear algebra algorithms based on the BLAS or ex tended BLAS do not achieve high performance on mul tivector processors with a hierarchical memory system because of a lack of data locality. For such machines, block linear algebra algorithms must be implemented in terms of matrix-matrix primitives BLAS3. Designing ef ficient linear algebra algorithms for these architectures requires analysis of the behavior of the matrix-matrix primitives and the resulting block algorithms as a func tion of certain system parameters. The analysis must identify the limits of performance improvement possible via blocking and any contradictory trends that require trade-off consideration. We propose a methodology that facilitates such an analysis and use it to analyze the per formance of the BLAS3 primitives used in block methods. A similar analysis of the block size-perfor mance relationship is also performed at the algorithm level for block versions of the LU decomposition and the Gram-Schmidt orthogonalization procedures.

Integer linear programming neural networks for job-shop scheduling

Abstract: The authors present an integer linear programming neural network (ILPNN) based on a modified Tank and Hopfield neural network model to solve job-shop scheduling, an NP-complete constraint satisfaction problem. The constraints of the job-shop problem are formulated as a set of integer linear equations. The cost function for minimization is the total starting times of all jobs subject to precedence constraints. In the authors' approach, the set of integer linear equations is solved by an iterative linear programming with integer adjustments (ILPIA) technique, without a branch-and-bound search. In particular, the linear and nonlinear zero-one variables are represented by linear sigmoid and nonlinear high-gain amplifiers with a response of a step function, respectively. Simulations based on solving a linear differential equation show that the ILPNN approach produces optimal or near-optimal solutions, although it does not guarantee optimal solutions. The authors also analyze the hardware implementation of ILPNNs and study the feasibility of this approach for large-scale problems. >

Structural optimization with frequency constraints

Abstract: This paper presents a design optimization algorithm for structural weight minimization with multiple frequency constraints. An optimality criterion method based on uniform Lagrangian density for resizing and a scaling procedure to locate the constraint boundary were used in optimization. Multiple frequency constraints of equality and inequality types were addressed. The effectiveness of the algorithm was demonstrated by designing a number of truss structures with as many as 489 design variables. No attempt was made to reduce the number of design variables by such procedures as linking and/or invoking symmetry conditions. The design examples include a 10-bar truss, 200-bar truss, a modified ACOSS-II, and COFS (Control of Flexible Structures) mast truss. All the structures contain nonstructural mass besides their own mass. The algorithm is extremely stable and, in all cases, the optimum designs were obtained in less than 20 iterations regardless of the size of the structure and the number of design variables.