Karmarkar's Linear Programming Algorithm Research Articles

FIR Filters with Punctured Radix-8 Symmetric Coefficients: Design and Multiplier-Free Realizations

In recent years, several approaches have been investigated to obtain multiplier-free realizations of digital filters. One approach makes use of periodically time-varying (PTV) structures. The idea is to distribute the computation in time and in space. Distribution in time provides reuse of the same hardware by means of PTV coefficients. Distribution in space increases the number of coefficients but simplifies the values of the coefficients. Computation distribution is based on radix-r number representation, and it can be carried out to the extent that each computation involves a simple coefficient that can be realized using only addition and shift (no hardware multiplier). Previous PTV realizations could not exploit the coefficient symmetry of finite impulse response (FIR) filters to reduce the number of coefficients. This paper proposes design and realizations of FIR filters with punctured radix-8 coefficients belonging to the septuple set {0, ±1, ±2, ±4), which can be implemented using only a shift operation without requiring any hardware multiplier. The realizations exploit the coefficient symmetry to reduce the hardware by about one-half. Due to a non-uniform grid of representation, we apply a modified Karmarkar's linear programming algorithm to find the optimum set of discrete coefficients that minimizes the weighted peak ripple error. Comparison with a conventional FIR filter with sum-of-powers-of-two (SOPOT) coefficients shows that the proposed filter is faster and uses less hardware than one with SOPOT coefficients. However, the punctured radix-8 system has a limit on the achievable ripple.

Combination of artificial neural-network forecasters for prediction of natural gas consumption

The focus of this paper is on combination of artificial neural-network (ANN) forecasters with application to the prediction of daily natural gas consumption needed by gas utilities. ANN forecasters can model the complex relationship between weather parameters and previous gas consumption with the future consumption. A two-stage system is proposed with the first stage containing two ANN forecasters, a multilayer feedforward ANN and a functional link ANN. These forecasters are initially trained with the error backpropagation algorithm, but an adaptive strategy is employed to adjust their weights during on-line forecasting. The second stage consists of a combination module to mix the two individual forecasts produced in the first stage. Eight different combination algorithms are examined, they are based on: averaging, recursive least squares, fuzzy logic, feedforward ANN, functional link ANN, temperature space approach, Karmarkar's linear programming algorithm and adaptive mixture of local experts (modular neural networks). The performance is tested on real data from six different gas utilities. The results indicate that combination strategies based on a single ANN outperform the other approaches.

Speeding up Karmarkar's algorithm for multicommodity flows

We show how to speed up Karmarkar's linear programming algorithm for the case of multicommodity flows. The special structure of the constraint matrix is exploited to obtain an algorithm for the multicommodity flow problem which requires O(s 3.5 v 2.5 eL) arithmetic operations, each operation being performed to a precision of O (L) bits. Herev is the number of vertices ande is the number of edges in the given network,s is the number of commodities, andL is bounded by the number of bits in the input. We obtain a speed up of the order of (e 0.5/v 0.5)+(e 2.5/v 2.5s2) over Karmarkar's modified algorithm which is substantial for dense networks. The techniques in the paper can also be used to speed up any interior point algorithm for any linear programming problem whose constraint matrix is structurally similar to the one in the multicommodity flow problem.

Karmarkar's linear programming algorithm and Newton's method

This paper describes a full-dimensional version of Karmarkar's linear programming algorithm, theprojective scaling algorithm, which is defined for any linear program in ℝ n having a bounded, full-dimensional polytope of feasible solutions. If such a linear program hasm inequality constraints, then it is equivalent under an injective affine mappingJ:ℝ n →ℝ m to Karmarkar's original algorithm for a linear program in ℝ m havingm—n equality constraints andm inequality constraints. Karmarkar's original algorithm minimizes a potential functiong(x), and the projective scaling algorithm is equivalent to that version of Karmarkar's algorithm whose step size minimizes the potential function in the step direction. The projective scaling algorithm is shown to be a global Newton method for minimizing a logarithmic barrier function in a suitable coordinate system. The new coordinate system is obtained from the original coordinate system by a fixed projective transformationy = Ф(x) which maps the hyperplaneH opt ={x:〈c, x〉 =c 0} specified by the optimal value of the objective function to the “hyperplane at infinity”. The feasible solution set is mapped underФ to anunbounded polytope. Letf LB(y) denote the logarithmic barrier function associated to them inequality constraints in the new coordinate system. It coincides up to an additive constant with Karmarkar's potential function in the new coordinate system. Theglobal Newton method iterate y * for a strictly convex functionf(y) defined on a suitable convex domain is that pointy * that minimizesf(y) on the search ray {y+λ v N(y): λ≧0} wherev N(y) =−(∇2 f(y))−1(∇f(y)) is the Newton's method vector. If {x (k)} are a set of projective scaling algorithm iterates in the original coordinate system andy (k) =Ф(x (k)) then {y (k)} are a set of global Newton method iterates forf LB(y) and conversely. Karmarkar's algorithm with step size chosen to minimize the potential function is known to converge at least at a linear rate. It is shown (by example) that this algorithm does not have a superlinear convergence rate.

The solution of large-scale least-squares problems on supercomputers

We discuss methods for solving medium to large-scale sparse least-squares problems on supercomputers, illustrating our remarks by experiments on the CRAY-2 supercomputer at Harwell. The method we are primarily concerned with is an augmented system approach which has the merit of both robustness and accuracy, in addition to a kernel operation that is just the solution of a symmetric indefinite system. We consider extensions to handle weighted and constrained problems, and include experiments on systems similar to those arising in the Karmarkar algorithm for linear programming. We indicate how recent improvements to the kernel software could greatly improve the performance of the least-squares code.

An extension of Karmarkar's projective algorithm for convex quadratic programming

We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln3) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.

L/sub 1/ and L/sub infinity / minimization via a variant of Karmarkar's algorithm

Simple iterative algorithms are presented for L/sub 1/ and L/sub infinity / minimization (regression) based on a variant of Karmarkar's linear programming algorithm. Although these algorithms are based on entirely different theoretical principles to the popular IRLS (iteratively reweighted least squares) algorithm, they have almost identical matrix operations. Also presented are the results of a Monte Carlo study comparing the numerical convergence properties of the Karmarkar algorithm for L/sub 1/ minimization to those of an IRLS and a simplex algorithm. The test problem involves L/sub 1/ estimation of AR (autoregressive) model parameters. The Karmarkar algorithm outperformed IRLS by achieving higher numerical accuracy in fewer iterations. Techniques for reducing the computational cost per iteration of the Karmarkar L/sub 1/ algorithm are discussed. >

Improved Bounds and Containing Ellipsoids in Karmarkar's Linear Programming Algorithm

Karmarkar's projective algorithm for linear programming provides not only primal solutions but dual solutions giving bounds on the optimal value. Here we show how improved bounds can be obtained at the expense of solving a two-dimensional linear programming problem at every iteration, and also how an ellipsoid containing all dual optimal solutions can be generated from available information. We also give the results of limited computational experiments related to these topics.

Exploiting special structure in Karmarkar's linear programming algorithm

We propose methods to take advantage of specially-structured constraints in a variant of Karmarkar's projective algorithm for standard form linear programming problems. We can use these constraints to generate improved bounds on the optimal value of the problem and also to compute the necessary projections more efficiently, while maintaining the theoretical bound on the algorithm's performance. It is shown how various upper-bounding constraints can be handled implicitly in this way. Unfortunately, the situation for network constraints appears less favorable.

A variable-metric variant of the Karmarkar algorithm for linear programming

The most time-consuming part of the Karmarkar algorithm for linear programming is the projection of a vector onto the nullspace of a matrix that changes at each iteration. We present a variant of the Karmarkar algorithm that uses standard variable-metric techniques in an innovative way to approximate this projection. In limited tests, this modification greatly reduces the number of matrix factorizations needed for the solution of linear programming problems.

A variant of Karmarkar's linear programming algorithm for problems in standard form

A variant of Karmarkar's linear programming algorithm for problems in standard form

A modification of karmarkar's linear programming algorithm

We present a modification of Karmarkar's linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviatinga priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.

Towards an asymptotic analysis of Karmarkar's algorithm

The Karmarkar algorithm for linear programming (1984) has generated excitement and controversy. This controversy has been the theme of many popular panel discussions, reporting on experimental results both supporting and refuting claims of substantial speed-ups. How does one resolve these conflicting reports? The thrust of the work here has been on implementation and experimentation. We argue that further theoretical analysis of the algorithm can shed light on this controversy. In particular we ask: What is the ‘average’ behavior of the algorithm, and what are its dynamical properties? In this paper we do not address the issue of work per iteration, perhaps the most controversial aspect of the algorithm in practice. We do look, however, at the iterative process, and here we give a new proof of the number of iterative steps required by the algorithm in the worst case. Our proof is considerably simpler than Karmarkar's, and provides new information and insight. Specifically, our proof identifies an explicitly geometric quantity that measures the algorithm's progress at each step. Analyzing the changing size of this quantity gives information about the dynamics of the algorithm.