Discrete L1 Research Articles

Computational experience with an algorithm for discrete L1 approximation

In a recent paper Wolfe [13] proposes and analyses an algorithm for discrete linearL p approximation with 1≤p<2. In particular he gives a local convergence result for the casep=1. For this case, we discuss the implementation of the method, illustrate its performance on some examples, and draw some comparisons with the linear programming based method of Barrodale and Roberts [3].

A discrete characterization theorem for the discrete L1 linear approximation problem

A discrete characterization theorem for the discrete L1 linear approximation problem

Stability inequalities for discrete nonlinear two-point boundary value problems

For discretizations of nonlinear two-point boundary value problems we prove estimates of the difference quotients up to order n-1 in the discrete L∞-norm by the discrete L1 -norm of the discretization. In the hierarchy of LÞ -norms these estimates are best possible. For the linear case deep results P of Grigorieff and Kreiss are generalized.

Generators for Discrete Polynomial L1 Approximation Problems.

Polynomial approximation problems represent a class of specially structured problems which are frequently encountered in empirical curve-fitting. Two generators for creating such problems have been developed, implemented and used in the testing of discrete L 1 approximation codes. Both generators permit automatic generation of problems with specified characteristics and (for one generator) having known, unique and controllable solutions.

Examples of best discrete l1 and l2 rational approximations

Examples of best discrete l1 and l2 rational approximations

Two linear programming algorithms for the linear discrete 𝐿₁ norm problem

Computational studies by several authors have indicated that linear programming is currently the most efficient procedure for obtaining L 1 {L_1} , norm estimates for a discrete linear problem. However, there are several linear programming algorithms, and the "best" approach may depend on the problem’s structure (e.g., sparsity, triangularity, stability). In this paper we shall compare two published simplex algorithms, one referred to as primal and the other referred to as dual, and show that they are conceptually equivalent.

Solutions of minimum time problem and minimum fuel problem for discrete linear admissible control systems

In this paper the minimum time problem and the minimum fuel problem for discrete linear admissible control systems arc formulated (is one linear programming problem with two different objective functions. It is found that the obtained problem is very similar to the dual form of the linear programming problem for the discrete linear L1 approximation problem. An efficient dual simplex algorithm for the latter problem is used with the necessary modification* to obtain the solution of either of the former problems. Numerical results show that the present, method compares favourably with other known methods.

A method of virtual displacements for the degenerate discrete 𝑙₁ approximation problem

Given the system of equations \[ ∑ j = 1 n a i j x j = b i , i = 1 , … , m , \sum \limits _{j = 1}^n {{a_{ij}}{x_j} = {b_i},\quad i = 1, \ldots ,m,} \] let A i = ( a i 1 , … , a i n ) {A_i} = ({a_{i1}}, \ldots ,{a_{in}}) . It is known that if the matrix A = ( a i j ) A = ({a_{ij}}) has rank k ⩽ n k \leqslant n , then there is a point X which provides a minimum of \[ R ( X ) = ∑ i = 1 m | r i ( X ) | = ∑ i = 1 m | ( A i , X ) − b i | R(X) = \sum \limits _{i = 1}^m {|{r_i}(X)| = } \sum \limits _{i = 1}^m {|({A_i},X) - {b_i}|} \] such that r i ( X ) = 0 {r_i}(X) = 0 for at least k values of the index i. If r i ( X ) = 0 {r_i}(X) = 0 for exactly k values of the index i, the point or vertex is called ordinary, while if r i ( X ) = 0 {r_i}(X) = 0 for more than k values of i, the vertex is termed degenerate. A necessary and sufficient condition to determine if X minimizes R is valid if X is an ordinary vertex but not if X is degenerate. A degeneracy at X can be removed by applying perturbations to an appropriate number of the b i {b_i} so that X becomes an ordinary vertex of a modified problem. By noting that the test uses only values of the A i {A_i} , it is possible to avoid actual introduction of the perturbations to the b i {b_i} with a resulting substantial improvement of the efficiency of the computation.

Methodology and analysis for comparing discrete linear l1approximation codes

This is the first of a projected series of papers dealing with computational experimentation in mathematical programming. This paper provides early results of a test case using four discrete linear L1 approximation codes. Variables influencing code behavior are identified and measures of performance are specified. More importantly, an experimental design is developed for assessing code performance and is illustrated using the variable “problem size”.

An efficient method for the discrete linear 𝐿₁ approximation problem

An improved dual simplex algorithm for the solution of the discrete linear L 1 {L_1} approximation problem is described. In this algorithm certain intermediate iterations are skipped. This method is comparable with an improved simplex method due to Barrodale and Roberts, in both speed and number of iterations. It also has the advantage that in case of ill-conditioned problems, the basis matrix can lend itself to triangular factorization and can thus ensure a stable solution. Numerical results are given.

On the discrete linear L1 approximation and L1 solutions of overdetermined linear equations

Usow's algorithm for solving the discrete linear L 1 approximation problem is generalized so that it can also solve an Overdetermined system of linear equations in the L 1 norm. It is then shown that this algorithm is completely equivalent to a dual simplex algorithm applied to a linear programming problem in nonnegative bounded variables. However, one iteration in the former is equivalent to one or more iterations in the latter. A dual simplex algorithm is described which seems to be the most efficient and capable method for solving these two problems. Its efficiency is due to the absence of artificial variables and to its simplicity. Its capability is due to the fact that the Haar condition associated with Usow's method is completely relaxed. Numerical results are given.

On an algorithm for discrete nonlinear L1 approximation

On an algorithm for discrete nonlinear L1 approximation

An interval programming algorithm for discrete linear L1 approximation problems

An interval programming algorithm for discrete linear L1 approximation problems

On $L_1 $ Approximation II: Computation for Discrete Functions and Discretization Effects

On $L_1 $ Approximation II: Computation for Discrete Functions and Discretization Effects