Class Of Linear Programming Problems Research Articles

TESTING AND VALIDATION PROBLEMS IN STOCHASTIC LINEAR PROGRAMMING

A class of linear programming problems is analyzed here, where some of the parameters are estimated by methods like least squares and are therefore stochastic. Since the decision vector may also be stochastic in this framework, this leads to bilinear problems in stochastic programming in suitable cases. Further, the conventional methods for applying programming models using regression estimates for its parameters appear to raise questions of validation of such models. Some aspects of the validation problems involving the risk aversion parameter are also analyzed here.

Dual existence theorems for a special class of linear programming problems in reflexive Banach lattices

In this work the existence of dual optimal solutions for a special class of linear programming problems in a reflexive Banach space is investigated. Then these statements are applied to linear optimization problems with Noethebian operator-constraints. Finally, a maximal condition for an optimal control problem with Noehebiari operator: constraints its Derived in L ∞[0,T].

Technical Note—Statistical Measures for Linear Functions on Polytopes

This note presents a technique for obtaining a random sample from an arbitrary N-dimensional bounded convex polyhedral set (polytope). The need for computing the expected value and variance measures for linear functions defined on such sets arises in multiattribute decision problems and in certain classes of linear programming problems. We develop the method and produce formulations for estimating the statistical measures for linear functions. A simple example is presented.

The Generalized Inverse in Linear Programming—An Intersection Projection Method and the Solution of a Class of Structured Linear Programming Problems

An iterative procedure yielding a nonnegative fixed point associated with the LP problem, maximize $( {x,c} )$ where $x \in \Lambda = \{ { x |Ax = b,x\geqq \theta } \}$, is developed. This procedure uses a perpendicular projection matrix formed from A, b, c, and $A^\dag $ (the Moore–Penrose–Bjerhammer generalized inverse of b). A class of linear programming problems is considered for which $A^\dag $ and related projection matrices can be obtained explicitly. This class of problems includes the Hitchcock–Koopmans transportation problem and certain of its multidimensional extensions as special cases.

Technical Note—Solutions of a Special Class of Linear Programming Problems

Explicit solutions for linear problems of the form: maximize (h, x), subject to Ax ≦ b, b ∈ R(A), are given, as well as explicit solutions for its dual problem. This note replaces the full row rank assumption on A of earlier results of Ben-Israel and Charnes by the assumption b ∈ R(A). With this assumption, explicit solutions are derived for the dual problem that are different from related results of Zlobec and Ben-Israel.

An "Out-of-Kilter" network tutorial

Network flow algorithms have been developed to handle a special class of linear programming problems. These problems may each be described by a network whose links carry flow. Network algorithms take advantage of this special structure to produce an optimal solution much more quickly, with less storage required, and with virtually no round off error in comparison with general linear programming codes. The objective of this paper is to introduce the reader to the basic ideas involved in formulating a network problem, and to explain in a non-technical manner what the "out-of-kilter" network algorithm (1) <u>does</u>. This algorithm is very general and extremely efficient in solving network flow cost problems (as opposed to CPM, PERT, and GERT type networks).

An economic interpretation of duality in linear programming

This paper develops new duality relations in linear programming which give new economic interpretations to the dual problem. These duality relations are part of a new closed-form solution to an economically interesting class of linear programming problems. A generalization of these results to geometric programming is announced, and potential extensions to integer programming as well as potential applications to stochastic programming are indicated.

A Decomposition Method for Interval Linear Programming

An interval linear program is [Formula: see text] where the matrix A, vectors b−, b+, and c are given. If A has full row rank, the optimal solutions of (IP) can be written explicitly (A. Ben-Israel and A. Charnes: “An explicit solution of a special class of linear programming problems,” Operations Research 16 (1968), 1166–1175). This result is used in conjunction with the Danteig-Wolfe decomposition principle to develop a finite iterative technique for solving the general (IP). Since any bounded linear program may be cast in form (IP) the technique may also be considered as an alternative method for linear programming.

Letter to the Editor—An Approach to Some Structured Linear Programming Problems

A recent paper by J. M. Bennett describes a decomposition algorithm for the class of linear programming problems commonly called “angular systems.” This note draws attention to a variant of the algorithm that is particularly suited to problems in which certain submatrices are sparse.

An Explicit Solution of a Special Class of Linear Programming Problems

The linear programs considered here are of the form: [Formula: see text] where A is of full row rank, and (LP) is feasible with bounded optimal solutions. The main result is an explicit representation of the general optimal solution of (LP) in terms of a generalized inverse of A. This explicit solution of (LP)—explicit in the sense that A−1b is an explicit solution of Ax = b—has obvious theoretical (and possibly computational) advantages over the well-known iterative methods of linear programming. The results are illustrated by a simple example, and extensions to general linear programs are discussed.

A class of linear programming problems requiring a large number of iterations

Abstract : A coordinate-free description of the simplex algorithm (for nondegenerate linear programming problems) is supplied, and is used to show that the number of iterations can be larger than was previously known. For 0 m n, there is constructed a nondegenerate linear programming problem whose bounded (n - m)-dimensional feasible region is defined by means of m linear equality constraints in n nonnegative variables, and in which, after starting from the worst choice of an initial feasible vertex, m(n - m - l) + 1 simplex iterations are required in order to reach the optimal vertex. It is conjectured that this is the maximum possible number of iterations (for arbitrary 0 m n), but the conjecture is proved only for n m + 4. (Author)