For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computatio...

We consider multi-target linear-quadratic control problem on semiinfinite interval. We show that the problem can be reduced to a simple convex optimization problem on the simplex

We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods.

Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. We consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problem...

The random simplex algorithm for linear programming proceeds as follows: at each step, it moves from a vertex u of the polytope to a randomly chosen neighbor of u, the random choice being made from those neighbors of u that improve the objective function. We exhibit a polytope defined by n constraints in three dimensions with height O(log n), for which the expected running time of the random si...

The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions – computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses. 1. Asyptotic convex geometry and Linear Programming Linear Programming studies the problem of ma...

An improved dual simplex algorithm for the solution of the discrete linear Lj 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 len...