The problem of finding a minimum l^1-norm solution to an underdetermined linear system is an important problem in compressed sensing, where it is also known as basis pursuit. We propose a heuristic optimality check as a general tool for l^1-minimization, which often allows for early termination by “guessing” a primal-dual optimal pair based on an … Read more
We propose a modification to the (generalized) bundle scheme for minimization of a convex nondifferentiable sum-function in the case where some of the components are “easy”, that is, they are Lagrangian functions of explicitly known convex programs with “few” variables and constraints. This happens in many practical cases, particularly within applications to combinatorial optimization. In … Read more
In this paper, we propose a simple variant of the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming problem (LP). Our variant executes a natural finite termination procedure at each iteration and it is easy to implement the algorithm. Our algorithm admits an objective-function free polynomial-time complexity when it is applied to LPs whose dual feasible region … Read more
In their 1972 paper ‘How good is the simplex algorithm ?’ Klee and Minty present a class of problems the simplex algorithm for linear programming (LP) is not able to solve in a polynomial way. Later developments have resulted in algorithms by Khachiyan and Karmarkar that do solve LP in a polynomial way, although the … Read more
We consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic center of a polytope. We show that the problem has a dual of the same form and give complexity results for several different … Read more
The paper proposes a polynomial algorithm for solving systems of linear inequalities. The algorithm uses a polynomial relaxation-type procedure which either finds a solution for Ax = b, 0
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the dual simplex method with the most negative pivoting rule for LP. The bound is comparable with the bound given by Kitahara and Mizuno (2010) for the primal simplex method. We apply the result to the … Read more
We give an upper bound for the number of different basic feasible solutions generated by Dantzig’s simplex method (the simplex method with the most negative pivoting rule) for LP with bounded variables by extending the result of Kitahara and Mizuno (2010). We refine the analysis by them and improve an upper bound for a standard … Read more
The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, … Read more
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the … Read more