In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods … Read more
We present a number of new algorithmic ideas for solving LP relaxations of extremely large precedence constrained production scheduling problems. These ideas are used to develop an implementation that is tested on a variety of real-life, large scale instances; yielding optimal solutions in very practicable CPU time. Citation Unpublished. Columbia University, BHP Billiton, August 2009. … Read more
Arc-search is developed in optimization algorithms. In this paper, simple analytic arcs are used to approximate the central path of the linear programming. The primal-dual path-following interior-point method is then used to search optimizers along the arcs for linear programming. They require fewer iterations to find the optimal solutions in all the tested problems in … Read more
The Farkas Lemma is extended to simultaneous linear operator and polyhedral sublinear operator inequalities by Boolean valued analysis. Citation Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia Article Download View The Farkas Lemma Revisited
We formalize an algorithm for solving the L1-norm best-fit hyperplane problem derived using first principles and geometric insights about L1 projection and L1 regression. The procedure follows from a new proof of global optimality and relies on the solution of a small number of linear programs. The procedure is implemented for validation and testing. This … Read more
In this paper we develop and analyze an efficient algorithm which solves seven related optimization problems simultaneously, in relative scale. Each iteration of our method is very cheap, with main work spent on matrix-vector multiplication. We prove that if a certain sequence generated by the algorithm remains bounded, then the method must terminate in $O(1/\delta)$ … Read more
It has recently been shown (Burer, 2006) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs (CPPs), i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are necessarily NP-hard. A basic tractable relaxation … Read more
Based on extensive computational evidence (hundreds of thousands of randomly generated problems) the second author conjectured that $\bar{\kappa}(\zeta)=1$, which is a factor of $\sqrt{2n}$ better than that has been proved, and which would yield an $O(\sqrt{n})$ iteration full-Newton step infeasible interior-point algorithm. In this paper we present an example showing that $\bar{\kappa}(\zeta)$ is in the … Read more
This paper proposes an infeasible interior-point algorithm with full-Newton step for linear programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110–1136, 2006). We introduce a kernel function in the algorithm. For $p\in[0,1)$, the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods, … Read more
In this paper we study linear optimization problems with multi-dimensional linear positive second-order stochastic dominance constraints. By using the polyhedral properties of the second- order linear dominance condition we present a cutting-surface algorithm, and show its finite convergence. The cut generation problem is a difference of convex functions (DC) optimization problem. We exploit the polyhedral … Read more