On First-Order Potential Reduction Algorithms for Linear Programmming∗

We describe primal, dual and primal-dual steepest-descent potential reduction methods for linear and convex minimization on the simplex in Rn and present their precise iteration complexity analyses depending on the problem dimension and the Lipschitz parameter of the constraint matrix or the convex objective function. In these methods, no matrix needs to be ever inversed so that they are pure firstorder methods. We also propose a reformulation of linear programming where a Gram matrix needs to be only inversed or factorized once, and present its iteration complexity bound that is independent of any Lipschitz constant. 1 Convex Optimization with the Simplex Constraint We consider the following optimization problem on the simplex: Minimize f(x) Subject to ex = 1, x ≥ 0; (1) where e is the vector of all ones. Such a problem in considered in [7] where a firs-order (primal) potential reduction algorithm was proposed. In this note we develop a primaldual first potential reduction method and prove its convergence speed is improved by a factor √ n, which method and its analysis resemble those in [9]. We assume that f(x) is a convex function in x ∈ R. Then, there is a (Wolf) dual problem Maximize λ+ f(z)−∇f(z) z Subject to ∇f(z)− eλ ≥ 0, e z = 1; z ≥ 0. (2) ∗This was a teaching note for course MS&E310, Linear Optimization