Oracle Based Optimization (OBO) conveniently designates an approach to handle a class of convex optimization problems in which the information pertaining to the function to be minimized and/or to the feasible set takes the form of a linear outer approximation revealed by an oracle. We show, through three representative examples, how difficult problems can be … Read more
The global convergence properties of a class of penalty methods for nonlinear programming are analyzed. These methods include successive linear programming approaches, and more specifically, the successive linear-quadratic programming approach presented by Byrd, Gould, Nocedal and Waltz (Math. Programming 100(1):27–48, 2004). Every iteration requires the solution of two trust-region subproblems involving piecewise linear and quadratic … Read more
The second-order cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Q-quadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples … Read more
We extend the Q method to the symmetric cone programming. An infeasible interior point algorithm and a Newton-type algorithm are given. We give convergence results of the interior point algorithm and prove that the Newton-type algorithm is good for Citation AdvOl-Report#2004/18 McMaster University, Advanced Optimization Laboratory Hamilton, Ontario, Canada October 2004 Article Download View The … Read more
The conjugate gradient (CG) algorithm is well-known to have excellent theoretical properties for solving linear systems of equations $Ax = b$ where the $n\times n$ matrix $A$ is symmetric positive definite. However, for extremely ill-conditioned matrices the CG algorithm performs poorly in practice. In this paper, we discuss an adaptive preconditioning procedure which improves the … Read more
This paper presents a new primal dual simplex method and investigates the duality formation implying in classical simplex methods. We reviews classical simplex methods for linear programming problems and give a detail discussion for the relation between modern and classical algorithms. The two modified versions are present. The advantages of the new algorithms are simplicity … Read more
We adapt a method due to Nesterov so as to obtain an algorithm for solving block-angular fractional packing or covering problems to relative tolerance epsilon, while using a number of iterations that grows polynomially in the size of the problem and whose dependency on epsilon is proportional to 1/epsilon. Citation CORC report TR-2004-09, Computational Optimization … Read more
The DIRECT algorithm is a deterministic sampling method for bound constrained Lipschitz continuous optimization. We prove a subsequential convergence result for the DIRECT algorithm that quantifies some of the convergence observations in the literature. Our results apply to several variations on the original method, including one that will handle general constraints. We use techniques from … Read more
Lagrangian relaxation is a popular technique to solve difficult optimization problems. However, the applicability of this technique depends on having a relatively low number of hard constraints to dualize. When there are many hard constraints, it may be preferable to relax them dynamically, according to some rule depending on which multipliers are active. From the … Read more
Given a convex body S and a point x \in S, let sym(x,S) denote the symmetry value of x in S: sym(x,S):= max{t : x + t(x – y) \in S for every y \in S}, which essentially measures how symmetric S is about the point x, and define sym(S):=\max{sym(x,S) : x \in S }. … Read more