In this paper we consider global optimization algorithms based on multiple local searches for the Molecular Distance Geometry Problem (MDGP). Three distinct approaches (Multistart, Monotonic Basin Hopping, Population Basin Hopping) are presented and for each of them a computational analysis is performed. The results are also compared with those of two other approaches in the … Read more
We present a two-phase algorithm for solving large-scale quadratic programs (QPs). In the first phase, gradient-projection iterations approximately minimize an augmented Lagrangian function and provide an estimate of the optimal active set. In the second phase, an equality-constrained QP defined by the current inactive variables is approximately minimized in order to generate a second-order search … Read more
In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization … Read more
This paper addresses the problem of finding cutting planes for multi-stage stochastic integer programs. We give a general method for generating cutting planes for multi-stage stochastic integer programs based on combining inequalities that are valid for the individual scenarios. We apply the method to generate cuts for a stochastic version of a dynamic knapsack problem … Read more
Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but-bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of {\sl robust counterpart} of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links … Read more
Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient … Read more
The aim of this paper is to study how efficiently we evaluate a system of multivariate polynomials and their partial derivatives in homotopy continuation methods. Our major tool is an extension of the Hornor scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there … Read more
Given an n by n symmetric possibly indefinite matrix A, a modified Cholesky algorithm computes a factorization of the positive definite matrix A+E, where E is a correction matrix. Since the factorization is often used to compute a Newton-like downhill search direction for an optimization problem, the goals are to compute the modification without much … Read more
A $k$-bit Hamming prefix code is a binary code with the following property: for any codeword $x$ and any prefix $y$ of another codeword, both $x$ and $y$ having the same length, the Hamming distance between $x$ and $y$ is at least $k$. Given an alphabet $A = [a_1,\ldots,a_n]$ with corresponding probabilities $[p_1,\ldots,p_n]$, the $k$-bit … Read more