We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different Integer Linear Programming (ILP) based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP-relaxation and the original problem and then introduce a new core concept for the MKP, … Read more
Recently the proximal bundle method for minimizing a convex function has been extended to an inexact oracle that delivers function and subgradient values of unknown accuracy. We adapt this method to a partially inexact oracle that becomes exact only when an objective target level for a descent step is met. In Lagrangian relaxation, such oracles … Read more
We introduce a general-purpose learning machine that we call the “Guaranteed Error Machine”, or GEM, and two learning algorithms, a “real GEM algorithm” and an “ideal GEM algorithm”. The real GEM algorithm is for use in real applications, while the ideal GEM algorithm is introduced as a theoretical tool; however, these two algorithms have identical … Read more
In this paper we analyze the iteration-complexity of the hybrid proximal extragradient (HPE) method for finding a zero of a maximal monotone operator recently proposed by Solodov and Svaiter. One of the key points of our analysis is the use of new termination criteria based on the $\varepsilon$-enlargement of a maximal monotone operator. The advantage … Read more
The mixing set with a knapsack constraint arises in deterministic equivalent of probabilistic programming problems with finite discrete distributions. We first consider the case that the probabilistic program has equal probabilities for each scenario. We study the resulting mixing set with a cardinality constraint and propose facet-defining inequalities that subsume known explicit inequalities for this … Read more
Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex ptimization problems. In this paper, we propose a simple and fast algorithm for signal reconstruction from partial Fourier data. The algorithm minimizes the sum of three terms corresponding to total variation, … Read more
This paper presents a biased random-keys genetic algorithm for the Resource Constrained Project Scheduling Problem. The chromosome representation of the problem is based on random keys. Active schedules are constructed using a priority-rule heuristic in which the priorities of the activities are defined by the genetic algorithm. A forward-backward improvement procedure is applied to all … Read more
We propose a fast algorithm for solving the l1-regularized least squares problem for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative method called “shrinkage” yields an estimate of the subset of components … Read more
In this paper a new algorithm to locally minimize nonsmooth, nonconvex functions is developed. We introduce the notion of secants and quasisecants for nonsmooth functions. The quasisecants are applied to find descent directions of locally Lipschitz functions. We design a minimization algorithm which uses quasisecants to find descent directions. We prove that this algorithm converges … Read more
A multivariate polynomial $p(x)=p(x_1,…,x_n)$ is sos-convex if its Hessian $H(x)$ can be factored as $H(x)= M^T(x) M(x)$ with a possibly nonsquare polynomial matrix $M(x)$. It is easy to see that sos-convexity is a sufficient condition for convexity of $p(x)$. Moreover, the problem of deciding sos-convexity of a polynomial can be cast as the feasibility of … Read more