Eliciting the utility of a decision maker is difficult. In this paper, we develop a flexible decision making framework, which uses the concept of utility robustness to address the problem of ambiguity and inconsistency in utility assessments. The ideas are developed by giving a probabilistic interpretation to utility and marginal utility functions. Boundary and additional … Read more
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most $k$. We show that this problem is $\NP$-hard for any fixed integer $k\ge 2$. Equivalently, for $k\ge 2$, it is $\NP$-hard to test membership in the … Read more
We consider solving the $\ell_1$-regularized least-squares ($\ell_1$-LS) problem in the context of sparse recovery, for applications such as compressed sensing. The standard proximal gradient method, also known as iterative soft-thresholding when applied to this problem, has low computational cost per iteration but a rather slow convergence rate. Nevertheless, when the solution is sparse, it often … Read more
After an introduction to main ideas of semi-infinite optimization, this article surveys recent developments in theory and numerical methods for standard and generalized semi-infinite optimization problems. Particular attention is paid to connections with mathematical programs with complementarity constraints, lower level Wolfe duality, semi-smooth approaches, as well as branch and bound techniques in adaptive convexification procedures. … Read more
In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is … Read more
Given the steady increase in cores per CPU, it is only a matter of time until supercomputers will have a million or more cores. In this article, we investigate the opportunities and challenges that will arise when trying to utilize this vast computing power to solve a single integer linear optimization problem. We also raise … Read more
We present a new tree-search algorithm for solving mixed-integer nonlinear programs (MINLPs). Rather than relying on computationally expensive nonlinear solves at every node of the branch-and-bound tree, our algorithm solves a quadratic approximation at every node. We show that the resulting algorithm retains global convergence properties for convex MINLPs, and we present numerical results on … Read more
In this paper, we address the problem of preconditioning sequences of large sparse nonsymmetric systems of linear equations and present two new strategies to construct approximate updates of factorized preconditioners. Both updates are based on the availability of an incomplete LU (ILU) factorization for one matrix of the sequence and differ in the approximation of … Read more
We introduce a natural strengthening of Gomory-Chvátal cutting planes for the important class of 0/1-integer programming problems and study the properties of the elementary closure that arises from the new class of cuts. Most notably, we prove that the new closure is polyhedral, we characterize the family of all facet-defining inequalities, and we compare it … Read more
In this paper we consider minimization problems with constraints. We show that if the set of constaints is a Riemannian manifold of non positive curvature and the objective function is lower semicontinuous and satisfi es the Kurdyka-Lojasiewicz property, then the alternating proximal algorithm in Euclidean space is naturally extended to solve that class of problems. We … Read more