We show that the most cost-efficient subset problem for linear programming games is NP-hard, and in fact inapproximable within a constant factor in polynomial time, unless P = NP. This in turn implies that computing the prices output by an egalitarian mechanism and computing a cost allocation in the equal split-off set for linear programming … Read more
Iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets. When the objective consists of a loss function plus a nonsmooth regularization term, the solution often lies on a low-dimensional manifold of parameter space along which the regularizer is smooth. … Read more
This short note considers and resolves the apparent contradiction between known worst-case complexity results for first and second-order methods for solving unconstrained smooth nonconvex optimization problems and a recent note by Jarre (2011) implying a very large lower bound on the number of iterations required to reach the solution’s neighbourhood for a specific problem with … Read more
The problem of finding a minimum l^1-norm solution to an underdetermined linear system is an important problem in compressed sensing, where it is also known as basis pursuit. We propose a heuristic optimality check as a general tool for l^1-minimization, which often allows for early termination by “guessing” a primal-dual optimal pair based on an … Read more
Mathematical programs whose formulation is symmetric often take a long time to solve using Branch-and-Bound type algorithms, because of the several symmetric optima. One of the techniques used to decrease the adverse effects of symmetry is adjoining symmetry breaking constraints to the formulation before solving the problem. These constraints aim to make some of the … Read more
In this paper we propose an algorithm for the global optimization of computationally expensive black–box functions. For this class of problems, no information, like e.g. the gradient, can be obtained and function evaluation is highly expensive. In many applications, however, a lower bound on the objective function is known; in this situation we derive a … Read more
Recently, Nemirovski’s analysis indicates that the extragradient method has the $O(1/t)$ convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, we have developed a class of Fej\’er monotone projection and contraction methods. Until now, only convergence results are available to these projection and contraction methods, though … Read more
Mixed-integer conic programming is a generalization of mixed-integer linear programming. In this paper, we present an extension of the duality theory for mixed-integer linear programming to the case of mixed-integer conic programming. In particular, we construct a subadditive dual for mixed-integer conic programming problems. Under a simple condition on the primal problem, we are able … Read more
We show that any nonconvex quadratically constrained quadratic program(QCQP) can be represented as a generalized copositive program. In fact,we provide two representations. The first is based on the concept of completely positive (CP) matrices over second order cones, while the second is based on CP matrices over the positive semidefinte cone. Our analysis assumes that … Read more
The rank function $\rank(\cdot)$ is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of $\rank(\cdot)$, and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the … Read more