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
In this paper, we propose a smoothing sequential quadratic programming (SSQP) algorithm for solving a class of nonsmooth nonconvex, perhaps even non-Lipschitz minimization problems, which has wide applications in statistics and sparse reconstruction. At each step, the SSQP algorithm solves a strongly convex quadratic minimization problem with a diagonal Hessian matrix, which has a simple … Read more
We consider a new class of huge-scale problems, the problems with {\em sparse subgradients}. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends {\em logarithmically} in the dimension. This technique is … Read more
In this paper, we are interested in the development of efficient first-order methods for convex optimization problems in the simultaneous presence of smoothness of the objective function and stochasticity in the first-order information. First, we consider the Stochastic Primal Gradient method, which is nothing else but the Mirror Descent SA method applied to a smooth … Read more
The contributions of the first half of this thesis are on the computational and algebraic aspects of convexity in polynomial optimization. We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves … Read more
It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this … Read more
We are given a digraph G = (N, A), where each arc is colored with one among k given colors. We look for a spanning arborescence T of G rooted at a given node and having minimum changeover cost. We call this the Minimum Changeover Cost Arborescence problem. To the authors’ knowledge, it is a … Read more
We consider the problem of deleting a limited number of nodes from a graph in order to minimize a connectivity measure between the surviving nodes. We prove that the problem is $NP$-complete even on quite particular types of graph, and define a dynamic programming recursion that solves the problem in polynomial time when the graph … Read more
In 1996, Laurent and Poljak introduced an extremely general class of cutting planes for the max-cut problem, called gap inequalities. We prove several results about them, including the following: (i) there must exist non-dominated gap inequalities with gap larger than 1, unless NP = co-NP; (ii) there must exist non-dominated gap inequalities with exponentially large … Read more
In this paper, we consider block-decomposition first-order methods for solving large-scale conic semidefinite programming problems. Several ingredients are introduced to speed-up the method in its pure form such as: an aggressive choice of stepsize for performing the extragradient step; use of scaled inner products in the primal and dual spaces; dynamic update of the scaled … Read more