Alberto Del Pia We consider the problem of minimizing a sparse quadratic function over the unit hypercube. In binary quadratic programming, treewidth of the interaction graph is a central parameter for tractability: bounded treewidth yields polynomial-time solvability. Motivated by this fact, we investigate whether treewidth plays a similar role when the binary domain is replaced by the unit … Read more
We show that every polytope $P\subseteq[0,1]^n$, and more generally every compact convex set, has Chv\’atal rank at most $12.22n^2+n\log_2 n+2n+4$. This improves the $O(n^2\log n)$ bound of Eisenbrand and Schulz and, together with the $\Omega(n^2)$ lower bound of Rothvo{\ss} and Sanit\`a, shows that the maximum Chv\’atal rank of a polytope in $[0,1]^n$ is $\Theta(n^2)$. More … Read more
We present an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem, and that runs on a Turing machine in time polynomial in the size of the instance and in 1/epsilon, provided that the number of integer variables and the number of negative eigenvalues of the Hessian of the objective … Read more
We study geometric characterizations of unbounded integer polynomial optimization problems. Unboundedness along a ray characterizes unbounded integer linear and quadratic optimization problems with rational coefficients. We show that this is no longer true in degree three, already in dimension three, in contrast with the continuous setting, where rays certify unboundedness up to degree three. To … Read more
While several classes of integer linear optimization problems are known to be solvable in polynomial time, far fewer tractability results exist for integer nonlinear optimization. In this work, we narrow this gap by identifying a broad class of discrete nonlinear optimization problems that admit polynomial-time algorithms. Central to our approach is the notion of projection-width, … Read more
We consider the multilinear polytope defined as the convex hull of the feasible region of a linearized binary polynomial optimization problem. We define a relaxation in an extended space for this polytope, which we refer to as the complete edge relaxation. The complete edge relaxation is stronger than several well-known relaxations of the multilinear polytope, … Read more
Sparse Principal Component Analysis (SPCA) is a fundamental technique for dimensionality reduction, and is NP-hard. In this paper, we introduce a randomized approximation algorithm for SPCA, which is based on the basic SDP relaxation. Our algorithm has an approximation ratio of at most the sparsity constant with high probability, if called enough times. Under a … Read more
In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial optimization. In this formulation, we assume that the variables are partitioned into a fixed number of sets, and … Read more
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