We propose an active-set algorithm for smooth multiobjective optimization problems subject to box constraints. The method works on one face of the feasible set at a time, treating it as a lower-dimensional region on which the problem simplifies. At each iteration, the algorithm decides whether to remain on the current face or to move to … Read more
This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming relaxations for which the size of the semidefinite blocks is determined by the canonical polyadic rank rather than the number of variables. As a result, LRPOP can … Read more
Suppose we wish to determine the quality of a candidate solution to a convex stochastic program in which the objective function is a statistical functional parameterized by the decision variable and known deterministic constraints may be present. Inspired by stopping criteria in primal-dual and interior-point methods, we develop cancellation theorems that characterize the convergence of … Read more
This work addresses the Close-Enough Traveling Salesman Problem (CETSP), a variant of the classic traveling salesman problem in which we seek to visit neighborhoods of points in the plane (defined as disks) rather than specific points. We present two exact formulations for this problem based on second-order cone programming (SOCP), along with approximated mixed-integer linear … Read more
Consider the primal problem of minimizing the sum of two closed proper convex functions \(f\) and \(g\). If the relative interiors of the domains of \(f\) and \(g\) intersect, then the primal problem and its corresponding Fenchel dual satisfy strong duality. When these relative interiors fail to intersect, pathologies and numerical difficulties may occur. In … Read more
We investigate questions about the cone \(\mathrm{SEP}_n\) of separable bipartite states, consisting of the Hermitian matrices acting on \(\mathbb{C}^n\otimes\mathbb{C}^n\) that can be written as conic combinations of rank one matrices of the form \(xx^*\otimes yy^*\) with \(x,y\in\mathbb{C}^n\). Bipartite states that are not separable are said to be entangled. Detecting quantum entanglement is a fundamental task … Read more
For solution mappings of parameterized models (such as optimization problems, variational inequalities, and generalized equations), standard stability inevitably fails as the parameter approaches the boundary of the feasible domain. One remedy is relative stability restricted to a constraint set (e.g., the feasible domain), which is our focus in this paper. We establish generalized differentiation criteria … Read more
In this paper, we propose an extension of the classical weight reduction inequalities for the binary knapsack polytope for settings where the maximum-weight item in the associated pack is not unique. We derive sufficient conditions under which the extended inequalities are facet-defining and identify conditions under which they strictly dominate the original weight reduction inequalities. … Read more
Graph-based machine learning has seen an increased interest over the last decade with many connections to other fields of applied mathematics. Learning based on partial differential equations, such as the phase-field Allen-Cahn equation, allows efficient handling of semi-supervised learning approaches on graphs. The numerical solution of the graph Allen-Cahn equation via a convexity splitting or … Read more
The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar’s Positivstellensatz, which requires the feasible set to satisfy the algebraic Archimedean property. In this paper, we introduce a regularized moment-SOS hierarchy capable of handling problems on unbounded sets or bounded sets violating the Archimedean … Read more