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Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP)…
Region extraction is necessary in a wide range of applications, from object detection in autonomous driving to analysis of subcellular morphology in cell biology. There exist two main approaches: convex hull extraction, for which exact and…
This paper introduces Roundabout Constrained Convex Generators (RCGs), a set representation framework for modeling multiply connected regions in control and verification applications. The RCG representation extends the constrained convex…
The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a similarity space and concepts are represented by convex regions in this space. After pointing…
Optimization problems with set-valued objective functions arise in contexts such as multi-stage optimization with vector-valued objectives. The aim is to identify an optimizer -- a feasible point with an optimal objective value -- based on…
High-order tensor methods for solving both convex and nonconvex optimization problems have generated significant research interest, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's…
A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere.…
This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex…
We introduce the Suggest-and-Improve framework for general nonconvex quadratically constrained quadratic programs (QCQPs). Using this framework, we generalize a number of known methods and provide heuristics to get approximate solutions to…
We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are…
Let $C$ be a proper convex cone generated by a compact set which supports a measure $\mu$. A construction due to A.Barvinok, E.Veomett and J.B. Lasserre produces, using $\mu$, a sequence $(P_k)_{k\in \mathbb{N}}$ of nested spectrahedral…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional…
The asymptotic analysis of a generic stochastic optimization algorithm mainly relies on the establishment of a specific descent condition. While the convexity assumption allows for technical shortcuts and generally leads to strict…
Statistical problems often involve linear equality and inequality constraints on model parameters. Direct estimation of parameters restricted to general polyhedral cones, particularly when one is interested in estimating low dimensional…
Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the…
We introduce a numerical framework to verify the finite step convergence of first-order methods for parametric convex quadratic optimization. We formulate the verification problem as a mathematical optimization problem where we maximize a…
The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of…
Adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity amongst second-order methods. In this paper, we propose and…
We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the…