Related papers: Lifting for Simplicity: Concise Descriptions of Co…

Necessary and sufficient conditions for convexity and strong convexity, respectively, of sublevel sets that are defined by finitely many real-valued $C^{1,1}$-maps are presented. A novel characterization of strongly convex sets in terms of…

Convex polyhedra are the basis for several abstractions used in static analysis and computer-aided verification of complex and sometimes mission critical systems. For such applications, the identification of an appropriate…

We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral…

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain…

In this paper we study convex subcomplexes of spherical buildings. We pay special attention to fixed point sets of type-preserving isometries of spherical buildings. This sets are also convex subcomplexes of the natural polyhedral structure…

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite…

Lifting is an efficient technique to scale up graphical models generalized to relational domains by exploiting the underlying symmetries. Concurrently, neural models are continuously expanding from grid-like tensor data into structured…

Convex polyhedral abstractions of logic programs have been found very useful in deriving numeric relationships between program arguments in order to prove program properties and in other areas such as termination and complexity analysis. We…

A set in the Euclidean plane is said to be biconvex if, for some angle $\theta\in[0,\pi/2)$, all its sections along straight lines with inclination angles $\theta$ and $\theta+\pi/2$ are convex sets (i.e, empty sets or segments).…

Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some non-convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets…

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance…

We propose an abstract definition of convex spaces as sets where one can take convex combinations in a consistent way. A priori, a convex space is an algebra over a finitary version of the Giry monad. We identify the corresponding Lawvere…

In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they…

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and…

In this work we deal with the so-called path convexities, defined over special collections of paths. For example, the collection of the shortest paths in a graph is associated with the well-known geodesic convexity, while the collection of…

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here projection methods are iterative…

A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of…

A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added…

Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic…

Convex clustering is a recent stable alternative to hierarchical clustering. It formulates the recovery of progressively coalescing clusters as a regularized convex problem. While convex clustering was originally designed for handling…