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We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a…
Bilevel optimization has gained prominence in various applications. In this study, we introduce a framework for solving bilevel optimization problems, where the variables in both the lower and upper levels are constrained on Riemannian…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…
Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…
This paper studies the problem of enumerating all maximal collinear subsets of size at least three in a given set of $n$ points. An algorithm for this problem, besides solving degeneracy testing and the exact fitting problem, can also help…
This paper proposes a fast and accurate surface normal estimation method which can be directly used on depth maps (organized point clouds). The surface normal estimation process is formulated as a closed-form expression. In order to reduce…
In computed tomography, the approximation quality of a scan of a physical object is typically limited by the acquisition modalities, especially the hardware including X-ray detectors. To improve upon this, we experiment with a…
Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By…
We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact…
With the rapid development of high-resolution 3D vision applications, the traditional way of manipulating surface detail requires considerable memory and computing time. To address these problems, we introduce an efficient surface detail…
The quest for regular models of arithmetic surfaces allows different viewpoints and approaches: using valuations or a covering by charts. In this article, we sketch both approaches and then show in a concrete example, how surprisingly…
It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization process towards local (and non-physical) minima, and by the…
Energies with high-order non-submodular interactions have been shown to be very useful in vision due to their high modeling power. Optimization of such energies, however, is generally NP-hard. A naive approach that works for small problem…
A basic issue in optimization, inverse theory,neural networks, computational chemistry and many other problems is the geometrical characterization of high dimensional functions. In inverse calculations one aims to characterize the set of…
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…
A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Al- though censuses are useful resources for mathematicians, constructing them is difficult: the best…
Hidden convexity is a powerful idea in optimization: under the right transformations, nonconvex problems that are seemingly intractable can be solved efficiently using convex optimization. We introduce the notion of a Lagrangian dual…
We focus on the robust principal component analysis (RPCA) problem, and review a range of old and new convex formulations for the problem and its variants. We then review dual smoothing and level set techniques in convex optimization,…
We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this…