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This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
In this two-part paper, we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints. The algorithm solves a sequence of (separable) strongly convex problems and…
Motivated by the fact that not all nonconvex optimization problems are difficult to solve, we survey in this paper three widely-used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems.…
Understanding the optimization dynamics of neural networks is necessary for closing the gap between theory and practice. Stochastic first-order optimization algorithms are known to efficiently locate favorable minima in deep neural…
We propose a novel method to fit and segment multi-structural data via convex relaxation. Unlike greedy methods --which maximise the number of inliers-- this approach efficiently searches for a soft assignment of points to models by…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
A novel approach is introduced to a very widely occurring problem, providing a complete, explicit resolution of it: minimisation of a convex quadratic under a general quadratic, equality or inequality, constraint. Completeness comes via…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
Symmetry -- invariance to certain operators -- is a fundamental concept in many branches of physics. We propose ways to measure symmetric properties of vertices, and their surroundings, in networks. To be stable to the randomness inherent…
Optimization is offered as an objective approach to resolving complex, real-world decisions involving uncertainty and conflicting interests. It drives business strategies as well as public policies and, increasingly, lies at the heart of…
Most of the optimal guidance problems can be formulated as nonconvex optimization problems, which can be solved indirectly by relaxation, convexification, or linearization. Although these methods are guaranteed to converge to the global…
This work is a geometrical approach to the optimization problem motivated by transportation system management. First, an attempt has been made to furnish a comprehensive account of geometric programming based on the elementary Finsler…
Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative…
Deep neural networks are workhorse models in machine learning with multiple layers of non-linear functions composed in series. Their loss function is highly non-convex, yet empirically even gradient descent minimisation is sufficient to…
Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric…
In many areas of applied mathematics, engineering, and social and natural sciences, decentralization of information is a key aspect determining how to approach a problem. In this review article, we study information structures in a…
Stochastic optimization has found wide applications in minimizing objective functions in machine learning, which motivates a lot of theoretical studies to understand its practical success. Most of existing studies focus on the convergence…
Signal processing is rich in inherently continuous and often nonlinear applications, such as spectral estimation, optical imaging, and super-resolution microscopy, in which sparsity plays a key role in obtaining state-of-the-art results.…
We consider solving large scale nonconvex optimisation problems with nonnegativity constraints. Such problems arise frequently in machine learning, such as nonnegative least-squares, nonnegative matrix factorisation, as well as problems…
In this paper we observe that information theoretical concepts are valuable tools for extracting information from images and, in particular, information on image symmetries. It is shown that the problem of detecting reflectional and…