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We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced…
In this paper we consider the problem of distributed nonlinear optimisation of a separable convex cost function over a graph subject to cone constraints. We show how to generalise, using convex analysis, monotone operator theory and…
We present two parallel optimization algorithms for a convex function $f$. The first algorithm optimizes over linear inequality constraints in a Hilbert space, $\mathbb H$, and the second over a non convex polyhedron in $\mathbb R^n$. The…
We study over a number field, the iterates of automorphisms of the affine space. More precisely, we are interested in the periodic and non-periodic points; for the former the questions are similar to the ones about torsion points on abelian…
A numerical method is developed to solve linear semi-infinite programming problem (LSIP) in which the iterates produced by the algorithm are feasible for the original problem. This is achieved by constructing a sequence of standard linear…
This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in…
The note focuses on the differential geometric approach to the study of nonlinear systems that are affine in control. We first develop normal forms for nonlinear system affine in control. Based on these normal forms, we then address the…
Non-commutative polynomial optimization is a powerful technique with numerous applications in quantum nonlocality, quantum key distribution, causal inference, many-body physics, amongst others. The standard approach is to reduce such…
We give sublinear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
In the last years many results in the area of semidefinite programming were obtained for invariant (finite dimensional, or infinite dimensional) semidefinite programs - SDPs which have symmetry. This was done for a variety of problems and…
We give a simple, fast algorithm for hyperparameter optimization inspired by techniques from the analysis of Boolean functions. We focus on the high-dimensional regime where the canonical example is training a neural network with a large…
This paper presents a new approach, based on polynomial optimization and the method of moments, to the problem of anomaly detection. The proposed technique only requires information about the statistical moments of the normal-state…
The decentralized optimization paradigm assumes that each term of a finite-sum objective is privately stored by the corresponding agent. Agents are only allowed to communicate with their neighbors in the communication graph. We consider the…
In this article, we present a method for approximating affine processes on the cone of positive Hilbert-Schmidt operators using matrix-valued affine processes. By leveraging results from the theory on affine processes with values in the…
Border basis schemes are open subschemes of the Hilbert scheme of $\mu$ points in an affine space $\mathbb{A}^n$. They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space…
It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are…
We present a fully pseudo-spectral scheme to solve axisymmetric hyperbolic equations of second order. With the Chebyshev polynomials as basis functions, the numerical grid is based on the Lobbato (for two spatial directions) and Radau (for…
We present a coordinate ascent method for a class of semidefinite programming problems that arise in non-convex quadratic integer optimization. These semidefinite programs are characterized by a small total number of active constraints and…