Related papers: Solving clustered low-rank semidefinite programs a…

In this paper, we propose a distributed algorithm for solving loosely coupled problems with chordal sparsity which relies on primal-dual interior-point methods. We achieve this by distributing the computations at each iteration, using…

In this work, we introduce an interior-point method that employs tensor decompositions to efficiently represent and manipulate the variables and constraints of semidefinite programs, targeting problems where the solutions may not be…

The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting…

In this paper, we study first-order methods on a large variety of low-rank matrix optimization problems, whose solutions only live in a low dimensional eigenspace. Traditional first-order methods depend on the eigenvalue decomposition at…

In this paper, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers,…

We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish…

Since more than three decades, interior-point methods proved very useful for optimization, from linear over semidefinite to conic (and partly beyond non-convex) programming; despite the fact that already in the semidefinite case (even when…

Primal-Dual Interior-Point methods are capable of solving constrained convex optimization problems to tight tolerances in a fast and robust manner. The derivatives of the primal-dual solution with respect to the problem matrices can be…

We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the…

We develop a first-order accelerated algorithm for a class of constrained bilinear saddle-point problems with applications to network systems. The algorithm is a modified time-varying primal-dual version of an accelerated mirror-descent…

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms we analyse are so-called short-step algorithms and they match the current best iteration complexity…

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations…

We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only…

In many high-dimensional problems, like sparse-PCA, planted clique, or clustering, the best known algorithms with polynomial time complexity fail to reach the statistical performance provably achievable by algorithms free of computational…

We use the lexicographic order to define a hierarchy of primal and dual bounds on the optimum of a bounded integer program. These bounds are constructed using lex maximal and minimal feasible points taken under different permutations. Their…

In this thesis, we settle the computational complexity of some fundamental questions in polynomial optimization. These include the questions of (i) finding a local minimum, (ii) testing local minimality of a point, and (iii) deciding…

Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context,…

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing…

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…

We show that sparsity constrained optimization problems over low dimensional spaces tend to have a small duality gap. We use the Shapley-Folkman theorem to derive both data-driven bounds on the duality gap, and an efficient primalization…