We prove that the Douglas-Rachford splitting method converges, almost surely, to local minimizers of semialgebraic weakly convex optimization problems, under the assumption of the strict saddle property. The approach consists of two steps: first, we prove a manifold identification result, and local smoothness of the involved iteration operator. Then, we proceed to show that strict … Read more
The cosine measure was introduced in 2003 to quantify the richness of a finite positive spanning sets of directions in the context of derivative-free directional methods. A positive spanning set is a set of vectors whose nonnegative linear combinations span the whole space. The present work extends the definition of cosine measure. In particular, the … Read more
This study establishes the convergence of a cutting-plane algorithm tailored for a specific non-convex optimization problem. The presentation begins with the problem definition, accompanied by the necessary hypotheses that substantiate the application of a cutting plane. Following this, we develop an algorithm designed to tackle the problem. Lastly, we provide a demonstration that the sequence … Read more
Decision trees are one of the most famous methods for solving classification problems, mainly because of their good interpretability properties. Moreover, due to advances in recent years in mixed-integer optimization, several models have been proposed to formulate the problem of computing optimal classification trees. The goal is, given a set of labeled points, to split … Read more
Quadratic optimization (QO) has been studied extensively in the literature due to its applicability in many practical problems. While practical, it is known that QO problems are generally NP-hard. So, researchers developed many approximation methods to find good solutions. In this paper, we analyze QO problems using robust optimization techniques. To this end, we first … Read more
A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding … Read more
In this paper, we propose AdaBB, an adaptive gradient method based on the Barzilai-Borwein stepsize. The algorithm is line-search-free and parameter-free, and essentially provides a convergent variant of the Barzilai-Borwein method for general unconstrained convex optimization. We analyze the ergodic convergence of the objective function value and the convergence of the iterates for solving general … Read more
We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new … Read more
We derive the error bounds for multilinear terms in $[0,1]^n$ using a proof methodology based on the polyhedral representation of the convex hull. We extend the result for multilinear terms in $[\boldsymbol{L},\boldsymbol{0}] \times [\boldsymbol{0},\boldsymbol{U}]\subset\mathbb{R}^n$. ArticleDownload View PDF
We propose a Block Majorization Minimization method with Extrapolation (BMMe) for solving a class of multi-convex optimization problems. The extrapolation parameters of BMMe are updated using a novel adaptive update rule. By showing that block majorization minimization can be reformulated as a block mirror descent method, with the Bregman divergence adaptively updated at each iteration, … Read more