Regularity of sets under a reformulation in a product space of reduced dimension

Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze how some regularity properties are preserved under a reformulation in a product space of reduced dimension. This allows us to establish local linear convergence of parallel projection methods which are constructed through this reformulation

A new projection method for finding the closest point in the intersection of convex sets

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces

Regularity of Sets Under a Reformulation in a Product Space with Reduced Dimension

Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze how some regularity properties are preserved under a reformulation in a product space of reduced dimension. This allows us to establish local linear convergence of parallel projection methods which are constructed through this reformulation.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The author was partially supported by the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PID2022-136399NB-C21, and by the Generalitat Valenciana (AICO/2021/165)

A feasibility approach for constructing combinatorial designs of circulant type

In this work, we propose an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelations. The problem is formulated as a so-called feasibility problem having three sets, to which the Douglas-Rachford projection algorithm is applied. The approach is illustrated on three different classes of circulant combinatorial designs: circulant weighing matrices, D-optimal matrices, and Hadamard matrices with two circulant cores. Furthermore, we explicitly construct two new circulant weighing matrices, a $CW(126,64)$ and a $CW(198,100)$, whose existence was previously marked as unresolved in the most recent version of Strassler's table

A product space reformulation with reduced dimension for splitting algorithms

In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra’s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments

An enhanced formulation for solving graph coloring problems with the Douglas–Rachford algorithm

We study the behavior of the Douglas–Rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In spite of the combinatorial nature of this problem, the Douglas–Rachford algorithm was recently shown to be a successful heuristic for solving a wide variety of graph coloring instances, when the problem was cast as a feasibility problem on binary indicator variables. In this work we consider a different formulation, based on semidefinite programming. The much improved performance of the Douglas–Rachford algorithm, with this new approach, is demonstrated through various numerical experiments.F. J. Aragón Artacho and R. Campoy were partially supported by MICINN of Spain and ERDF of EU, Grants MTM2014-59179-C2-1-P and PGC2018-097960-B-C22. F. J. Aragón Artacho was supported by the Ramón y Cajal program by MINECO of Spain and ERDF of EU (RYC-2013-13327) and R. Campoy was supported by MINECO of Spain and ESF of EU (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”

The Boosted DC Algorithm for Linearly Constrained DC Programming

The Boosted Difference of Convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical Difference of Convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions programs with linear constraints, and prove that every cluster point of the sequence generated by this algorithm is a Karush–Kuhn–Tucker point of the problem if the feasible set has a Slater point. When the objective function is quadratic, we prove that any sequence generated by the algorithm is bounded and R-linearly (geometrically) convergent. Finally, we present some numerical experiments where we compare the performance of DCA and BDCA on some challenging problems: to test the copositivity of a given matrix, to solve one-norm and infinity-norm trust-region subproblems, and to solve piecewise quadratic problems with box constraints. Our numerical results demonstrate that this new extension of BDCA outperforms DCA.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission (PGC2018-097960-B-C22), and by the Generalitat Valenciana (AICO/2021/165). PTV was supported by Vietnam Ministry of Education and Training Project hosting by the University of Technology and Education, Ho Chi Minh City Vietnam (2023-2024)