Enlargements of positive sets

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set $A$ we associate a family of enlargements \E(A) and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of \E(A) and the family of so-called representative functions of $A$. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.Comment: 16 page

Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of them delivering interesting byproducts.Comment: 11 pages, no figure

Alternating proximal-gradient steps for (stochastic) nonconvex-concave minimax problems

Minimax problems of the form $\min_x \max_y \Psi(x,y)$ have attracted increased interest largely due to advances in machine learning, in particular generative adversarial networks. These are typically trained using variants of stochastic gradient descent for the two players. Although convex-concave problems are well understood with many efficient solution methods to choose from, theoretical guarantees outside of this setting are sometimes lacking even for the simplest algorithms. In particular, this is the case for alternating gradient descent ascent, where the two agents take turns updating their strategies. To partially close this gap in the literature we prove a novel global convergence rate for the stochastic version of this method for finding a critical point of $g(\cdot) := \max_y \Psi(\cdot,y)$ in a setting which is not convex-concave

SSDB spaces and maximal monotonicity

In this paper, we develop some of the theory of SSD spaces and SSDB spaces, and deduce some results on maximally monotone multifunctions on a reflexive Banach space.Comment: 16 pages. Written version of the talk given at IX ISORA in Lima, Peru, October 200

Convergence Rates of First and Higher Order Dynamics for Solving Linear Ill-posed Problems

Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov's algorithm and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), respectively. In this paper we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residuum of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. We prove that the proposed flows for minimising the residuum of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions. In particular we show that in comparison to convex analysis results the rates can be significantly higher, which is possible by constraining the investigations to the particular convex energy functional, which is the squared norm of the residuum

A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.FJAA and DTB were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. FJAA was partially supported by the Generalitat Valenciana (AICO/2021/165). RIB was partially supported by FWF (Austrian Science Fund), project P 34922-N. DTB was supported by MINECO and European Social Fund (PRE2019-090751) under the program “Ayudas para contratos predoctorales para la formación de doctores” 2019

A variable smoothing algorithm for solving convex optimization problems

Abstract. In this article we propose a method for solving unconstrained optimization problems with convex and Lipschitz continuous objective functions. By making use of the Moreau envelopes of the functions occurring in the objective, we smooth the latter to a convex and differentiable function with Lipschitz continuous gradient by using both variable and constant smoothing parameters. The resulting problem is solved via an accelerated first-order method and this allows us to recover approximately the optimal solutions to the initial optimization problem with a rate of convergence of order O

Identification of Land Suitability for Agricultural Use by Applying Morphometric and Risk Parameters Based on GIS Spatial Analysis

Agricultural land is one of the main resources for the development of rural communities and the peripheries of urban centres. An area of 936 km2, belonging to Intercommunity Association for Development Alba-Iulia, Transylvania region, Romania, was analysed in order to identify suitable land for agricultural use. This approach represents the stage preceding the identification of crops favourability for agricultural land, thus reducing the time and resources needed for the proper land evaluation mark. The extension of suitable surfaces for agricultural crops was realized using a GIS model based on spatial analysis, taking into account morphometric parameters (slope, altitude, slope orientation, the density of fragmentation) and the risk factors (probability of landslides, flooding, temperature and rainfall). The outcome of the case study was an agricultural land suitability map of the investigated area, which provides valuable information regarding areas suitable for crops. By applying this model, a better management of agricultural lands can be assured, representing an alternative to the classic method of evaluation marks. The proposed model was validated by comparing the results with the grades of crop suitability, method achieved through the land evaluation mark