arXiv subjects

Luis M. Briceño-Arias

Publications and source records attributed to Luis M. Briceño-Arias.

17 recordsLinked to original sources

arXiv2026

Lyapunov analysis for FISTA under strong convexity

In this paper, we conduct a theoretical and numerical study of the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) under strong convexity assumptions. We propose an autonomous Lyapunov function that reflects the strong convexity of the objective function, whether it arises from the smooth or non-smooth component. This Lyapunov function decreases monotonically at a linear rate along the iterations of the algorithm for a fixed inertial parameter, provided that the inertial parameter is built from the total strong convexity modulus of the objective function. Our analysis shows that the resulting algorithm with optimal parameters, its Lyapunov function, and its linear convergence rate are invariant under any transfer of strong convexity between the smooth and non-smooth components. This invariance also holds for forward-backward splitting (FBS). Within this framework, we compare the performance of FBS and some FISTA variants, and find that this strategy leads FISTA to outperform all other configurations, including FBS. Moreover, we identify parameter regimes in which FBS yields better performance than FISTA when the strong convexity of the non-smooth part is not leveraged appropriately.

math.OC↗

arXiv2026

Bregman proximal gradient method for linear optimization under entropic constraints

In this paper, we present an efficient algorithm for solving a linear optimization problem with entropic constraints, a class of problems that arises in game theory and information theory. Our analysis distinguishes between the cases of active and inactive constraints, addressing each using a Bregman proximal gradient method with entropic Legendre functions, for which we establish a convergence rate of $O(1/n)$ in objective values. For a specific cost structure, our framework provides a theoretical justification for the well-known Blahut-Arimoto algorithm and the uniqueness of the Lagrange multiplier associated with the entropic constraint. In the active constraint setting, we include a bisection procedure to approximate the strictly positive Lagrange multiplier. The efficiency of the proposed method is illustrated through comparisons with standard optimization solvers on a representative example from game theory, including extensions to higher-dimensional settings.

math.OC↗

arXiv2026

A Proximal Primal-Dual Approach to Generalized JKO Schemes for Doubly Nonlinear Parabolic Equations

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $p$-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.

math.NA↗

arXiv2024

Projection onto cones generated by epigraphs of perspective functions

In this paper we provide an efficient computation of the projection onto the cone generated by the epigraph of the perspective of any convex lower semicontinuous function. Our formula requires solving only two scalar equations involving the proximity operator of the function. This enables the computation of projections, for instance, onto exponential and power cones, and extends to previously unexplored conic projections, such as the projection onto the hyperbolic cone. We compare numerically the efficiency of the proposed approach in the case of exponential cones with an open source available method in the literature, illustrating its efficiency.

math.OC↗

arXiv2024

Enhanced computation of the proximity operator for perspective functions

In this paper we provide an explicit expression for the proximity operator of a perspective of any proper lower semicontinuous convex function defined on a Hilbert space. Our computation enhances and generalizes known formulae for the case when the Fenchel conjugate of the convex function has open domain or when it is radial. We provide several examples of non-radial functions for which the domain of its conjugate is not open and we compute the proximity operators of their perspectives.

math.OC↗

arXiv2024

Proximity Operators of Perspective Functions with Nonlinear Scaling

A perspective function is a construction which combines a base function defined on a given space with a nonlinear scaling function defined on another space and which yields a lower semicontinuous convex function on the product space. Since perspective functions are typically nonsmooth, their use in first-order algorithms necessitates the computation of their proximity operator. This paper establishes closed-form expressions for the proximity operator of a perspective function defined on a Hilbert space in terms of a proximity operator involving its base function and one involving its scaling function.

math.OC↗

arXiv2023

Perspective Functions with Nonlinear Scaling

The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented.

math.FA↗

arXiv2023

A Perturbation Framework for Convex Minimization and Monotone Inclusion Problems with Nonlinear Compositions

We introduce a framework based on Rockafellar's perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such problems have been investigated only from a primal perspective and only for nonlinear compositions of smooth functions in finite-dimensional spaces in the absence of linear compositions. In the context of Banach spaces, the proposed perturbation analysis serves as a foundation for the construction of a dual problem and of a maximally monotone Kuhn--Tucker operator which is decomposable as the sum of simpler monotone operators. In the Hilbertian setting, this decomposition leads to a block-iterative primal-dual algorithm that fully splits all the components of the problem and appears to be the first proximal splitting algorithm for handling nonlinear composite problems. Various applications are discussed.

math.OC↗

arXiv2022

Resolvent splitting with minimal lifting for composite monotone inclusions

In this paper we propose a resolvent splitting with minimal lifting for finding a zero of the sum of $n\ge 2$ maximally monotone operators involving the composition with a linear bounded operator. The resolvent of each monotone operator, the linear operator, and its adjoint are computed exactly once in the proposed algorithm. In the case when the linear operator is the identity, we recover the resolvent splitting with minimal lifting developed in Malitsky-Tam (2021). We also derive a new resolvent splitting for solving the composite monotone inclusion in the case $n=2$ with minimal $1-$fold lifting.

math.OC↗

arXiv2022

Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions

In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward splitting, among other methods. At each iteration, our algorithm needs two computations of the Lipschitzian operator while the cocoercive operator is activated only once. By using product space techniques, we derive a method for solving a composite monotone primal-dual inclusions including linear operators and we apply it to solve constrained composite convex optimization problems. Finally, we apply our algorithm to a constrained total variation least-squares problem and we compare its performance with efficient methods in the literature.

math.OC↗

arXiv2021

Split-Douglas-Rachford algorithm for composite monotone inclusions and Split-ADMM

In this paper we provide a generalization of the Douglas-Rachford splitting (DRS) and the primal-dual algorithm (Vu 2013, Condat 2013) for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual non-standard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn-Tucker point is guaranteed, generalizing the main result in Svaiter (2011). Inspired by Gabay (1983), we also derive a new Split-ADMM (SADMM) by applying our method to the dual of a convex optimization problem involving a linear operator which can be expressed as the composition of two linear operators. The proposed SADMM activates one linear operator implicitly and the other one explicitly, and we recover ADMM when the latter is set as the identity. Connections and comparisons of our theoretical results with respect to the literature are provided for the main algorithm and SADMM. The flexibility and efficiency of both methods is illustrated via a numerical simulations in total variation image restoration and a sparse minimization problem.

math.OC↗

arXiv2018

Forward-Backward-Half Forward Algorithm for Solving Monotone Inclusions

Tseng's algorithm finds a zero of the sum of a maximally monotone operator and a monotone continuous operator by evaluating the latter twice per iteration. In this paper, we modify Tseng's algorithm for finding a zero of the sum of three operators, where we add a cocoercive operator to the inclusion. Since the sum of a cocoercive and a monotone-Lipschitz operator is monotone and Lipschitz, we could use Tseng's method for solving this problem, but implementing both operators twice per iteration and without taking into advantage the cocoercivity property of one operator. Instead, in our approach, although the {continuous monotone} operator must still be evaluated twice, we exploit the cocoercivity of one operator by evaluating it only once per iteration. Moreover, when the cocoercive or {continuous-monotone} operators are zero it reduces to Tseng's or forward-backward splittings, respectively, unifying in this way both algorithms. In addition, we provide a {preconditioned} version of the proposed method including non self-adjoint linear operators in the computation of resolvents and the single-valued operators involved. This approach allows us to {also} extend previous variable metric versions of Tseng's and forward-backward methods and simplify their conditions on the underlying metrics. We also exploit the case when non self-adjoint linear operators are triangular by blocks in the primal-dual product space for solving primal-dual composite monotone inclusions, obtaining Gauss-Seidel type algorithms which generalize several primal-dual methods available in the literature. Finally we explore {applications to the obstacle problem, Empirical Risk Minimization, distributed optimization and nonlinear programming and we illustrate the performance of the method via some numerical simulations.

math.OC↗

arXiv2014

A Strongly Convergent Primal-Dual Method for Nonoverlapping Domain Decomposition

We propose a primal-dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling constraints which model various properties of the solution at the interfaces. The proposed method can handle a wide range of linear and nonlinear problems, with flexible, possibly nonlinear, transmission conditions across the interfaces. Strong convergence in the energy spaces is established in this general setting, and without any additional assumption on the energy functions or the geometry of the problem. Several examples are presented.

math.NA↗

arXiv2014

Forward--partial inverse--forward splitting for solving monotone inclusions

In this paper we provide a splitting method for finding a zero of the sum of a maximally monotone operator, a lipschitzian monotone operator, and a normal cone to a closed vectorial subspace of a real Hilbert space. The problem is characterized by a simpler monotone inclusion involving only two operators: the partial inverse of the maximally monotone operator with respect to the vectorial subspace and a suitable lipschitzian monotone operator. By applying the Tseng's method in this context we obtain a splitting algorithm that exploits the whole structure of the original problem and generalizes partial inverse and Tseng's methods. Connections with other methods available in the literature and applications to inclusions involving $m$ maximally monotone operators, to primal-dual composite monotone inclusions, and to zero-sum games are provided.

math.OC↗

arXiv2012

Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions

We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas-Rachford iteration involving the maximally monotone operator and the normal cone. In the second method it is a proximal step involving the partial inverse of the maximally monotone operator with respect to the vector subspace. Connections between the proposed methods and other methods in the literature are provided. Applications to monotone inclusions with finitely many maximally monotone operators and optimization problems are examined.

math.OC↗

arXiv2012

A Douglas-Rachford Splitting Method for Solving Equilibrium Problems

We propose a splitting method for solving an equilibrium problem involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of two appropriate maximally monotone operators. Our algorithm is a consequence of the Douglas--Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.

math.OC↗

arXiv2011

Outer approximation method for constrained composite fixed point problems involving Lipschitz pseudo contractive operators

We propose a method for solving constrained fixed point problems involving compositions of Lipschitz pseudo contractive and firmly nonexpansive operators in Hilbert spaces. Each iteration of the method uses separate evaluations of these operators and an outer approximation given by the projection onto a closed half-space containing the constraint set. Its convergence is established and applications to monotone inclusion splitting and constrained equilibrium problems are demonstrated.

math.OC↗