arXiv subjects

Jefferson G. Melo

Publications and source records attributed to Jefferson G. Melo.

16 recordsLinked to original sources

arXiv2025

A Cubic Regularization Method for Multiobjective Optimization

This work introduces a new cubic regularization method for nonconvex unconstrained multiobjective optimization problems. At each iteration of the method, a model associated with the cubic regularization of each component of the objective function is minimized. This model allows approximations for the first- and second-order derivatives, which must satisfy suitable error conditions. One interesting feature of the proposed algorithm is that the regularization parameter of the model and the accuracy of the derivative approximations are jointly adjusted using a nonmonotone line search criterion. Implementations of the method, where derivative information is computed using finite difference strategies, are discussed. It is shown that, under the assumption that the Hessians of the objectives are globally Lipschitz continuous, the method requires at most $\mathcal{O}(Cε^{-3/2})$ iterations to generate an $ε$-approximate Pareto critical. In particular, if the first- and second-order derivative information is computed using finite differences based solely on function values, the method requires at most $\mathcal{O}(n^{1-β}ε^{-3/2})$ iterations, corresponding to $\mathcal{O}\left(mn^{3-β}\varepsilon^{-\frac{3}{2}}\right)$ function evaluations, where $n$ is the dimension of the domain of the objective function, $m$ is the number of objectives, and $β\in [0,1]$ is a constant associated with the stepsize used in the finite-difference approximation. We further discuss the global convergence and local convergence rate of the method. Specifically, under the local convexity assumption, we show that the method achieves superlinear convergence when the first derivative is computed exactly, and quadratic convergence when both first- and second-order derivatives are exact.

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arXiv2025

A Relative Inexact Proximal Gradient Method with an Explicit Linesearch

This paper presents and investigates an inexact proximal gradient method for solving composite convex optimization problems characterized by an objective function composed of a sum of a full-domain differentiable convex function and a non-differentiable convex function. We introduce an explicit line search applied specifically to the differentiable component of the objective function, requiring only a relative inexact solution of the proximal subproblem per iteration. We prove the convergence of the sequence generated by our scheme and establish its iteration complexity, considering both the functional values and a residual associated with first-order stationary solutions. Additionally, we provide numerical experiments to illustrate the practical efficacy of our method.

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arXiv2024

Improved convergence rates for the multiobjective Frank-Wolfe method

This paper analyzes the convergence rates of the {\it Frank-Wolfe } method for solving convex constrained multiobjective optimization. We establish improved convergence rates under different assumptions on the objective function, the feasible set, and the localization of the limit point of the sequence generated by the method. In terms of the objective function values, we firstly show that if the objective function is strongly convex and the limit point of the sequence generated by the method lies in the relative interior of the feasible set, then the algorithm achieves a linear convergence rate. Next, we focus on a special class of problems where the feasible constraint set is $(α,q)$-uniformly convex for some $α>0$ and $q \geq 2$, including, in particular, $\ell_p$-balls for all $p>1$. In this context, we prove that the method attains: (i) a rate of $\mathcal{O}(1/k^\frac{q}{q-1})$ when the objective function is strongly convex; and (ii) a linear rate (if $q=2$) or a rate of $\mathcal{O}(1/k^{\frac{q}{q-2}})$ (if $q>2$) under an additional assumption, which always holds if the feasible set does not contain an unconstrained weak Pareto point. We also discuss enhanced convergence rates for the algorithm in terms of an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.

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arXiv2022

Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear $\cal K$-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone $\cal K$. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in ${\cal O}(\log(1/ρ)/ρ^{3})$ inner iterations, where $ρ>0$ is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.

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arXiv2022

Iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian method based on the classical Lagrangian function

This paper establishes the iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly-constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in ${\cal O}(\varepsilon^{-5/2}\log^2 \varepsilon^{-1})$ ACG iterations where $\varepsilon>0$ is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.

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arXiv2021

FISTA and Extensions -- Review and New Insights

The purpose of this technical report is to review the main properties of an accelerated composite gradient (ACG) method commonly referred to as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In addition, we state a version of FISTA for solving both convex and strongly convex composite minimization problems and derive its iteration complexities to generate iterates satisfying various stopping criteria, including one which arises in the course of solving other composite optimization problems via inexact proximal point schemes. This report also discusses different reformulations of the convex version of FISTA and how they relate to other formulations in the literature.

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arXiv2020

Iteration-complexity of an inexact proximal accelerated augmented Lagrangian method for solving linearly constrained smooth nonconvex composite optimization problems

This paper proposes and establishes the iteration-complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. It is shown that IPAAL generates an approximate stationary solution in at most ${\cal O}(\log(1/ρ)/ρ^{3})$ ACG iterations, where $ρ>0$ is the given tolerance. It is also shown that the previous complexity bound can be sharpened to ${\cal O}(\log(1/ρ)/ρ^{2.5})$ under additional mildly stronger assumptions. The above bounds are derived assuming that the initial point is neither feasible nor the domain of the composite term of the objective function is bounded. Some preliminary numerical results are presented to illustrate the performance of the IPAAL method.

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arXiv2019

An efficient adaptive accelerated inexact proximal point method for solving linearly constrained nonconvex composite problems

This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly set constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP method. The variant, in turn, solves a given penalized subproblem by generating a sequence of proximal subproblems which are then solved by an accelerated composite gradient algorithm. The main difference between AIPP and its variant is that the proximal subproblems in the former are always convex while the ones in the latter are not necessarily convex due to the fact that their prox parameters are chosen as aggressively as possible so as to improve efficiency. The possibly nonconvex proximal subproblems generated by the AIPP variant are also tentatively solved by a novel adaptive accelerated composite gradient algorithm based on the validity of some key convergence inequalities. As a result, the variant generates a sequence of proximal subproblems where the stepsizes are adaptively changed according to the responses obtained from the calls to the accelerated composite gradient algorithm. Finally, numerical results are given to demonstrate the efficiency of the proposed AIPP and QP-AIPP variants.

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arXiv2019

Complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs

This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form $f + h$ where $f$ is a differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient (ACG) method. It is shown that the proposed scheme generates a $ρ$-approximate stationary point in at most ${\cal{O}}(ρ^{-3})$ ACG iterations. Finally, numerical results showing the efficiency of the proposed method are also given.

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arXiv2018

A Partially Inexact Alternating Direction Method of Multipliers and its Iteration-Complexity Analysis

This paper proposes a partially inexact alternating direction method of multipliers for computing approximate solution of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic interation-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact ADMM with relative error criteria has been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.

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arXiv2017

Convergence rate bounds for a proximal ADMM with over-relaxation stepsize parameter for solving nonconvex linearly constrained problems

This paper establishes convergence rate bounds for a variant of the proximal alternating direction method of multipliers (ADMM) for solving nonconvex linearly constrained optimization problems. The variant of the proximal ADMM allows the inclusion of an over-relaxation stepsize parameter belonging to the interval $(0,2)$. To the best of our knowledge, all related papers in the literature only consider the case where the over-relaxation parameter lies in the interval $(0,(1+\sqrt{5})/2)$.

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arXiv2017

Iteration-complexity of a Jacobi-type non-Euclidean ADMM for multi-block linearly constrained nonconvex programs

This paper establishes the iteration-complexity of a Jacobi-type non-Euclidean proximal alternating direction method of multipliers (ADMM) for solving multi-block linearly constrained nonconvex programs. The subproblems of this ADMM variant can be solved in parallel and hence the method has great potential to solve large scale multi-block linearly constrained nonconvex programs. Moreover, our analysis allows the Lagrange multiplier to be updated with a relaxation parameter in the interval (0, 2).

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arXiv2017

Pointwise and ergodic convergence rates of a variable metric proximal ADMM

In this paper, we obtain global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates for a variable metric proximal alternating direction method of multipliers(VM-PADMM) for solving linearly constrained convex optimization problems. The VM-PADMM can be seen as a class of ADMM variants, allowing the use of degenerate metrics (defined by noninvertible linear operators). We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient (VM-HPE) framework for solving monotone inclusions. Then, the above-mentioned convergence rates for the VM-PADMM are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the VM-PADMM (resp. VM-HPE framework).

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arXiv2017

Improved pointwise iteration-complexity of a regularized ADMM and of a regularized non-Euclidean HPE framework

This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity of the latter method. Our analysis is based on first presenting and establishing the pointwise iteration-complexity of a regularized non-Euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error. It is then shown that the new method is a special instance of the latter framework where the sequence of summable errors is identically zero when the ADMM stepsize is less than one or a nontrivial sequence when the stepsize is in the interval [1, (1 +\sqrt{5})/2).

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arXiv2016

Extending the ergodic convergence rate of the proximal ADMM

Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in(0,(1+\sqrt{5})/2) have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to (1+\sqrt{5})/$. As far as we know, this is the first ergodic iteration-complexity for the stepsize (1+\sqrt{5})/2 obtained in the ADMM literature. These results are obtained by showing that the proximal ADMM is an instance of a non-Euclidean hybrid proximal extragradient framework whose pointwise and ergodic convergence rate are also studied.

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arXiv2016

A Newton conditional gradient method for constrained nonlinear systems

In this paper, we consider the problem of solving a constrained system of nonlinear equations. We propose an algorithm based on a combination of the Newton and conditional gradient methods, and establish its local convergence analysis. Our analysis is set up by using a majorant condition technique, allowing us to prove in a unified way convergence results for two large families of nonlinear functions. The first one includes functions whose derivative satisfies a Holder-like condition, and the second one consists of a substantial subclass of analytic functions. Numerical experiments illustrating the applicability of the proposed method are presented, and comparisons with some other methods are discussed.

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