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A generalized conditional gradient method for multiobjective composite optimization problems

OPTIMIZATION 2025年第2期74卷 473-503页

作者： Assuncao, P. B. Ferreira, O. P. Prudente, L. F. Univ Fed Goias Inst Matemat & Estat Goiania GO Brazil

This article deals with multiobjective composite optimization problems that consist of simultaneously minimizing several objective functions, each of which is composed of a combination of smooth and non-smooth functions. To tackle these problems, we propose a generalized version of the conditional gradient method, also known as Frank-Wolfe method. The method is analysed with three step size strategies, including Armijo-type, adaptive, and diminishing step sizes. We establish asymptotic convergence properties and iteration-complexity bounds, with and without convexity assumptions on the objective functions. Numerical experiments illustrating the practical behaviour of the methods are presented.

关键词： conditional gradient method Frank-Wolfe method multiobjective optimization Pareto optimality constrained optimization problem

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conditional gradient method for multiobjective optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2021年第3期78卷 741-768页

作者： Assuncao, P. B. Ferreira, O. P. Prudente, L. F. Univ Fed Goias Inst Matemat & Estat BR-74001970 Goiania Go Brazil

We analyze the conditional gradient method, also known as Frank-Wolfe method, for constrained multiobjective optimization. The constraint set is assumed to be convex and compact, and the objectives functions are assumed to be continuously differentiable. The method is considered with different strategies for obtaining the step sizes. Asymptotic convergence properties and iteration-complexity bounds with and without convexity assumptions on the objective functions are stablished. Numerical experiments are provided to illustrate the effectiveness of the method and certify the obtained theoretical results.

关键词： conditional gradient method Multiobjective optimization Pareto optimality Constrained optimization problem

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conditional gradient method for vector optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第3期85卷 857-896页

作者： Chen, Wang Yang, Xinmin Zhao, Yong Sichuan Univ Coll Math Chengdu 610065 Sichuan Peoples R China Chongqing Normal Univ Natl Ctr Appl Math Chongqing Chongqing 401331 Peoples R China Chongqing Normal Univ Sch Math Sci Chongqing 401331 Peoples R China Chongqing Jiaotong Univ Coll Math & Stat Chongqing 610101 Peoples R China

In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimization recently proposed by Assuncao et al. (Comput Optim Appl 78(3):741-768, 2021). In our method, the construction of the auxiliary subproblem is based on the well-known oriented distance function. Three different types of step size strategies (Armijo, adaptative and nonmonotone) are considered. Without convexity assumption related to the objective function, we obtain the stationarity of accumulation points of the sequences produced by the proposed method equipped with the Armijo or the nonmonotone step size rule. To obtain the convergence result of the method with the adaptative step size strategy, we introduce a useful cone convexity condition which allows us to circumvent the intricate question of the Lipschitz continuity of Jocabian for the objective function. This condition helps us to generalize the classical descent lemma to the vector optimization case. Under convexity assumption for the objective function, it is proved that all accumulation points of any generated sequences obtained by our method are weakly efficient solutions. Numerical experiments illustrating the practical behavior of the methods are presented.

关键词： Vector optimization conditional gradient method Stationary point Convergence C-convexity

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conditional gradient method for Optimization Problems with a Constraint in the Form of the Intersection of a Convex Smooth Surface and a Convex Compact Set

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2023年第7期63卷 1191-1198页

作者： Chernyaev, Yu. A. Kazan Natl Res Tech Univ Kazan 420111 Tatarstan Russia

The conditional gradient method is generalized to nonconvex sets of constraints representing the set-theoretic intersection of a convex smooth surface and a convex compact set. Necessary optimality conditions are stud... 详细信息

关键词： convex smooth surface convex compact set minimization of a smooth function conditional gradient method

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conditional gradient method for Double-Convex Fractional Programming Matrix Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第1期176卷 163-177页

作者： Bouhamidi, Abderrahman Bellalij, Mohammed Enkhbat, Rentsen Jbilou, Khalid Raydan, Marcos Univ Littoral Cote dOpale Batiment H Poincare50 Rue F BuissonBP 699 F-62228 Calais France Univ Valenciennes Lab Math & Leurs Applicat F-59313 Le Mt Houy Valenciennes France Natl Univ Mongolia POB 46-635 Ulaanbaatar Mongolia Univ Simon Bolivar Dept Comp Cient & Estadist Ap 89000 Caracas 1080A Venezuela

We consider the problem of optimizing the ratio of two convex functions over a closed and convex set in the space of matrices. This problem appears in several applications and can be classified as a double-convex fractional programming problem. In general, the objective function is nonconvex but, nevertheless, the problem has some special features. Taking advantage of these features, a conditional gradient method is proposed and analyzed, which is suitable for matrix problems. The proposed scheme is applied to two different specific problems, including the well-known trace ratio optimization problem which arises in many engineering and data processing applications. Preliminary numerical experiments are presented to illustrate the properties of the proposed scheme.

关键词： Fractional programming conditional gradient method Trace ratio problem

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conditional gradient method Without Line-Search

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RUSSIAN MATHEMATICS 2018年第1期62卷 82-85页

作者： Konnov, I. V. Kazan Fed Univ Ul Kremlyovskaya 18 Kazan 420008 Russia

We propose a simple rule for the step-size choice in the conditional gradient method, which does not require any line-search procedure. It takes into account the current behavior of the method. Its convergence is esta... 详细信息

关键词： optimization problems conditional gradient method step-size choice

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Alternating conditional gradient method for convex feasibility problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2021年第1期80卷 245-269页

作者： Diaz Millan, R. Ferreira, O. P. Prudente, L. F. Deakin Univ Sch Informat Technol Melbourne Vic Australia Univ Fed Goias Inst Matemat & Estat BR-74001970 Goiania Go Brazil

The classical convex feasibility problem in a finite dimensional Euclidean space consists of finding a point in the intersection of two convex sets. In the present paper we are interested in two particular instances of this problem. First, we assume to know how to compute an exact projection onto one of the sets involved and the other set is compact such that the conditional gradient (CondG) method can be used for computing efficiently an inexact projection on it. Second, we assume that both sets involved are compact such that the CondG method can be used for computing efficiently inexact projections on them. We combine alternating projection method with CondG method to design a new method, which can be seen as an inexact feasible version of alternate projection method. The proposed method generates two different sequences belonging to each involved set, which converge to a point in the intersection of them whenever it is not empty. If the intersection is empty, then the sequences converge to points in the respective sets whose distance between them is equal to the distance between the sets in consideration. Numerical experiments are provided to illustrate the practical behavior of the method.

关键词： Convex feasibility problem Alternating projection method conditional gradient method Inexact projections

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A generalized conditional gradient method and its connection to an iterative shrinkage method

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2009年第2期42卷 173-193页

作者： Bredies, Kristian Lorenz, Dirk A. Maass, Peter Univ Bremen Fachbereich 03 D-28334 Bremen Germany Technion Israel Inst Technol Dept Elect Engn IL-32000 Haifa Israel

This article combines techniques from two fields of applied mathematics: optimization theory and inverse problems. We investigate a generalized conditional gradient method and its connection to an iterative shrinkage method, which has been recently proposed for solving inverse problems. The iterative shrinkage method aims at the solution of non-quadratic minimization problems where the solution is expected to have a sparse representation in a known basis. We show that it can be interpreted as a generalized conditional gradient method. We prove the convergence of this generalized method for general class of functionals, which includes non-convex functionals. This also gives a deeper understanding of the iterative shrinkage method.

关键词： conditional gradient method Sparsity constraints Inverse problems Non-convex optimization

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A Generalized conditional gradient method for Dynamic Inverse Problems with Optimal Transport Regularization

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FOUNDATIONS OF COMPUTATIONAL MATHEMATICS 2023年第3期23卷 833-898页

作者： Bredies, Kristian Carioni, Marcello Fanzon, Silvio Romero, Francisco Karl Franzens Univ Graz Inst Math & Sci Comp Heinrichstr 36 A-8010 Graz Austria Univ Cambridge Dept Appl Math & Theoret Phys Wilberforce Rd Cambridge CB3 0WA England

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351-2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou-Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou-Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436-1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise.

关键词： conditional gradient method Dynamic inverse problems Benamou-Brenier energy Optimal transport regularization Continuity equation

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A Newton conditional gradient method for constrained nonlinear systems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年第0期311卷 473-483页

作者： Goncalves, Max L. N. Melo, Jefferson G. Univ Fed Goias Inst Math & Stat Campus II Caixa Postal 131 BR-74001970 Goiania Go Brazil

In this paper, we consider the problem of solving constrained systems of nonlinear equations. We propose an algorithm based on a combination of 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. Some preliminary numerical experiments are reported. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Constrained nonlinear systems Newton method conditional gradient method Local convergence

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