We introduce a proximalalgorithm using quasidistances for multiobjective minimization problems with quasiconvex functions defined in arbitrary Riemannian manifolds. The reason of using quasidistances instead of the c...
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We introduce a proximalalgorithm using quasidistances for multiobjective minimization problems with quasiconvex functions defined in arbitrary Riemannian manifolds. The reason of using quasidistances instead of the classical Riemannian distance comes from the applications in economy, computer science and behavioral sciences, where the quasidistances represent a non symmetric measure. Under some appropriate assumptions on the problem and using tools of Riemannian geometry we prove that accumulation points of the sequence generated by the algorithm satisfy the critical condition of Pareto-Clarke. If the functions are convex then these points are Pareto efficient solutions.
Most recently, a balanced augmented Lagrangian method (ALM) has been proposed by He and Yuan for the canonical convex minimization problem with linear constraints, which advances the original ALM by balancing its subp...
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Most recently, a balanced augmented Lagrangian method (ALM) has been proposed by He and Yuan for the canonical convex minimization problem with linear constraints, which advances the original ALM by balancing its subproblems, improving its implementation and enlarging its applicable range. In this paper, we propose a dual-primal version of the newly developed balanced ALM, which updates the new iterate via a conversely dual-primal iterative order formally. The new algorithm inherits all advantages of the prototype balanced ALM, and it can be extended to more general separable convex programming problems with both linear equality and inequality constraints. The convergence analysis of the proposed method can be well conducted in the context of variational inequalities. In particular, by some application problems, we numerically validate that these balanced ALM type methods can outperform existing algorithms of the same kind significantly.
Square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimi...
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Square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimization counterparts of the traditional least-squares regularized models. In this paper, we give a unified proof to show that any square-root regularized model whose penalty function being the sum of a simple norm and a seminorm can be interpreted as the distributionally robust optimization (DRO) formulation of the corresponding least-squares problem. In particular, the optimal transport cost in the DRO formulation is given by a certain dual form of the penalty. To solve the resulting square -root regularized model whose loss function and penalty function are both nonsmooth, we design a proximalpoint dual semismooth Newton algorithm and demonstrate its efficiency when the penalty is the sparse group Lasso penalty or the fused Lasso penalty. Extensive experiments demonstrate that our algorithm is highly efficient for solving the square-root sparse group Lasso problems and the square-root fused Lasso problems.
We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear o...
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We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function. The two operations are linked by the fact that, under mild assumptions, the subdifferential of the proximal composition of a convex function is the resolvent composition of its subdifferential. The resolvent and proximal compositions are shown to encapsulate known concepts, such as the resolvent and proximal averages, as well as new operations pertinent to the analysis of equilibrium problems. A large core of properties of these compositions is established and several instantiations are discussed. Applications to the relaxation of monotone inclusion and convex optimization problems are presented.
This paper presents an improved Lagrangian-PPA based predic-tion correction method to solve linearly constrained convex optimization prob-lem. At each iteration, the predictor is achieved by minimizing the proximal La...
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This paper presents an improved Lagrangian-PPA based predic-tion correction method to solve linearly constrained convex optimization prob-lem. At each iteration, the predictor is achieved by minimizing the proximal Lagrangian function with respect to the primal and dual variables. These op-timization subproblems involved either admit analytical solutions or can be solved by a fast algorithm. The new update is generated by using the infor-mation of the current iterate and the predictor, as well as an appropriately chosen stepsize. Compared with the existing PPA based method, the param-eters are relaxed. We also establish the convergence and convergence rate of the proposed method. Finally, numerical experiments are conducted to show the efficiency of our Lagrangian-PPA based prediction correction method.
Some algorithms in signal and image processing may be formulated in the Krasnoselski-Mann ( KM) iteration form and the KM theorem asserts the convergence of this iteration under certain assumptions. We give more gener...
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Some algorithms in signal and image processing may be formulated in the Krasnoselski-Mann ( KM) iteration form and the KM theorem asserts the convergence of this iteration under certain assumptions. We give more general iterative schemes which include the KM iteration as a special case and establish the convergence of extended iterations. Based on the generalized KM theorems, some algorithms with a broader scope are analysed and treated in new settings.
We propose to use the proximal point algorithm to regularize a "dual" problem of generalized fractional programs (GFP). The proposed technique leads to a new dual algorithm that generates a sequence which co...
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We propose to use the proximal point algorithm to regularize a "dual" problem of generalized fractional programs (GFP). The proposed technique leads to a new dual algorithm that generates a sequence which converges from below to the minimal value of the considered problem. At each step, the proposed algorithm solves approximately an auxiliary problem with a unique dual solution whose every cluster point gives a solution to the dual problem. In the exact minimization case, the sequence of dual solutions converges to an optimal dual solution. For a class of functions, including the linear case, the convergence of the dual values is at least linear.
The notion of quasi-Fejer monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notio...
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The notion of quasi-Fejer monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of variable metric algorithms, whereby the underlying norm is allowed to vary at each iteration. Applications to convex optimization and inverse problems are demonstrated. (c) 2012 Elsevier Ltd. All rights reserved.
Until now, a few bundle methods for general maximal monotone operators exist and they were only employed with one polyhedral approximation of the -enlargement of the maximal monotone operator considered. However, we f...
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Until now, a few bundle methods for general maximal monotone operators exist and they were only employed with one polyhedral approximation of the -enlargement of the maximal monotone operator considered. However, we find in the literature several hybrid-proximal methods which could be adapted with a great deal of bundle techniques in order to find a zero of a maximal monotone operator;yet, we could also consider the use of two polyhedral approximations. The method developed in this study has used a double polyhedral approximation at each iteration. Besides, as an application, we give a bundle method for a forward-backward type algorithm.
In this paper, a common zero of a finite family of monotone operators is approximated in Hadamard spaces. Some applications in convex minimization and fixed point theory are also presented.
In this paper, a common zero of a finite family of monotone operators is approximated in Hadamard spaces. Some applications in convex minimization and fixed point theory are also presented.
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