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Dias J.P.,University of Lisbon | Figueira M.,University of Lisbon | Frid H.,Institute Matematica Pura e Aplicada
Archive for Rational Mechanics and Analysis | Year: 2010

Motivated by Benney's general theory, we propose new models for short wave-long wave interactions when the long waves are described by nonlinear systems of conservation laws. We prove the strong convergence of the solutions of the vanishing viscosity and short wave-long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems. We analyze several of the representative examples, such as scalar conservation laws, general symmetric systems, nonlinear elasticity and nonlinear electromagnetism. © 2010 Springer-Verlag.


Fernandez D.,IMPA | Fernandez D.,University of Campinas | Solodov M.,Institute Matematica Pura e Aplicada
Mathematical Programming | Year: 2010

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush-Kuhn-Tucker systems for variational problems that holds under an appropriate second-order condition. © 2008 Springer-Verlag.


Guigues V.,Institute Matematica Pura e Aplicada | Romisch W.,Humboldt University of Berlin
SIAM Journal on Optimization | Year: 2012

We define a risk-averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk-averse formulation of the problem. This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of such risk functionals are given and used to derive conditions of coherence. In the one-period case, conditions for convexity and consistency with second order stochastic dominance are also provided. The risk-averse dynamic programming equations are specialized considering convex combinations of one-period extended polyhedral risk measures such as spectral risk measures. To implement the proposed policy, the approximation of the risk-averse recourse functions for stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk-averse problem. © 2012 Society for Industrial and Applied Mathematics.


Monteiro R.D.C.,Georgia Institute of Technology | Svaiter B.F.,Institute Matematica Pura e Aplicada
SIAM Journal on Optimization | Year: 2010

In this paper we analyze the iteration complexity of the hybrid proximal extragradient (HPE) method for finding a zero of a maximal monotone operator recently proposed by Solodov and Svaiter. One of the key points of our analysis is the use of new termination criteria based on the ε-enlargement of a maximal monotone operator. The advantage of using these termination criteria is that their definition do not depend on the boundedness of the domain of the operator. We then show that Korpelevich's extragradient method for solving monotone variational inequalities falls in the framework of the HPE method. As a consequence, using the complexity analysis of the HPE method, we obtain new complexity bounds for Korpelevich's extragradient method which do not require the feasible set to be bounded, as assumed in a recent paper by Nemirovski. Another feature of our analysis is that the derived iteration-complexity bounds are proportional to the distance of the initial point to the solution set. The HPE framework is also used to obtain the first iterationcomplexity result for Tseng's modified forward-backward splitting method for finding a zero of the sum of a monotone Lipschitz continuous map with an arbitr ary maximal monotone operator whose resolvent is assumed to be easily computable. Also using the framework of the HPE method, we study the complexity of a variant of a Newton-type extragradient algorithm proposed by Solodov and Svaiter for finding a zero of a smooth monotone function with Lipschitz continuous Jacobian. © 2010 Society for Industrial and Applied Mathematics.


Movasati H.,Institute Matematica Pura e Aplicada
Nuclear Physics B | Year: 2011

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the q-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of (quasi) modular forms attached to the one parameter family of Calabi-Yau varieties. Our result is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series. © 2011 Elsevier B.V.


Iusem A.N.,Institute Matematica Pura e Aplicada | Nasri M.,Institute Matematica Pura e Aplicada
RAIRO - Operations Research | Year: 2010

We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problems whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method for constrained optimization. At each iteration, primal variables are updated by solving an unconstrained variational inequality problem, and then dual variables are updated through a closed formula. A full convergence analysis is provided, allowing for inexact solution of the subproblems. © 2010 EDP Sciences, ROADEF, SMAI.


Iusem A.N.,Institute Matematica Pura e Aplicada | Nasri M.,Institute Matematica Pura e Aplicada
Journal of Global Optimization | Year: 2011

We propose a variant of Korpelevich's method for solving variational inequality problems with operators in Banach spaces. A full convergence analysis of the method is presented under reasonable assumptions on the problem data. © Springer Science+Business Media, LLC. 2010.


Bello Cruz J.Y.,Federal University of Goais | Iusem A.N.,Institute Matematica Pura e Aplicada
Numerical Functional Analysis and Optimization | Year: 2011

In this article, we propose a strongly convergent variant on the projected subgradient method for constrained convex minimization problems in Hilbert spaces. The advantage of the proposed method is that it converges strongly when the problem has solutions, without additional assumptions. The method also has the following desirable property: the sequence converges to the solution of the problem which lies closest to the initial iterate. Copyright © Taylor & Francis Group, LLC.


Izmailov A.F.,Moscow State University | Solodov M.V.,Institute Matematica Pura e Aplicada
Mathematical Programming | Year: 2012

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91-124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47-73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, for the problem with equality constraints only, we prove superlinear convergence of sSQP under the assumption that the dual starting point is close to a noncritical multiplier. Since noncritical multipliers include all those satisfying SOSC but are not limited to them, we believe this gives the first superlinear convergence result for any Newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than SOSC. In the general case when inequality constraints are present, we show that such a relaxation of assumptions is not possible. We also consider applying sSQP to the problem where inequality constraints are reformulated into equalities using slack variables, and discuss the assumptions needed for convergence in this approach. We conclude with consequences for local regularization methods proposed in (Izmailov and Solodov SIAM J Optim 16:210-228, 2004; Wright SIAM J. Optim. 15:673-676, 2005). In particular, we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally. © 2010 Springer and Mathematical Optimization Society.


Martinez N.,Institute Matematica Pura e Aplicada
Letters in Mathematical Physics | Year: 2015

We introduce poly-symplectic groupoids, which are natural extensions of symplectic groupoids to the context of poly-symplectic geometry, and define poly-Poisson structures as their infinitesimal counterparts. We present equivalent descriptions of poly-Poisson structures, including one related with AV-Dirac structures. We also discuss symmetries and reduction in the setting of poly-symplectic groupoids and poly-Poisson structures, and use our viewpoint to revisit results and develop new aspects of the theory initiated in Iglesias et al. (Lett Math Phys 103:1103–1133, 2013). © 2015, Springer Science+Business Media Dordrecht.

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